| Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/GAP/pkg/liealgdb/gap/slac/ (Algebra von RWTH Aachen Version 4.15.1©) Datei vom 24.8.2025 mit Größe 177 kB |
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Quelle slac.gi Sprache: unbekannt |
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Spracherkennung für: .gi vermutete Sprache: Unknown {[0] [0] [0]} [Methode: Schwerpunktbildung, einfache Gewichte, sechs Dimensionen] InstallMethod( SolvableLieAlgebra, "for a field and a list", true,
[ IsField, IsList ], 0, function( F, data ) local d, no, a, b, S, L; if not Length( data ) >= 2 then Error("<data> has to have length at least two"); fi; if not IsPosInt( data[1] ) then Error("the first element of <data> has to be a positive integer"); else d:= data[1]; if not d in [1,2,3,4] then Error("the dimension has to be 1,2,3, or 4"); fi; fi; no:= data[2]; if d=1 then if no > 1 then Error( "the second element of <data> has to be <= 1" ); fi; elif d=2 then if no > 2 then Error("the second element of <data> has to be <= 2"); fi; elif d=3 then if no > 4 then Error("the second element of <data> has to be <= 4"); fi; else if not no in [1..14] then Error("the second element of <data> has to be <= 14"); fi; fi; if Length( data ) >=3 then a:= data[3]; fi; if Length( data ) >= 4 then b:= data[4]; fi; if d = 1 then S:= EmptySCTable( 1, Zero(F), "antisymmetric" ); L := LieAlgebraByStructureConstants( F, S ); L!.arg := data; return L; elif d = 2 then if no=1 then S:= EmptySCTable( 2, Zero(F), "antisymmetric" ); L := LieAlgebraByStructureConstants( F, S ); L!.arg := data; return L; else S:= EmptySCTable( 2, Zero(F), "antisymmetric" ); SetEntrySCTable( S, 2, 1, [One(F),1] ); L := LieAlgebraByStructureConstants( F, S ); L!.arg := data; return L; fi; elif d=3 then if no=1 then S:= EmptySCTable( 3, Zero(F), "antisymmetric" ); L := LieAlgebraByStructureConstants( F, S ); L!.arg := data; return L; elif no=2 then S:= EmptySCTable( 3, Zero(F), "antisymmetric" ); SetEntrySCTable( S, 3, 1, [One(F),1] ); SetEntrySCTable( S, 3, 2, [One(F),2] ); L := LieAlgebraByStructureConstants( F, S ); L!.arg := data; return L; elif no=3 then S:= EmptySCTable( 3, Zero(F), "antisymmetric" ); SetEntrySCTable( S, 3, 1, [One(F),2] ); SetEntrySCTable( S, 3, 2, [a,1,One(F),2] ); L := LieAlgebraByStructureConstants( F, S ); L!.arg := data; return L; else S:= EmptySCTable( 3, Zero(F), "antisymmetric" ); SetEntrySCTable( S, 3, 1, [One(F),2] ); SetEntrySCTable( S, 3, 2, [a,1] ); L := LieAlgebraByStructureConstants( F, S ); L!.arg := data; return L; fi; else if no=1 then S:= EmptySCTable( 4, Zero(F), "antisymmetric" ); L := LieAlgebraByStructureConstants( F, S ); L!.arg := data; return L; elif no=2 then S:= EmptySCTable( 4, Zero(F), "antisymmetric" ); SetEntrySCTable( S, 4, 1, [One(F),1] ); SetEntrySCTable( S, 4, 2, [One(F),2] ); SetEntrySCTable( S, 4, 3, [One(F),3] ); L := LieAlgebraByStructureConstants( F, S ); L!.arg := data; return L; elif no=3 then S:= EmptySCTable( 4, Zero(F), "antisymmetric" ); SetEntrySCTable( S, 4, 1, [One(F),1] ); SetEntrySCTable( S, 4, 2, [One(F),3] ); SetEntrySCTable( S, 4, 3, [-a,2,a+One(F),3] ); L := LieAlgebraByStructureConstants( F, S ); L!.arg := data; return L; elif no = 4 then S:= EmptySCTable( 4, Zero(F), "antisymmetric" ); SetEntrySCTable( S, 4, 2, [One(F),3] ); SetEntrySCTable( S, 4, 3, [One(F),3] ); L := LieAlgebraByStructureConstants( F, S ); L!.arg := data; return L; elif no = 5 then S:= EmptySCTable( 4, Zero(F), "antisymmetric" ); SetEntrySCTable( S, 4, 2, [One(F),3] ); L := LieAlgebraByStructureConstants( F, S ); L!.arg := data; return L; elif no = 6 then S:= EmptySCTable( 4, Zero(F), "antisymmetric" ); SetEntrySCTable( S, 4, 1, [One(F),2] ); SetEntrySCTable( S, 4, 2, [One(F),3] ); SetEntrySCTable( S, 4, 3, [a,1,b,2,One(F),3] ); L := LieAlgebraByStructureConstants( F, S ); L!.arg := data; return L; elif no = 7 then S:= EmptySCTable( 4, Zero(F), "antisymmetric" ); SetEntrySCTable( S, 4, 1, [One(F),2] ); SetEntrySCTable( S, 4, 2, [One(F),3] ); SetEntrySCTable( S, 4, 3, [a,1,b,2] ); L := LieAlgebraByStructureConstants( F, S ); L!.arg := data; return L; elif no=8 then S:= EmptySCTable( 4, Zero(F), "antisymmetric" ); SetEntrySCTable( S, 1, 2, [One(F),2] ); SetEntrySCTable( S, 3, 4, [One(F),4] ); L := LieAlgebraByStructureConstants( F, S ); L!.arg := data; return L; elif no=9 then S:= EmptySCTable( 4, Zero(F), "antisymmetric" ); SetEntrySCTable( S, 4, 1, [One(F),1,a,2] ); SetEntrySCTable( S, 4, 2, [One(F),1] ); SetEntrySCTable( S, 3, 1, [One(F),1] ); SetEntrySCTable( S, 3, 2, [One(F),2] ); L := LieAlgebraByStructureConstants( F, S ); L!.arg := data; return L; elif no=10 then S:= EmptySCTable( 4, Zero(F), "antisymmetric" ); SetEntrySCTable( S, 4, 1, [One(F),2] ); SetEntrySCTable( S, 4, 2, [a,1] ); SetEntrySCTable( S, 3, 1, [One(F),1] ); SetEntrySCTable( S, 3, 2, [One(F),2] ); L := LieAlgebraByStructureConstants( F, S ); L!.arg := data; return L; elif no=11 then if Characteristic(F) <> 2 then Error( "The characteristic of F has to be 2 for L4_11"); fi; S:= EmptySCTable( 4, Zero(F), "antisymmetric" ); SetEntrySCTable( S, 4, 1, [One(F),1] ); SetEntrySCTable( S, 4, 2, [b,2] ); SetEntrySCTable( S, 4, 3, [One(F)+b,3] ); SetEntrySCTable( S, 3, 1, [One(F),2] ); SetEntrySCTable( S, 3, 2, [a,1] ); L := LieAlgebraByStructureConstants( F, S ); L!.arg := data; return L; elif no=12 then S:= EmptySCTable( 4, Zero(F), "antisymmetric" ); SetEntrySCTable( S, 4, 1, [One(F),1] ); SetEntrySCTable( S, 4, 2, [2*One(F),2] ); SetEntrySCTable( S, 4, 3, [One(F),3] ); SetEntrySCTable( S, 3, 1, [One(F),2] ); L := LieAlgebraByStructureConstants( F, S ); L!.arg := data; return L; elif no=13 then S:= EmptySCTable( 4, Zero(F), "antisymmetric" ); SetEntrySCTable( S, 4, 1, [One(F),1,a,3] ); SetEntrySCTable( S, 4, 2, [One(F),2] ); SetEntrySCTable( S, 4, 3, [One(F),1] ); SetEntrySCTable( S, 3, 1, [One(F),2] ); L := LieAlgebraByStructureConstants( F, S ); L!.arg := data; return L; else S:= EmptySCTable( 4, Zero(F), "antisymmetric" ); SetEntrySCTable( S, 4, 1, [a,3] ); SetEntrySCTable( S, 4, 3, [One(F),1] ); SetEntrySCTable( S, 3, 1, [One(F),2] ); L := LieAlgebraByStructureConstants( F, S ); L!.arg := data; return L; fi; fi; end ); InstallMethod( NilpotentLieAlgebra, "for a field and a list", true, [ IsField, IsList ], 0, function( F, data ) local L, d, no, a, ff, arg, N1_1, N2_1, N3_1, N3_2, N4_1, N4_2, N4_3, N5_1, N5_2, N5_3, N5_4, N5_5, N5_6, N5_7, N5_8, N5_9, N6_1, N6_2, N6_3, N6_4, N6_5, N6_6, N6_7, N6_8, N6_9, N6_10, N6_11, N6_12, N6_13, N6_14, N6_15, N6_16, N6_17, N6_18, N6_19, N6_20, N6_21, N6_22, N6_23, N6_24, N6_25, N6_26, N6_27, N6_28, N6_29, N6_30, N6_31, N6_32, N6_33, N6_34, N6_35, N6_36 ; N1_1 := function( F ) local T; T := EmptySCTable( 1, Zero(F), "antisymmetric" ); return LieAlgebraByStructureConstants( F, T ); end; N2_1 := function( F ) local T; T := EmptySCTable( 2, Zero(F), "antisymmetric" ); return LieAlgebraByStructureConstants( F, T ); end; N3_1 := function( F ) local T; T := EmptySCTable( 3, Zero(F), "antisymmetric" ); return LieAlgebraByStructureConstants( F, T ); end; N3_2 := function( F ) local T; T := EmptySCTable( 3, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); return LieAlgebraByStructureConstants( F, T ); end; N4_1 := function( F ) local T; T := EmptySCTable( 4, Zero(F), "antisymmetric" ); return LieAlgebraByStructureConstants( F, T ); end; N4_2 := function( F ) local T; T := EmptySCTable( 4, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); return LieAlgebraByStructureConstants( F, T ); end; N4_3 := function( F ) local T; T := EmptySCTable( 4, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); return LieAlgebraByStructureConstants( F, T ); end; N5_1:= function( F ) local T; T:= EmptySCTable( 5, Zero(F), "antisymmetric" ); return LieAlgebraByStructureConstants( F, T ); end; N5_2:= function( F ) local T; T:= EmptySCTable( 5, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); return LieAlgebraByStructureConstants( F, T ); end; N5_3:= function( F ) local T; T:= EmptySCTable( 5, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); return LieAlgebraByStructureConstants( F, T ); end; N5_4:= function( F ) local T; T:= EmptySCTable( 5, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,5] ); SetEntrySCTable( T, 3, 4, [1,5] ); return LieAlgebraByStructureConstants( F, T ); end; N5_5:= function( F ) local T; T:= EmptySCTable( 5, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,5] ); SetEntrySCTable( T, 2, 4, [1,5] ); return LieAlgebraByStructureConstants( F, T ); end; N5_6:= function( F ) local T; T:= EmptySCTable( 5, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); SetEntrySCTable( T, 1, 4, [1,5] ); SetEntrySCTable( T, 2, 3, [1,5] ); return LieAlgebraByStructureConstants( F, T ); end; N5_7:= function( F ) local T; T:= EmptySCTable( 5, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); SetEntrySCTable( T, 1, 4, [1,5] ); return LieAlgebraByStructureConstants( F, T ); end; N5_8:= function( F ) local T; T:= EmptySCTable( 5, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,4] ); SetEntrySCTable( T, 1, 3, [1,5] ); return LieAlgebraByStructureConstants( F, T ); end; N5_9:= function( F ) local T; T:= EmptySCTable( 5, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); SetEntrySCTable( T, 2, 3, [1,5] ); return LieAlgebraByStructureConstants( F, T ); end; N6_1:= function( F ) local T; T:= EmptySCTable( 6, Zero(F), "antisymmetric" ); return LieAlgebraByStructureConstants( F, T ); end; N6_2:= function( F ) local T; T:= EmptySCTable( 6, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); return LieAlgebraByStructureConstants( F, T ); end; N6_3:= function( F ) local T; T:= EmptySCTable( 6, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); return LieAlgebraByStructureConstants( F, T ); end; N6_4:= function( F ) local T; T:= EmptySCTable( 6, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,5] ); SetEntrySCTable( T, 3, 4, [1,5] ); return LieAlgebraByStructureConstants( F, T ); end; N6_5:= function( F ) local T; T:= EmptySCTable( 6, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,5] ); SetEntrySCTable( T, 2, 4, [1,5] ); return LieAlgebraByStructureConstants( F, T ); end; N6_6:= function( F ) local T; T:= EmptySCTable( 6, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); SetEntrySCTable( T, 1, 4, [1,5] ); SetEntrySCTable( T, 2, 3, [1,5] ); return LieAlgebraByStructureConstants( F, T ); end; N6_7:= function( F ) local T; T:= EmptySCTable( 6, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); SetEntrySCTable( T, 1, 4, [1,5] ); return LieAlgebraByStructureConstants( F, T ); end; N6_8:= function( F ) local T; T:= EmptySCTable( 6, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,4] ); SetEntrySCTable( T, 1, 3, [1,5] ); return LieAlgebraByStructureConstants( F, T ); end; N6_9:= function( F ) local T; T:= EmptySCTable( 6, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); SetEntrySCTable( T, 2, 3, [1,5] ); return LieAlgebraByStructureConstants( F, T ); end; N6_10:= function( F ) local T; T:= EmptySCTable( 6, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,6] ); SetEntrySCTable( T, 4, 5, [1,6] ); return LieAlgebraByStructureConstants( F, T ); end; N6_11:= function( F ) local T; T:= EmptySCTable( 6, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); SetEntrySCTable( T, 1, 4, [1,6] ); SetEntrySCTable( T, 2, 3, [1,6] ); SetEntrySCTable( T, 2, 5, [1,6] ); return LieAlgebraByStructureConstants( F, T ); end; N6_12:= function( F ) local T; T:= EmptySCTable( 6, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); SetEntrySCTable( T, 1, 4, [1,6] ); SetEntrySCTable( T, 2, 5, [1,6] ); return LieAlgebraByStructureConstants( F, T ); end; N6_13:= function( F ) local T; T:= EmptySCTable( 6, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,5] ); SetEntrySCTable( T, 1, 5, [1,6] ); SetEntrySCTable( T, 2, 4, [1,5] ); SetEntrySCTable( T, 3, 4, [1,6] ); return LieAlgebraByStructureConstants( F, T ); end; N6_14:= function( F ) local T; T:= EmptySCTable( 6, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); SetEntrySCTable( T, 1, 4, [1,5] ); SetEntrySCTable( T, 2, 3, [1,5] ); SetEntrySCTable( T, 2, 5, [1,6] ); SetEntrySCTable( T, 3, 4, [-1,6] ); return LieAlgebraByStructureConstants( F, T ); end; N6_15:= function( F ) local T; T:= EmptySCTable( 6, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); SetEntrySCTable( T, 1, 4, [1,5] ); SetEntrySCTable( T, 1, 5, [1,6] ); SetEntrySCTable( T, 2, 3, [1,5] ); SetEntrySCTable( T, 2, 4, [1,6] ); return LieAlgebraByStructureConstants( F, T ); end; N6_16:= function( F ) local T; T:= EmptySCTable( 6, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); SetEntrySCTable( T, 1, 4, [1,5] ); SetEntrySCTable( T, 2, 5, [1,6] ); SetEntrySCTable( T, 3, 4, [-1,6] ); return LieAlgebraByStructureConstants( F, T ); end; N6_17:= function( F ) local T; T:= EmptySCTable( 6, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); SetEntrySCTable( T, 1, 4, [1,5] ); SetEntrySCTable( T, 1, 5, [1,6] ); SetEntrySCTable( T, 2, 3, [1,6] ); return LieAlgebraByStructureConstants( F, T ); end; N6_18:= function( F ) local T; T:= EmptySCTable( 6, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); SetEntrySCTable( T, 1, 4, [1,5] ); SetEntrySCTable( T, 1, 5, [1,6] ); return LieAlgebraByStructureConstants( F, T ); end; N6_19:= function( F, a ) local T; T:= EmptySCTable( 6, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,4] ); SetEntrySCTable( T, 1, 3, [1,5] ); SetEntrySCTable( T, 1, 5, [1,6] ); SetEntrySCTable( T, 2, 4, [1,6] ); SetEntrySCTable( T, 3, 5, [a,6] ); return LieAlgebraByStructureConstants( F, T ); end; N6_20:= function( F ) local T; T:= EmptySCTable( 6, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,4] ); SetEntrySCTable( T, 1, 3, [1,5] ); SetEntrySCTable( T, 1, 5, [1,6] ); SetEntrySCTable( T, 2, 4, [1,6] ); return LieAlgebraByStructureConstants( F, T ); end; N6_21:= function( F, a ) local T; T:= EmptySCTable( 6, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); SetEntrySCTable( T, 1, 4, [1,6] ); SetEntrySCTable( T, 2, 3, [1,5] ); SetEntrySCTable( T, 2, 5, [a,6] ); return LieAlgebraByStructureConstants( F, T ); end; N6_22:= function( F, a ) local T; T:= EmptySCTable( 6, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,5] ); SetEntrySCTable( T, 1, 3, [1,6] ); SetEntrySCTable( T, 2, 4, [a,6] ); SetEntrySCTable( T, 3, 4, [1,5] ); return LieAlgebraByStructureConstants( F, T ); end; N6_23:= function( F ) local T; T:= EmptySCTable( 6, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,5] ); SetEntrySCTable( T, 1, 4, [1,6] ); SetEntrySCTable( T, 2, 4, [1,5] ); return LieAlgebraByStructureConstants( F, T ); end; N6_24:= function( F, a ) local T; T:= EmptySCTable( 6, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,5] ); SetEntrySCTable( T, 1, 4, [a,6] ); SetEntrySCTable( T, 2, 3, [1,6] ); SetEntrySCTable( T, 2, 4, [1,5] ); return LieAlgebraByStructureConstants( F, T ); end; N6_25:= function( F ) local T; T:= EmptySCTable( 6, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,5] ); SetEntrySCTable( T, 1, 4, [1,6] ); return LieAlgebraByStructureConstants( F, T ); end; N6_26:= function( F ) local T; T:= EmptySCTable( 6, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,4] ); SetEntrySCTable( T, 1, 3, [1,5] ); SetEntrySCTable( T, 2, 3, [1,6] ); return LieAlgebraByStructureConstants( F, T ); end; N6_27:= function( F ) local T; T:= EmptySCTable( 6, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,5] ); SetEntrySCTable( T, 2, 4, [1,6] ); return LieAlgebraByStructureConstants( F, T ); end; N6_28:= function( F ) local T; T:= EmptySCTable( 6, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); SetEntrySCTable( T, 1, 4, [1,5] ); SetEntrySCTable( T, 2, 3, [1,6] ); return LieAlgebraByStructureConstants( F, T ); end; N6_29:= function( F ) local T; T:= EmptySCTable( 6, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,5] ); SetEntrySCTable( T, 1, 5, [1,6] ); SetEntrySCTable( T, 2, 4, [1,5,1,6] ); SetEntrySCTable( T, 3, 4, [1,6] ); return LieAlgebraByStructureConstants( F, T ); end; N6_30:= function( F ) local T; T:= EmptySCTable( 6, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); SetEntrySCTable( T, 1, 4, [1,5] ); SetEntrySCTable( T, 1, 5, [1,6] ); SetEntrySCTable( T, 2, 3, [1,5,1,6] ); SetEntrySCTable( T, 2, 4, [1,6] ); return LieAlgebraByStructureConstants( F, T ); end; N6_31:= function( F, a ) local T; T:= EmptySCTable( 6, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); SetEntrySCTable( T, 1, 4, [1,5] ); SetEntrySCTable( T, 2, 3, [1,5,a,6] ); SetEntrySCTable( T, 2, 5, [1,6] ); SetEntrySCTable( T, 3, 4, [1,6] ); return LieAlgebraByStructureConstants( F, T ); end; N6_32:= function( F, a ) local T; T:= EmptySCTable( 6, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); SetEntrySCTable( T, 1, 4, [1,5] ); SetEntrySCTable( T, 2, 3, [a,6] ); SetEntrySCTable( T, 2, 5, [1,6] ); SetEntrySCTable( T, 3, 4, [1,6] ); return LieAlgebraByStructureConstants( F, T ); end; N6_33:= function( F ) local T; T:= EmptySCTable( 6, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,4] ); SetEntrySCTable( T, 1, 3, [1,5] ); SetEntrySCTable( T, 2, 5, [1,6] ); SetEntrySCTable( T, 3, 4, [1,6] ); return LieAlgebraByStructureConstants( F, T ); end; N6_34:= function( F ) local T; T:= EmptySCTable( 6, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); SetEntrySCTable( T, 1, 5, [1,6] ); SetEntrySCTable( T, 2, 3, [1,5] ); SetEntrySCTable( T, 2, 4, [1,6] ); return LieAlgebraByStructureConstants( F, T ); end; N6_35:= function( F, a ) local T; T:= EmptySCTable( 6, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,5] ); SetEntrySCTable( T, 1, 3, [1,6] ); SetEntrySCTable( T, 2, 4, [a,6] ); SetEntrySCTable( T, 3, 4, [1,5,1,6] ); return LieAlgebraByStructureConstants( F, T ); end; N6_36:= function( F, a ) local T; T:= EmptySCTable( 6, Zero(F), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,5] ); SetEntrySCTable( T, 1, 4, [a,6] ); SetEntrySCTable( T, 2, 3, [1,6] ); SetEntrySCTable( T, 2, 4, [1,5,1,6] ); return LieAlgebraByStructureConstants( F, T ); end; if not Length( data ) >= 2 then Error("<data> has to have length at least two"); fi; if not IsPosInt( data[1] ) then Error("the first element of <data> has to be a positive integer"); else d:= data[1]; if not d in [1,2,3,4,5,6] then Error("the dimension has to be an integer from 1 to 6"); fi; fi; no:= data[2]; if d=1 then if no > 1 then Error( "the second element of <data> has to be <= 1" ); fi; elif d=2 then if no > 1 then Error( "the second element of <data> has to be <= 1" ); fi; elif d=3 then if no > 2 then Error( "the second element of <data> has to be <= 2" ); fi; elif d=4 then if no > 3 then Error( "the second element of <data> has to be <= 3" ); fi; elif d=5 then if no > 9 then Error("the second element of <data> has to be <= 9"); fi; else if no > 36 then Error("the second element of <data> has to be <= 36"); fi; fi; if Length( data ) >= 3 then a:= data[3]; fi; if d = 1 then ff := [N1_1]; L := CallFuncList( ff[no], [ F ] ); L!.arg := data; return L; elif d = 2 then ff := [N2_1]; L := CallFuncList( ff[no], [ F ] ); L!.arg := data; return L; elif d = 3 then ff := [N3_1,N3_2]; L := CallFuncList( ff[no], [ F ] ); L!.arg := data; return L; elif d = 4 then ff := [N4_1,N4_2,N4_3]; L := CallFuncList( ff[no], [ F ] ); L!.arg := data; return L; elif d = 5 then ff:= [ N5_1, N5_2, N5_3, N5_4, N5_5, N5_6, N5_7, N5_8, N5_9 ]; L := CallFuncList( ff[no], [ F ] ); L!.arg := data; return L; else ff:= [ N6_1, N6_2, N6_3, N6_4, N6_5, N6_6, N6_7, N6_8, N6_9, N6_10, N6_11, N6_12, N6_13, N6_14, N6_15, N6_16, N6_17, N6_18, N6_19, N6_20, N6_21, N6_22, N6_23, N6_24, N6_25, N6_26, N6_27, N6_28, N6_29, N6_30, N6_31, N6_32, N6_33, N6_34, N6_35, N6_36]; if no in [19,21,22,24,31,32,35,36] then arg:= [F,a]; else arg:= [F]; fi; L := CallFuncList( ff[no], arg ); L!.arg := data; return L; fi; end ); BindGlobal("LieAlgDBHelper", rec()); LieAlgDBHelper.isomN:= function( L, x1, x2, x3, x4 ) # Here the xi satisfy the commutation relations of N; # we produce the isomorphism with M_0^13... local F, y1, y2, y3, y4, name, K, f; F:= LeftActingDomain( L ); if Characteristic(F) <> 3 then y1:= x1+x2; y2:= 3*x1-(3/2*One(F))*x2; y3:= x1+x2-x3+2*x4; y4:= x3; else y1:= x2; y2:= x1+x2; y3:= x1-x2+x3+x4; y4:= x1-x2+x3; fi; name:= "L4_13( "; Append( name, LieAlgDBField2String( F ) ); Append( name, ", " ); Append( name, String( Zero(F) ) ); Append( name, " )" ); K:= SolvableLieAlgebra( F, [4,13, Zero(F)] ); f:= AlgebraHomomorphismByImages(K,L,Basis(K),[y1,y2,y3,y4]); return rec( name:= [ name, [ Zero(F) ] ], isom:= f ); end; LieAlgDBHelper.isomM9:= function( L, x1, x2, x3, x4, a ) # The xi satisfy the commutation rules of M9a; we return # the isomorphism with either N, M8 or M9a local F, T, pol, facs, dd, id, f, name, K, c, exp, prim, ef, b, q, i; F:= LeftActingDomain( L ); if Characteristic(F) <> 2 and a = (-1/4)*One(F) then return LieAlgDBHelper.isomN( L, x1, x2, x3, x4 ); fi; T:= Indeterminate( F); pol:= T^2-T-a; facs:= Factors( pol ); if Length( facs ) = 2 then #direct sum... dd:= DirectSumDecomposition( L ); id:= LieAlgebraIdentification( dd[1] ); f:= id.isomorphism; x1:= Image( f, Basis( Source(f) )[2] ); x2:= Image( f, Basis( Source(f) )[1] ); id:= LieAlgebraIdentification( dd[2] ); f:= id.isomorphism; x3:= Image( f, Basis( Source(f) )[2] ); x4:= Image( f, Basis( Source(f) )[1] ); name:= "L4_8( "; Append( name, LieAlgDBField2String( F ) ); Append( name, " )" ); K:= SolvableLieAlgebra( F, [4,8] ); f:= AlgebraHomomorphismByImages(K,L,Basis(K),[x1,x2,x3,x4]); return rec( name:= [ name, [] ], isom:= f ); else if HasIsFinite(F) and IsFinite(F) then # for M9 we let b be the smallest power of the primitive root # such that T^2-T-b has no roots in F q:= Size( F ); prim:= PrimitiveRoot( F ); for i in [1..q-1] do if Length(Factors( T^2-T-prim^i ) ) = 1 then b:= prim^i; break; fi; od; if Characteristic(F) > 2 then c:= (b+One(F)/4)/(a+One(F)/4); exp:= LogFFE( c, PrimitiveRoot(F) ); c:= PrimitiveRoot(F)^(exp/2); x1:= c*x1+((One(F)-c)/2)*x2; x4:= ((One(F)-c)/2)*x3+c*x4; else facs:= Factors( T^2+T+a+b ); ef:= ExtRepPolynomialRatFun( facs[1] ); if Length( ef[1] ) = 0 then c:= -ef[2]; else c:= Zero( F ); fi; x1:= x1+c*x2; x4:= c*x3+x4; fi; else b:= a; fi; name:= "L4_9( "; Append( name, LieAlgDBField2String( F ) ); Append( name, ", " ); Append( name, String( b ) ); Append( name, " )" ); K:= SolvableLieAlgebra( F, [4,9, b] ); f:= AlgebraHomomorphismByImages(K,L,Basis(K),[x1,x2,x3,x4]); return rec( name:= [ name, [ b ] ], isom:= f ); fi; end; LieAlgDBHelper.isomM10:= function( L, x1, x2, x3, x4, a ) # the xi satisfy the commutation rels of M10(a) local f, b, y1, y2, y3, y4, name, K, F; F:= LeftActingDomain( L ); if HasIsFinite(F) and IsFinite(F) then f:= Inverse( FrobeniusAutomorphism(F) ); b:= Image( f, a ); y1:= x1; y2:= b*x1+x2; y3:= x3; y4:= b*x3+x4; name:= "L4_13( "; Append( name, LieAlgDBField2String( F ) ); Append( name, ", " ); Append( name, String( Zero(F) ) ); Append( name, " )" ); K:= SolvableLieAlgebra( F, [4,13, Zero(F)] ); f:= AlgebraHomomorphismByImages(K,L,Basis(K), [y1,y2,y1+y2+y4,y1+y2+y3]); return rec( name:= [ name, [Zero(F)] ], isom:= f ); else name:= "L4_10( "; Append( name, LieAlgDBField2String( F ) ); Append( name, ", " ); Append( name, String( a ) ); Append( name, " )" ); K:= SolvableLieAlgebra( F, [4,10, a] ); f:= AlgebraHomomorphismByImages(K,L,Basis(K),[x1,x2,x3,x4]); return rec( name:= [ name, [a] ], isom:= f ); fi; end; LieAlgDBHelper.isomM11:= function( L, x1, x2, x3, x4, a, b ) local name, K, f, gam, eps, del, y1, y2, y3, y4, F; F:= LeftActingDomain(L); if b = Zero(F) then name:= "L4_11( "; Append( name, LieAlgDBField2String( F ) ); Append( name, ", " ); Append( name, String( a ) ); Append( name, ", " ); Append( name, String( b ) ); Append( name, " )" ); K:= SolvableLieAlgebra( F, [4,11,a, b] ); f:= AlgebraHomomorphismByImages(K,L,Basis(K),[x1,x2,x3,x4]); return rec( name:= [ name, [a,b] ], isom:= f ); elif HasIsFinite(F) and IsFinite(F) then f:= Inverse( FrobeniusAutomorphism(F) ); gam:= Image( f, b/(a*(b^2+One(F))) ); eps:= Image( f, 1/a ); del:= 1/(1+b); y1:= a*gam*x1+b*del*x2; y2:= a*eps*b*del*x1+a*gam*eps*x2; y3:= eps*x3; y4:= gam*x3+del*x4; name:= "L4_11( "; Append( name, LieAlgDBField2String( F ) ); Append( name, ", " ); Append( name, String( One(F) ) ); Append( name, ", " ); Append( name, String( Zero(F) ) ); Append( name, " )" ); K:= SolvableLieAlgebra( F, [4,11, One(F), Zero(F)] ); f:= AlgebraHomomorphismByImages(K,L,Basis(K),[y1,y2,y3,y4]); return rec( name:= [ name, [One(F),Zero(F)] ],isom:= f); else name:= "L4_11( "; Append( name, LieAlgDBField2String( F ) ); Append( name, ", " ); Append( name, String( a ) ); Append( name, ", " ); Append( name, String( b ) ); Append( name, " )" ); K:= SolvableLieAlgebra( F, [4,11, a, b] ); f:= AlgebraHomomorphismByImages(K,L,Basis(K),[x1,x2,x3,x4]); return rec( name:= [ name, [a,b] ], isom:= f ); fi; end; LieAlgDBHelper.solv_type:= function( L ) local n, F, sp, k, BL, b, x1, x2, x3, x4, c1, c2, K, f, name, par, id, num, c3, c4, mat, is_diag, i, j, facs, ev, s, t, sol, sevec, tevec, cfs, u, v, w, found, y1, y2, y3, y4, u1, u3, v1, v3, a, D, adM, adK, dd, bK, b_adK, ef, ev1, ev2, mp, q, C, R, exp; n:= Dimension( L ); F:= LeftActingDomain( L ); sp:= LieDerivedSubalgebra( L ); k:= 1; BL:= Basis( L ); while Dimension( sp ) < n-1 do if not BL[k] in sp then b:= ShallowCopy( BasisVectors( Basis(sp) ) ); Add( b, BL[k] ); sp:= Subspace( L, b ); fi; k:= k+1; od; b:= Basis( sp ); if n = 2 then x1:= b[1]; for k in [1,2] do if not BL[k] in sp then x2:= BL[k]; break; fi; od; if x1*x2 = Zero(L) then name:= "L2_1( "; Append( name, LieAlgDBField2String( F ) ); Append( name, " )" ); K:= SolvableLieAlgebra( F, [2,1] ); f:= AlgebraHomomorphismByImages( K, L, Basis(K), Basis(L) ); return rec( name:= [ name, [] ], isom:= f ); fi; c1:= Coefficients( b, x2*x1 ); x2:= x2/c1[1]; name:= "L2_2( "; Append( name, LieAlgDBField2String( F ) ); Append( name, " )" ); K:= SolvableLieAlgebra( F, [2,2] ); f:= AlgebraHomomorphismByImages( K, L, Basis(K), [x1,x2] ); return rec( name:= [ name, [] ], isom:= f ); elif n = 3 then x1:= b[1]; x2:= b[2]; if x1*x2 <> Zero(L) then # Here [L,L] has dimension 1, and its centralizer # has dimension 2, and it contains [L,L]. sp:= LieCentralizer( L, LieDerivedSubalgebra(L) ); b:= Basis( sp ); x1:= Basis(sp)[1]; x2:= Basis(sp)[2]; fi; for k in [1..3] do if not BL[k] in sp then x3:= BL[k]; break; fi; od; if x1*x3 = Zero(L) and x2*x3 = Zero(L) then # Abelian! name:= "L3_1( "; Append( name, LieAlgDBField2String( F ) ); Append( name, " )" ); K:= SolvableLieAlgebra( F, [3,1] ); f:= AlgebraHomomorphismByImages( K, L, Basis(K), Basis(L) ); return rec( name:= [ name, [] ], isom:= f ); fi; c1:= Coefficients( b, x3*x1 ); c2:= Coefficients( b, x3*x2 ); if c1[1] = c2[2] and c1[2] = Zero(F) and c2[1] = Zero(F) then x3:= x3/c1[1]; name:= "L3_2( "; Append( name, LieAlgDBField2String( F ) ); Append( name, " )" ); K:= SolvableLieAlgebra( F, [3,2] ); f:= AlgebraHomomorphismByImages( K, L, Basis(K), [x1,x2,x3] ); return rec( name:= [ name, [] ], isom:= f ); fi; if c1[2] = Zero(F) then # i.e., x1 is an eigenvector of ad x3, not good. if c2[1] = Zero(F) then # x2 is an eigenvector as well, so x1+x2 cannot be # (by previous case)... x1:= x1+x2; else x1:= x2; fi; fi; x2:= x3*x1; b:= Basis( Subspace( L, [x1,x2] ), [x1,x2] ); c2:= Coefficients( b, x3*x2 ); if c2[2] <> Zero(F) then x2:= x2/c2[2]; x3:= x3/c2[2]; par:= c2[1]/(c2[2]^2); name:= "L3_3( "; Append( name, LieAlgDBField2String( F ) ); Append( name, ", " ); Append( name, String( par ) ); Append( name, " )" ); K:= SolvableLieAlgebra( F, [3,3,par] ); f:= AlgebraHomomorphismByImages( K, L, Basis(K), [x1,x2,x3] ); return rec( name:= [ name, [par] ], isom:= f ); fi; par:= c2[1]; if HasIsFinite(F) and IsFinite(F) and not par in [Zero(F),One(F)] then if Characteristic(F) = 2 then f:= Inverse( FrobeniusAutomorphism( F ) ); a:= Image( f, par ); x2:= a*x2; x3:= a*x3; par:= a^2*par; else exp:= LogFFE( 1/par, PrimitiveRoot(F) ); if IsEvenInt( exp ) then a:= PrimitiveRoot(F)^(exp/2); x2:= a*x2; x3:= a*x3; par:= a^2*par; else exp:= LogFFE( PrimitiveRoot(F)/par, PrimitiveRoot(F) ); a:= PrimitiveRoot(F)^(exp/2); x2:= a*x2; x3:= a*x3; par:= a^2*par; fi; fi; fi; name:= "L3_4( "; Append( name, LieAlgDBField2String( F ) ); Append( name, ", " ); Append( name, String( par ) ); Append( name, " )" ); K:= SolvableLieAlgebra( F, [3,4,par] ); f:= AlgebraHomomorphismByImages( K, L, Basis(K), [x1,x2,x3] ); return rec( name:= [ name, [par] ], isom:= f ); elif n = 4 then x1:= b[1]; x2:= b[2]; x3:= b[3]; for k in [1..4] do if not BL[k] in sp then x4:= BL[k]; break; fi; od; K:= Subalgebra( L, [x1,x2,x3] ); adM:= List( Basis(K), x -> AdjointMatrix(Basis(K),x ) ); adK:= MutableBasis( F, [ 0*adM[1] ] ); bK:= [ ]; b_adK:= [ ]; for i in [1..Length(adM)] do if not IsContainedInSpan( adK, adM[i] ) then CloseMutableBasis( adK, adM[i] ); Add( bK, Basis(K)[i] ); Add( b_adK, adM[i] ); fi; od; adK:= VectorSpace( F, b_adK, 0*adM[1] ); mat:= List( Basis(K), x -> Coefficients( Basis(K), x4*x ) ); mat:= TransposedMat( mat ); if mat in adK then c1:= Coefficients( Basis( adK, b_adK ), mat ); if c1 = [ ] then # L is abelian name:= "L4_1( "; Append( name, LieAlgDBField2String( F ) ); Append( name, " )" ); K:= SolvableLieAlgebra( F, [4,1] ); f:= AlgebraHomomorphismByImages( K, L, Basis(K), Basis(L) ); return rec( name:= [ name, [] ], isom:= f ); else x3:= x4 - LinearCombination( c1, bK ); id:= LieAlgebraIdentification( K ); x4:= Image( id.isomorphism, Basis( Source( id.isomorphism ) )[3] ); x1:= Image( id.isomorphism, Basis( Source( id.isomorphism ) )[1] ); x2:= Image( id.isomorphism, Basis( Source( id.isomorphism ) )[2] ); K:= Subalgebra( L, [x1,x2,x3] ); fi; fi; id:= LieAlgebraIdentification( K ); x1:= Image( id.isomorphism, Basis( Source( id.isomorphism ) )[1] ); x2:= Image( id.isomorphism, Basis( Source( id.isomorphism ) )[2] ); x3:= Image( id.isomorphism, Basis( Source( id.isomorphism ) )[3] ); b:= Basis( K, [x1,x2,x3] ); c1:= Coefficients( b, x4*x1 ); c2:= Coefficients( b, x4*x2 ); c3:= Coefficients( b, x4*x3 ); mat:= [ c1, c2, c3 ]; num:= id.name[4]; if num = '1' then # K is abelian R:= PolynomialRing( F, 1 ); mat:= TransposedMat( mat ); mp:= MinimalPolynomial( F, mat ); is_diag:= true; for i in [1..3] do if not is_diag then break; fi; for j in [1..3] do if not is_diag then break; fi; if i <> j and mat[i][j] <> Zero(F) then is_diag:= false; fi; od; od; if Degree(mp) = 1 then if c1[1] = Zero(F) then name:= "L4_1( "; Append( name, LieAlgDBField2String( F ) ); Append( name, " )" ); K:= SolvableLieAlgebra( F, [4,1] ); f:= AlgebraHomomorphismByImages( K, L, Basis(K), Basis(L) ); return rec( name:= [ name, [] ], isom:= f ); else x4:= x4/c1[1]; name:= "L4_2( "; Append( name, LieAlgDBField2String( F ) ); Append( name, " )" ); K:= SolvableLieAlgebra( F, [4,2] ); f:= AlgebraHomomorphismByImages(K,L,Basis(K),[x1,x2,x3,x4]); return rec( name:= [ name, [] ], isom:= f ); fi; fi; if Degree(mp) = 2 then # compute rational canonical form; rather clumsy algorithm # which however works for the 3x3 case. facs:= Factors( R, mp ); ev:= [ ]; for f in facs do ef:= ExtRepPolynomialRatFun( f ); if Length( ef[1] ) = 0 then Add( ev, -ef[2] ); else Add( ev, Zero( F ) ); fi; od; if ev[1] <> ev[2] then sol:= NullspaceMat( TransposedMat(mat) - ev[1]*IdentityMat( 3, F ) ); ev1:= List( sol, x -> LinearCombination( x, [x1,x2,x3] ) ); sol:= NullspaceMat( TransposedMat(mat) - ev[2]*IdentityMat( 3, F ) ); ev2:= List( sol, x -> LinearCombination( x, [x1,x2,x3] ) ); if Length( ev1 ) = 2 then x1:= ev1[1]; x2:= ev1[2] + ev2[1]; x3:= x4*x2; s:= ev[1]; t:= ev[2]; else x1:= ev2[1]; x2:= ev2[2] + ev1[1]; x3:= x4*x2; s:= ev[2]; t:= ev[1]; fi; else s:= ev[1]; t:= s; sol:= NullspaceMat( TransposedMat(mat) - ev[1]*IdentityMat( 3, F ) ); ev1:= List( sol, x -> LinearCombination( x, [x1,x2,x3] ) ); sp:= Subspace( L, ev1 ); if not x1 in sp then x2:= x1; elif not x3 in sp then x2:= x3; fi; x3:= x4*x2; sp:= Subspace( L, [x2,x3] ); if not ev1[1] in sp then x1:= ev1[1]; else x1:= ev1[2]; fi; fi; if s <> Zero(F) then x3:= x3/s; x4:= x4/s; name:= "L4_3( "; Append( name, LieAlgDBField2String( F ) ); Append( name, ", " ); Append( name, String( t/s ) ); Append( name, " )" ); K:= SolvableLieAlgebra( F, [4,3,t/s] ); f:= AlgebraHomomorphismByImages(K,L,Basis(K), [x1,x2,x3,x4]); return rec( name:= [ name, [t/s] ], isom:= f ); elif t <> Zero(F) then x3:= x3/t; x4:= x4/t; name:= "L4_4( "; Append( name, LieAlgDBField2String( F ) ); Append( name, " )" ); K:= SolvableLieAlgebra( F, [4,4] ); f:= AlgebraHomomorphismByImages(K,L,Basis(K), [x1,x2,x3,x4]); return rec( name:= [ name, [] ], isom:= f ); else name:= "L4_5( "; Append( name, LieAlgDBField2String( F ) ); Append( name, " )" ); K:= SolvableLieAlgebra( F, [4,5] ); f:= AlgebraHomomorphismByImages(K,L,Basis(K), [x1,x2,x3,x4]); return rec( name:= [ name, [] ], isom:= f ); fi; fi; # look for a cyclic vector... # we take a random element, maybe better algorithm needed... if IsFinite(F) then cfs:= List( [1..10], x -> PrimitiveRoot(F)^x ); else cfs:= List( [1..10], x -> x*One(F) ); fi; Add( cfs, Zero(F) ); found:= false; while not found do u:= Random(cfs)*x1+Random(cfs)*x2+Random(cfs)*x3; v:= x4*u; w:= x4*v; sp:= Subspace( L, [u,v,w] ); if Dimension(sp) = 3 then found:= true; fi; od; x1:= u; x2:= v; x3:= w; b:= Basis( sp, [x1,x2,x3] ); c1:= Coefficients( b, x4*x3 ); if c1[3] <> Zero(F) then x2:= x2/c1[3]; x3:= x3/(c1[3]^2); x4:= x4/c1[3]; name:= "L4_6( "; Append( name, LieAlgDBField2String( F ) ); Append( name, ", " ); Append( name, String( c1[1]/(c1[3]^3) ) ); Append( name, ", "); Append( name, String( c1[2]/(c1[3]^2) ) ); Append( name, " )" ); K:= SolvableLieAlgebra( F, [4,6, c1[1]/(c1[3]^3), c1[2]/(c1[3]^2) ] ); f:= AlgebraHomomorphismByImages(K,L,Basis(K),[x1,x2,x3,x4]); return rec( name:= [ name, [c1[1]/(c1[3]^3), c1[2]/(c1[3]^2)] ], isom:= f ); else if c1[1] <> Zero(F) and c1[2] <> Zero(F) then s:= c1[2]/c1[1]; x2:= x2*s; x3:= x3*s^2; x4:= x4*s; t:= s^2*c1[2]; name:= "L4_7( "; Append( name, LieAlgDBField2String( F ) ); Append( name, ", " ); Append( name, String( t ) ); Append( name, ", "); Append( name, String( t ) ); Append( name, " )" ); K:= SolvableLieAlgebra( F, [4,7, t, t] ); f:= AlgebraHomomorphismByImages(K,L,Basis(K),[x1,x2,x3,x4]); return rec( name:= [ name, [t,t] ], isom:= f ); else if HasIsFinite(F) and IsFinite(F) then q:= Size(F); if c1[1] <> Zero(F) then par:= c1[1]; a:= PrimitiveRoot(F); if q mod 6 = 1 or q mod 6 = 4 then exp:= LogFFE( par, a ); b:= One(F); if exp mod 3 <> 0 then exp:= LogFFE( par/a, a ); b:= a; if exp mod 3 <> 0 then exp:= LogFFE( par/(a^2), a ); b:= a^2; fi; fi; a:= a^(exp/3); else exp:= LogFFE( par, a ); if exp mod 3 = 1 then for b in F do if b^3 = a then a:= a^((exp-1)/3)*b; break; fi; od; elif exp mod 3 = 2 then for b in F do if b^3 = a^2 then a:= a^((exp-2)/3)*b; break; fi; od; else a:= a^(exp/3); fi; b:= One(F); fi; c1:= [ b, Zero(F) ]; elif c1[2] <> Zero(F) then par:= c1[2]; a:= PrimitiveRoot(F); if IsEvenInt( q ) then exp:= LogFFE( par, a ); if exp mod 2 = 1 then a:= a^((exp-1)/2)*a^(q/2); else a:= a^(exp/2); fi; b:= One(F); else exp:= LogFFE( par, a ); b:= One(F); if not IsEvenInt( exp ) then exp:= LogFFE( par/a, a ); b:= a; fi; a:= a^(exp/2); fi; c1:= [ Zero(F), b ]; else a:= One(F); fi; x2:= x2/a; x3:= x3/(a^2); x4:= x4/a; fi; name:= "L4_7( "; Append( name, LieAlgDBField2String( F ) ); Append( name, ", " ); Append( name, String( c1[1] ) ); Append( name, ", "); Append( name, String( c1[2] ) ); Append( name, " )" ); K:= SolvableLieAlgebra( F, [4,7, c1[1], c1[2] ] ); f:= AlgebraHomomorphismByImages(K,L,Basis(K),[x1,x2,x3,x4]); return rec( name:= [ name, [c1[1],c1[2]] ], isom:= f ); fi; fi; elif num = '2' then D:= TransposedMat( mat ); x4:= x4+D[1][3]*x1+D[2][3]*x2-D[2][2]*x3; mat:= List( [x1,x2,x3], x -> Coefficients( b, x4*x ) ); D:= TransposedMat( mat ); if D[1][1] <> Zero(F) then x4:= x4/D[1][1]; D:= D/D[1][1]; w:= D[1][2]; v:= D[2][1]; if w <> Zero(F) then x1:= w*x1; v:= v*w; return LieAlgDBHelper.isomM9( L, x1, x2, x3, x4, v ); else #direct sum... dd:= DirectSumDecomposition( L ); id:= LieAlgebraIdentification( dd[1] ); f:= id.isomorphism; x1:= Image( f, Basis( Source(f) )[2] ); x2:= Image( f, Basis( Source(f) )[1] ); id:= LieAlgebraIdentification( dd[2] ); f:= id.isomorphismorphism; x3:= Image( f, Basis( Source(f) )[2] ); x4:= Image( f, Basis( Source(f) )[1] ); name:= "L4_8( "; Append( name, LieAlgDBField2String( F ) ); Append( name, " )" ); K:= SolvableLieAlgebra( F, [4,8] ); f:= AlgebraHomomorphismByImages(K,L,Basis(K), [x1,x2,x3,x4]); return rec( name:= [ name, [] ], isom:= f ); fi; else if D[2][1] <> Zero(F) then x4:= x4/D[2][1]; a:= D[1][2]/D[2][1]; if Characteristic(F) <> 2 then a:= a - (1/4)*One(F); return LieAlgDBHelper.isomM9( L, x1/(4*One(F))+x2/(2*One(F)), x1/(2*One(F)), x3, x3/(2*One(F))+x4, a ); else return LieAlgDBHelper.isomM10( L, x1, x2, x3, x4, a ); fi; else if D[1][2] <> Zero(F) then x4:= x4/D[1][2]; if Characteristic(F) <> 2 then y1:= (x1+x2)/(2*One(F)); y2:= x2; y3:= x3; y4:= (x3+x4)/(2*One(F)); return LieAlgDBHelper.isomN( L, y1, y2, y3, y4 ); else name:= "L4_10( "; Append( name, LieAlgDBField2String( F ) ); Append( name, ", " ); Append( name, String( Zero(F) ) ); Append( name, " )" ); K:= SolvableLieAlgebra( F, [4,10, Zero(F) ] ); f:= AlgebraHomomorphismByImages(K,L,Basis(K), [x2,x1,x3,x4]); return rec( name:= [ name, [Zero(F)] ], isom:= f ); fi; fi; fi; fi; elif num = '3' then D:= TransposedMat( mat ); a:= id.parameters[1]; if a <> Zero(F) then x4:= x4-(D[1][3]/a-D[2][3])*x1+D[1][3]/a*x2-D[2][1]*x3; x4:= x4/D[1][1]; return LieAlgDBHelper.isomM9( L, x2, x1, x4, x3, a ); else x4:= x4+D[2][3]*x1-D[2][1]*x3; mat:= List( [x1,x2,x3], x -> Coefficients( b, x4*x ) ); D:= TransposedMat( mat ); if D[1][3] = Zero(F) then x4:= x4/D[1][1]; return LieAlgDBHelper.isomM9( L, x2, x1, x4, x3, a ); elif D[1][1] = Zero(F) then x4:= x4/D[1][3]; name:= "L4_6( "; Append( name, LieAlgDBField2String( F ) ); Append( name, ", " ); Append( name, String( Zero(F) ) ); Append( name, ", "); Append( name, String( Zero(F) ) ); Append( name, " )" ); K:= SolvableLieAlgebra( F, [4,6, Zero(F), Zero(F)] ); f:= AlgebraHomomorphismByImages(K,L,Basis(K), [-x4,x1,x2,x3]); return rec( name:= [ name, [Zero(F),Zero(F)] ], isom:= f ); else x4:= x4/D[1][1]; x3:= -D[1][3]/D[1][1]*x1+x3; return LieAlgDBHelper.isomM9( L, x2, x1, x4, x3, a ); fi; fi; else # i.e., num = '4'... D:= TransposedMat( mat ); a:= id.parameters[1]; if a <> Zero(F) and Characteristic(F) <> 2 then x4:= x4+D[2][3]*x1+D[1][3]/a*x2-D[2][1]*x3; x4:= x4/D[1][1]; return LieAlgDBHelper.isomM9( L, x1/(4*One(F))+x2/(2*One(F)), x1/(2*One(F)), x4, x3+x4/(2*One(F)), a-(1/4*One(F)) ); elif a <> Zero(F) and Characteristic(F) = 2 then x4:= x4+D[2][3]*x1+D[1][3]/a*x2-D[2][1]*x3; if D[3][3] = Zero(F) then x4:= x4/D[1][1]; return LieAlgDBHelper.isomM10( L, x1, x2, x4, x3, a ); elif D[1][1] <> Zero(F) then x4:= x4/D[1][1]; b:= One(F) + D[3][3]/D[1][1]; return LieAlgDBHelper.isomM11( L, x1, x2, x3, x4, a, b ); else if a <> One(F) then x4:= x4/D[3][3]; y1:= (a*x1+x2)/(a+One(F)); y2:= a*(x1+x2)/(a+One(F)); y3:= x3; y4:= x3+(a+1)*x4; return LieAlgDBHelper.isomM11( L, y1, y2, y3, y4, a, a ); else return LieAlgDBHelper.isomM11( L, x2, x1, x3, x4, One(F), Zero(F) ); fi; fi; else x4:= x4+D[2][3]*x1-D[2][1]*x3; if D[1][3]=Zero(F) and D[3][1]=Zero(F) and D[1][1]=D[3][3] then x4:= x4/D[1][1]; name:= "L4_12( "; Append( name, LieAlgDBField2String( F ) ); Append( name, " )" ); K:= SolvableLieAlgebra( F, [4,12] ); f:= AlgebraHomomorphismByImages(K,L,Basis(K), [x1,x2,x3,x4]); return rec( name:= [ name, [] ], isom:= f ); else if D[3][3] <> Zero(F) then u1:= D[1][1]; u3:= D[3][1]; v1:= D[1][3]; v3:= D[3][3]; if v1 <> Zero(F) then x1:= x1+(v3/v1)*x3; elif u3 <> Zero(F) then x3:= -(v3/u3)*x1+x3; elif u1 = Zero(F) then y1:= x1; x1:= x3; x2:= -x2; x3:= y1; else y1:= x1; x1:= x1/(One(F) - v3/u1)-x3; x3:= -(v3/u1)*y1/(One(F) - v3/u1)+x3; fi; K:= Subalgebra( L, [x1,x2,x3] ); b:= Basis( K, [x1,x2,x3] ); c1:= Coefficients( b, x4*x1 ); c2:= Coefficients( b, x4*x2 ); c3:= Coefficients( b, x4*x3 ); mat:= [ c1, c2, c3 ]; D:= TransposedMat( mat ); fi; # now D[3][3] is zero... u1:= D[1][1]; u3:= D[3][1]; v1:= D[1][3]; if u1 <> Zero(F) and v1 <> Zero(F) then x4:= x4/u1; x1:= (1/u1)*x1; x2:= (1/(u1*v1))*x2; x3:= (1/v1)*x3; a:= (v1/u1)*(u3/u1); name:= "L4_13( "; Append( name, LieAlgDBField2String( F ) ); Append( name, ", " ); Append( name, String( a ) ); Append( name, " )" ); K:= SolvableLieAlgebra( F, [4,13, a] ); f:= AlgebraHomomorphismByImages(K,L,Basis(K), [x1,x2,x3,x4]); return rec( name:= [ name, [a] ], isom:= f ); elif u1 <> Zero(F) and v1 = Zero(F) then if u3 <> Zero(F) then x4:= x4/u1; x1:= (1/u3)*x1; x2:= (1/(u1*u3))*x2; x3:= (1/u1)*x3; y1:= x1; x1:= x1+x3; x2:= -x2; x3:= y1; else x4:= x4/u1; x3:= x1+x3; fi; name:= "L4_13( "; Append( name, LieAlgDBField2String( F ) ); Append( name, ", " ); Append( name, String( Zero(F) ) ); Append( name, " )" ); K:= SolvableLieAlgebra( F, [4,13, Zero(F)] ); f:= AlgebraHomomorphismByImages(K,L,Basis(K), [x1,x2,x3,x4]); return rec( name:= [ name, [Zero(F)] ], isom:= f ); elif u1 = Zero(F) and v1 <> Zero(F) then x4:= x4/v1; par:= u3/v1; if HasIsFinite(F) and IsFinite(F) and not par in [Zero(F),One(F)] then if Characteristic(F) = 2 then f:= Inverse( FrobeniusAutomorphism( F ) ); a:= Image( f, 1/par ); else exp:= LogFFE( 1/par, PrimitiveRoot(F) ); if IsEvenInt( exp ) then a:= PrimitiveRoot(F)^(exp/2); else exp:= LogFFE( PrimitiveRoot(F)/par, PrimitiveRoot(F) ); a:= PrimitiveRoot(F)^(exp/2); fi; fi; x1:= a*x1; x2:= a*x2; x4:= a*x4; par:= a^2*par; fi; if par = Zero(F) then name:= "L4_7( "; Append( name, LieAlgDBField2String( F ) ); Append( name, ", " ); Append( name, String( Zero(F) ) ); Append( name, ", " ); Append( name, String( Zero(F) ) ); Append( name, " )" ); K:= SolvableLieAlgebra( F, [4,7, Zero(F), Zero(F)] ); f:= AlgebraHomomorphismByImages( K, L, Basis(K), [-x4,x1,x2,x3]); return rec( name:= [ name, [Zero(F),Zero(F)] ], isom:= f ); else name:= "L4_14( "; Append( name, LieAlgDBField2String( F ) ); Append( name, ", " ); Append( name, String( par ) ); Append( name, " )" ); K:= SolvableLieAlgebra( F, [4,14, par] ); f:= AlgebraHomomorphismByImages( K, L, Basis(K), [x1,x2,x3,x4] ); return rec( name:= [ name, [par] ], isom:= f ); fi; else x4:= x4/u3; y1:= x3; y2:= -x2; y3:= x1; y4:= x4; name:= "L4_7( "; Append( name, LieAlgDBField2String( F ) ); Append( name, ", " ); Append( name, String( Zero(F) ) ); Append( name, ", " ); Append( name, String( Zero(F) ) ); Append( name, " )" ); K:= SolvableLieAlgebra( F, [4, 7, Zero(F), Zero(F)] ); f:= AlgebraHomomorphismByImages( K, L, Basis(K), [-y4,y1,y2,y3] ); return rec( name:= [ name, [Zero(F),Zero(F)] ], isom:= f ); fi; fi; fi; fi; else Error( "dim > 4 not yet implemented" ); fi; end; #======================================================================== LieAlgDBHelper.liealg_hom:= function( K, L, p, i ) return AlgebraHomomorphismByImagesNC( K, L, p, i ); end; LieAlgDBHelper.class_dim_le4:= function( L ) # finds an isomorphism of the nilpotent Lie algebra L of dim <= 4, # to a "normal form". local F, C, C1, D, i, x, y, z, u, T, V, K, A, cc, b, a; F:= LeftActingDomain( L ); if Dimension( L ) = 3 then if IsLieAbelian( L ) then return rec( type:= [ 3, 1 ], f:= LieAlgDBHelper.liealg_hom( L, L, Basis( L ), Basis( L ) ) ); else C:= LieCentre( L ); for i in [1..3] do if not Basis( L )[i] in C then x:= Basis( L )[i]; break; fi; od; C:= Subalgebra( L, [x,Basis( C )[1]] ); for i in [1..3] do if not Basis( L )[i] in C then y:= Basis( L )[i]; break; fi; od; T:= EmptySCTable( 3, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); K:= LieAlgebraByStructureConstants( F, T ); return rec( type:= [ 3, 2 ], f:= LieAlgDBHelper.liealg_hom( L, K, [x,y,x*y], Basis( K ) ) ); fi; elif Dimension( L ) = 4 then if IsLieAbelian( L ) then return rec( type:= [ 4, 1 ], f:= LieAlgDBHelper.liealg_hom( L, L, Basis( L ), Basis( L ) ) ); else C:= LieCentre( L ); if Dimension( C ) = 2 then for i in [1..4] do if not Basis( L )[i] in C then x:= Basis( L )[i]; break; fi; od; C1:= Subalgebra( L, [ Basis( C )[1], Basis( C )[2], x ] ); for i in [1..4] do if not Basis( L )[i] in C1 then y:= Basis( L )[i]; break; fi; od; A:= Subalgebra( L, [ x*y ] ); if Basis( C )[1] in A then u:= Basis( C )[2]; else u:= Basis( C )[1]; fi; T:= EmptySCTable( 4, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); K:= LieAlgebraByStructureConstants( F, T ); return rec( type:= [ 4, 2 ], f:= LieAlgDBHelper.liealg_hom( L, K, [x,y,x*y,u], Basis( K ) ) ); elif Dimension( C ) = 1 then D:= LieDerivedSubalgebra( L ); b:= ShallowCopy( Basis( D ) ); cc:= [ ]; for i in [1..4] do x:= Basis( L )[i]; if not x in D then Add( cc, x ); Add( b, x ); D:= Subalgebra( L, b ); fi; od; z:= cc[1]*cc[2]; if cc[1]*z <> Zero( L ) then x:= cc[1]; y:= cc[2]; else x:= cc[2]; y:= cc[1]; fi; z:= x*y; u:= x*z; # we have to change y to make sure that [x,y]= z, and # [y,u]= 0. V:= Basis( Subalgebra( L, [u] ), [u] ); a:= Coefficients( V, y*z )[1]; y:= -a*x+y; T:= EmptySCTable( 4, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); K:= LieAlgebraByStructureConstants( F, T ); return rec( type:= [ 4, 3 ], f:= LieAlgDBHelper.liealg_hom( L, K, [x,y,z,u], Basis( K ) ) ); fi; fi; fi; end; LieAlgDBHelper.skew_symm_NF:= function( V, f ) # here V is a vector space, and f : VxV -> F a skew-symmetric # bilinear function; we compute a basis of V such that f has standard form. # Here f is just given as a GAP function. local b, dim, found, i, j, c, W, bU, u, v, bb, done, x, w; b:= Basis(V); dim:= Dimension(V); found:= false; for i in [1..dim] do if found then break; fi; for j in [i+1..dim] do c:= f(b[i],b[j]); if not IsZero(c) then u:= [b[i]]; v:= [b[j]/c]; found:= true; break; fi; od; od; done:= false; while not done do # consider the linear map T: V --> V defined by # T(w) = \sum_i f(w,v[i])u[i] - f(w,u[i])v[i], # then T is projection onto the space spanned by the elts in u,v. # set U = (1-T)V; then T is zero on U, and V is the direct # sum of U and the space spanned by u,v. We compute U. W:= MutableBasis( LeftActingDomain(V), [], Zero(V) ); for x in b do w:= x-Sum( List( [1..Length(u)], k -> f(x,v[k])*u[k]-f(x,u[k])*v[k] ) ); if not IsContainedInSpan( W, w ) then CloseMutableBasis( W, w ); fi; od; bU:= BasisVectors(W); found:= false; for i in [1..Length(bU)] do if found then break; fi; for j in [i+1..Length(bU)] do c:= f(bU[i],bU[j]); if not IsZero(c) then Add( u, bU[i] ); Add( v, bU[j]/c ); found:= true; break; fi; od; od; if not found then done:= true; bb:= [ ]; for i in [1..Length(u)] do Add( bb, u[i] ); Add( bb, v[i] ); od; Append( bb, bU ); elif 2*Length(u) = dim then done:= true; bb:= [ ]; for i in [1..Length(u)] do Add( bb, u[i] ); Add( bb, v[i] ); od; fi; od; return bb; end; LieAlgDBHelper.class_dim_5:= function( K ) local F, C, D, ind, i, bL, bsp, L, W, t, T, d, N, imgs, p, s, coc1, coc2, tau, M, coc21, coc22, bM, cz, cx, type, a, b, c, m, mat, cf, c1, c2, x, mt, sp, y, y1, r; if IsLieAbelian( K ) then return rec( type:= [ 5, 1 ], f:= LieAlgDBHelper.liealg_hom( K, K, Basis( K ), Basis( K ) ) ); fi; F:= LeftActingDomain( K ); # see if there is an abelian component ( necessarily of dim 1 ) C:= LieCentre( K ); D:= LieDerivedSubalgebra( K ); ind:= 0; for i in [1..Dimension( C )] do if not Basis( C )[i] in D then ind:= i; break; fi; od; if ind > 0 then bL:= ShallowCopy( Basis( D ) ); bsp:= ShallowCopy( bL ); Add( bsp, Basis( C )[ind] ); sp:= MutableBasis( F, bsp ); for i in [1..Dimension( K )] do if not IsContainedInSpan( sp, Basis( K )[i] ) then Add( bL, Basis( K )[i] ); CloseMutableBasis( sp, Basis( K )[i] ); fi; od; # now K = bL \oplus C[ind]. L:= Subalgebra( K, bL ); bL:= ShallowCopy( Basis( L ) ); Add( bL, Basis( C )[ind] ); W:= Basis( K, bL ); # so now bL is a basis of K, the first n-1 elements form # a basis of an ideal, the last element of an abelian ideal. # We have an isomorphism of K to the direct sum of L with a # 1-dim ideal; write x\in K on the basis bL. r:= LieAlgDBHelper.class_dim_le4( L ); type:= r.type; tau:= r.f; T:= ShallowCopy( StructureConstantsTable( Basis( Range( tau ) ) ) ); d:= Dimension( Range( tau ) ); for i in [1..d] do T[i]:= ShallowCopy( T[i] ); Add( T[i], [ [], [] ] ); od; T[d+3]:= T[d+2]; T[d+2]:= T[d+1]; T[d+1]:= List( [1..d+1], x -> [ [], [] ] ); N:= LieAlgebraByStructureConstants( F, T ); imgs:= [ ]; for x in Basis( K ) do cx:= Coefficients( W, x ); y:= Image( tau, LinearCombination( Basis( L ), cx{[1..Length( cx )-1]} ) ); y:= ShallowCopy( Coefficients( Basis( Range( tau ) ), y ) ); Add( y, cx[ Length( cx ) ] ); if type[2]= 2 then # isom y1:= ShallowCopy( y ); y1[1]:= y[1]; y1[2]:= y[2]; y1[3]:= y[3]; y1[4]:= y[5]; y1[5]:= y[4]; y:= ShallowCopy( y1 ); fi; Add( imgs, LinearCombination( Basis( N ), y ) ); od; return rec( type:= [ 5, type[2] ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) ); fi; # if there are no central components, then we look at the # cocycle we get by dividing by the centre, and so on. C:= LieCentre( K ); p:= NaturalHomomorphismByIdeal( K, C ); L:= Range( p ); s:= function( v ) return PreImagesRepresentative( p, v ); end; coc1:= function( u, v ) return s( u )*s( v )-s( u*v ); end; r:= LieAlgDBHelper.class_dim_le4( L ); type:= r.type; tau:= r.f; M:= Range( tau ); coc2:= function( u, v ) return coc1( PreImage( tau,u ), PreImage( tau,v ) ); end; # i.e., the cocycle for M # now we need to map coc2 to normal form, according to the type of M if type = [ 4, 1 ] then coc21:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[1]; end; bM:= Basis( M, LieAlgDBHelper.skew_symm_NF( M, coc21 ) ); T:= EmptySCTable( 5, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,5] ); SetEntrySCTable( T, 3, 4, [1,5] ); N:= LieAlgebraByStructureConstants( F, T ); imgs:= [ ]; for x in Basis( K ) do # the coordinate of the central part of x: # this will be the coordinate of the central part of the # image of x. cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) )[1]; cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p,x ) ) ) ); Add( cx, cz ); Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 5, 4 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) ); elif type = [ 4, 2 ] then coc21:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[1]; end; bM:= ShallowCopy( BasisVectors( Basis( M ) ) ); a:= coc21( bM[1], bM[3] ); b:= coc21( bM[1], bM[4] ); c:= coc21( bM[2], bM[3] ); d:= coc21( bM[2], bM[4] ); mat:= IdentityMat( 4, F ); if IsZero( a ) then # c is non zero m:= IdentityMat( 4, F ); m[2][1]:= 1; mat:= m; a:= c; b:= b+d; fi; m:= IdentityMat( 4, F ); m[1][1]:= 1/a; m[1][2]:= -c; m[2][2]:= a; m[4][4]:= 1/( a*d-b*c ); m[3][4]:= -m[4][4]*b/a; mat:= mat*m; mat:= TransposedMat( mat ); bM:= Basis( M, List( mat, x -> LinearCombination( Basis( M ), x ) ) ); T:= EmptySCTable( 5, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,5] ); SetEntrySCTable( T, 2, 4, [1,5] ); N:= LieAlgebraByStructureConstants( F, T ); imgs:= [ ]; for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) )[1]; cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) ); Add( cx, cz ); # effectively subtract a coboundary ( i.e., apply one further # isomorphism ): # Here we have ( in the normal form Lie algebra ) # [x1,x2]= x3. Hence c( x1,x2 )= 1, rest 0 is a coboundary. # The isomorphism corresponding to the subtraction of c # from the cocycle, subtracts coc21( x1, x2 )*x5 from # the element. ( Rather vague comment... ). cf:= coc21( bM[1], bM[2] ); cx[5]:= cx[5] - cf*cx[3]; Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 5, 5 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) ); elif type = [ 4, 3 ] then coc21:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[1]; end; bM:= Basis( M ); a:= coc21( bM[1], bM[4] ); b:= coc21( bM[2], bM[3] ); if not IsZero( b ) then T:= EmptySCTable( 5, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); SetEntrySCTable( T, 1, 4, [1,5] ); SetEntrySCTable( T, 2, 3, [1,5] ); N:= LieAlgebraByStructureConstants( F, T ); imgs:= [ ]; for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) )[1]; cx:= ShallowCopy( Coefficients( Basis( M ), Image( tau, Image( p, x ) ) ) ); cf:= a/b; cx[1]:= cx[1]*cf; cx[2]:= cx[2]*cf; cx[3]:= cx[3]*cf^2; cx[4]:= cx[4]*cf^3; Add( cx, cf^4*cz/a ); # take care of the coboundary... c1:= coc21( cf^-1*Basis( M )[1],cf^-1*Basis( M )[2] ); cx[5]:= cx[5] - c1*( cf^4/a )*cx[3]; c1:= coc21( cf^-1*Basis( M )[1],cf^-2*Basis( M )[3] ); cx[5]:= cx[5] - c1*( cf^4/a )*cx[4]; Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 5, 6 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) ); else T:= EmptySCTable( 5, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); SetEntrySCTable( T, 1, 4, [1,5] ); N:= LieAlgebraByStructureConstants( F, T ); imgs:= [ ]; for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p, x ) ) )[1]; cx:= ShallowCopy( Coefficients( Basis( M ), Image( tau, Image( p,x ) ) ) ); Add( cx, cz/a ); # take care of the coboundary... cf:= coc21( Basis( M )[1], Basis( M )[2] ); cx[5]:= cx[5] - cf/a*cx[3]; cf:= coc21( Basis( M )[1], Basis( M )[3] ); cx[5]:= cx[5] - cf/a*cx[4]; Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 5, 7 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) ); fi; elif type = [ 3, 1 ] then coc21:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[1]; end; coc22:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[2]; end; bM:= LieAlgDBHelper.skew_symm_NF( M, coc21 ); b:= coc22( bM[1], bM[3] ); c:= coc22( bM[2],bM[3] ); if IsZero( b ) then # change the basis bM so that it becomes nonzero. bM[1]:= bM[1]+bM[2]; b:= coc22( bM[1],bM[3] ); c:= coc22( bM[2],bM[3] ); fi; # change the basis so that c becomes zero... bM[2]:= bM[2]-c/b*bM[1]; bM:= Basis( M, bM ); a:= coc22( bM[1],bM[2] ); b:= coc22( bM[1],bM[3] ); T:= EmptySCTable( 5, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,4] ); SetEntrySCTable( T, 1, 3, [1,5] ); N:= LieAlgebraByStructureConstants( F, T ); imgs:= [ ]; for x in Basis( K ) do # the coordinate of the central part of x: cz:= Coefficients( Basis( C ), x-s( Image( p, x ) ) ); cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) ); # account for the base change in the centre: cz[2]:= -cz[1]*a/b+cz[2]/b; Append( cx, cz ); Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 5, 8 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) ); elif type = [ 3, 2 ] then coc21:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[1]; end; coc22:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[2]; end; bM:= Basis( M ); m:= [[coc21( bM[1],bM[3] ),coc21( bM[2],bM[3] )], [coc22( bM[1],bM[3] ),coc22( bM[2],bM[3] )] ]; m:= m^-1; mt:= TransposedMat( m ); T:= EmptySCTable( 5, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); SetEntrySCTable( T, 2, 3, [1,5] ); N:= LieAlgebraByStructureConstants( F, T ); c1:= mt[1][1]*coc21( bM[1],bM[2] )+mt[2][1]*coc22( bM[1],bM[2] ); c2:= mt[1][2]*coc21( bM[1],bM[2] )+mt[2][2]*coc22( bM[1],bM[2] ); imgs:= [ ]; for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p, x ) ) ); cx:= ShallowCopy( Coefficients( Basis( M ), Image( tau, Image( p, x ) ) ) ); Append( cx, cz*mt ); # take care of the coboundary... cx[4]:= cx[4] - c1*cx[3]; cx[5]:= cx[5] - c2*cx[3]; Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 5, 9 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) ); fi; end; LieAlgDBHelper.comp_mat:= function( x, y, det ) local a,b,c,d; if IsZero(y) then c:= 0; a:= 1/x; d:= x*det;b:= 0; else a:= -y*det; c:= -y*det; d:= x*det; b:= (1+x*y*det)/y; fi; return [[a,b],[c,d]]; end; LieAlgDBHelper.sq_free:= function( r ) # r is a rational number, we output a list of two rationals [a,b], # such that b is square free, r = a^2b and b is an integer. local sign, f, u, v, k, n1, n2, d1, d2; if r >= 0 then sign:= 1; else sign:= -1; r:= -r; fi; f:= Factors( NumeratorRat( r ) ); u:= [ ]; v:= [ ]; k:= 1; while k <= Length( f ) do if k < Length( f ) then if f[k]= f[k+1] then Add( u, f[k] ); k:= k+2; else Add( v, f[k] ); k:= k+1; fi; else Add( v, f[k] ); k:= k+1; fi; od; n1:= Product( u ); n2:= Product( v ); f:= Factors( DenominatorRat( r ) ); u:= [ ]; v:= [ ]; k:= 1; while k <= Length( f ) do if k < Length( f ) then if f[k]= f[k+1] then Add( u, f[k] ); k:= k+2; else Add( v, f[k] ); k:= k+1; fi; else Add( v, f[k] ); k:= k+1; fi; od; d1:= Product( u ); d2:= Product( v ); return [n1/(d1*d2), sign*n2*d2 ]; end; LieAlgDBHelper.class_dim_6:= function( K ) local F, C, D, ind, i, bL, bsp, sp, W, r, T, d, N, imgs, x, cx, g, P, p, L, s, coc1, coc2, coc21, coc22, coc23, type, tau, M, bM, aa, a, bb, b, c, m, v1, v2, e, f, c1, c2, c3, c4, a21, tt, cz1, cx1, s13, s24, s14, t, uu, u, v, cf, y, y1, cz, a32, a41, a42, store, q, cc, f1, f2, f3, f4, f5, ff, cf1, xi, ex, eps,root; if IsLieAbelian( K ) then return rec( type:= [ 6, 1 ], f:= LieAlgDBHelper.liealg_hom( K, K, Basis( K ), Basis( K ) ) ); fi; F:= LeftActingDomain( K ); # see if there is an abelian component ( necessarily of dim 1 ) C:= LieCentre( K ); D:= LieDerivedSubalgebra( K ); ind:= 0; for i in [1..Dimension( C )] do if not Basis( C )[i] in D then ind:= i; break; fi; od; if ind > 0 then bL:= ShallowCopy( Basis( D ) ); bsp:= ShallowCopy( bL ); Add( bsp, Basis( C )[ind] ); sp:= MutableBasis( F, bsp ); for i in [1..Dimension( K )] do if not IsContainedInSpan( sp, Basis( K )[i] ) then Add( bL, Basis( K )[i] ); CloseMutableBasis( sp, Basis( K )[i] ); fi; od; # now K = bL \oplus C[ind]. L:= Subalgebra( K, bL ); bL:= ShallowCopy( Basis( L ) ); Add( bL, Basis( C )[ind] ); W:= Basis( K, bL ); # so now bL is a basis of K, the first n-1 elements form # a basis of an ideal, the last element of an abelian ideal. # We have an isomorphism of K to the direct sum of L with a # 1-dim ideal; write x\in K on the basis bL. r:= LieAlgDBHelper.class_dim_5( L ); type:= r.type; tau:= r.f; T:= ShallowCopy( StructureConstantsTable( Basis( Range( tau ) ) ) ); d:= Dimension( Range( tau ) ); for i in [1..d] do T[i]:= ShallowCopy( T[i] ); Add( T[i], [ [], [] ] ); od; T[d+3]:= T[d+2]; T[d+2]:= T[d+1]; T[d+1]:= List( [1..d+1], x -> [ [], [] ] ); N:= LieAlgebraByStructureConstants( F, T ); imgs:= [ ]; for x in Basis( K ) do cx:= Coefficients( W, x ); y:= Image( tau, LinearCombination( Basis( L ), cx{[1..Length( cx )-1]} ) ); y:= ShallowCopy( Coefficients( Basis( Range( tau ) ), y ) ); Add( y, cx[ Length( cx ) ] ); if type[2]= 2 then # isom y1:= ShallowCopy( y ); y1[1]:= y[1]; y1[2]:= y[2]; y1[3]:= y[3]; y1[4]:= y[6]; y1[5]:= y[4]; y1[6]:= y[5]; y:= ShallowCopy( y1 ); elif type[2]= 3 then # isom y1:= ShallowCopy( y ); y1[1]:= y[1]; y1[2]:= y[2]; y1[3]:= y[3]; y1[4]:= y[4]; y1[5]:= y[6]; y1[6]:= y[5]; y:= ShallowCopy( y1 ); fi; Add( imgs, LinearCombination( Basis( N ), y ) ); od; return rec( type:= [ 6, type[2] ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) ); fi; # if there are no central components, then we look at the # cocycle we get by dividing by the centre, and so on. C:= LieCentre( K ); p:= NaturalHomomorphismByIdeal( K, C ); L:= Range( p ); s:= function( v ) return PreImagesRepresentative( p, v ); end; coc1:= function( u, v ) return s( u )*s( v )-s( u*v ); end; if Dimension( L ) <= 4 then r:= LieAlgDBHelper.class_dim_le4( L ); else r:= LieAlgDBHelper.class_dim_5( L ); fi; type:= r.type; tau:= r.f; M:= Range( tau ); coc2:= function( u, v ) return coc1( PreImage( tau,u ), PreImage( tau,v ) ); end; # i.e., the cocycle for M # now we need to map coc2 to normal form, according to the type of M if type = [ 5, 2 ] then coc21:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[1]; end; bM:= ShallowCopy( BasisVectors( Basis( ( M ) ) ) ); # we change the basis bM so that the cocycle is in "normal form": a:= coc21( bM[1],bM[3] ); d:= coc21( bM[2],bM[3] ); m:= LieAlgDBHelper.comp_mat( a, d, One( F ) ); v1:= m[1][1]*bM[1]+m[1][2]*bM[2]; v2:= m[2][1]*bM[1]+m[2][2]*bM[2]; bM[1]:= v1; bM[2]:= v2; b:= coc21( bM[1],bM[4] ); c:= coc21( bM[1],bM[5] ); bM[4]:= bM[4]-b*bM[3]; bM[5]:= bM[5]-c*bM[3]; e:= coc21( bM[2],bM[4] ); f:= coc21( bM[2],bM[5] ); g:= coc21( bM[4],bM[5] ); if not ( IsZero( e ) and IsZero( f ) ) then m:= LieAlgDBHelper.comp_mat( e, f, 1/g ); v1:= m[1][1]*bM[4]+m[1][2]*bM[5]; v2:= m[2][1]*bM[4]+m[2][2]*bM[5]; bM[4]:= v1; bM[5]:= v2; T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,6] ); SetEntrySCTable( T, 4, 5, [1,6] ); N:= LieAlgebraByStructureConstants( F, T ); imgs:= [ ]; c1:= coc21( bM[1],bM[2] ); bM:= Basis( M, bM ); for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p, x ) ) ); cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) ); Append( cx, cz ); # take care of the coboundary... cx[6]:= cx[6] - c1*cx[3]; # isomorphism to M_2 ( see paper ): cx[5]:= cx[5]-cx[2]; Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 6, 10 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) ); else bM[4]:= bM[4]/g; T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,6] ); SetEntrySCTable( T, 4, 5, [1,6] ); N:= LieAlgebraByStructureConstants( F, T ); imgs:= [ ]; c1:= coc21( bM[1],bM[2] ); bM:= Basis( M, bM ); for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) ); Append( cx, cz ); # take care of the coboundary... cx[6]:= cx[6] - c1*cx[3]; Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 6, 10 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) ); fi; elif type = [ 5, 3 ] then coc21:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[1]; end; bM:= ShallowCopy( BasisVectors( Basis( M ) ) ); # we change the basis bM so that the cocycle is in "normal form": a:= coc21( bM[1],bM[4] ); bM[2]:= bM[2]/a; bM[3]:= bM[3]/a; bM[4]:= bM[4]/a; b:= coc21( bM[1],bM[5] ); d:= coc21( bM[2],bM[5] ); bM[1]:= bM[1]-b/d*bM[2]; bM[5]:= bM[5]/d; c:= coc21( bM[2],bM[3] ); if not IsZero( c ) then T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); SetEntrySCTable( T, 1, 4, [1,6] ); SetEntrySCTable( T, 2, 3, [1,6] ); SetEntrySCTable( T, 2, 5, [1,6] ); N:= LieAlgebraByStructureConstants( F, T ); imgs:= [ ]; c1:= coc21( bM[1],bM[2] ); c2:= coc21( bM[1],bM[3] ); bM:= Basis( M, bM ); for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) ); Append( cx, cz ); # take care of the coboundary... cx[6]:= cx[6] - c1*cx[3]-c2*cx[4]; # make c equal to 1... cx[1]:= cx[1]/c; cx[2]:= cx[2]/c; cx[3]:= cx[3]/c^2; cx[4]:= cx[4]/c^3; cx[5]:= cx[5]/c^3; cx[6]:= cx[6]/c^4; Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 6, 11 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) ); else T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); SetEntrySCTable( T, 1, 4, [1,6] ); SetEntrySCTable( T, 2, 5, [1,6] ); N:= LieAlgebraByStructureConstants( F, T ); imgs:= [ ]; c1:= coc21( bM[1],bM[2] ); c2:= coc21( bM[1],bM[3] ); bM:= Basis( M,bM ); for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) ); Append( cx, cz ); # take care of the coboundary... cx[6]:= cx[6] - c1*cx[3]-c2*cx[4]; Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 6, 12 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) ); fi; elif type = [ 5, 5 ] then coc21:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[1]; end; bM:= ShallowCopy( BasisVectors( Basis( M ) ) );; # we change the basis bM so that the cocycle is in "normal form": b:= coc21( bM[1],bM[5] ); bM[2]:= bM[2]/b; bM[3]:= bM[3]/b; bM[5]:= bM[5]/b; a:= coc21( bM[1],bM[4] ); c:= coc21( bM[2],bM[3] ); bM[1]:= bM[1]-a*bM[3]; bM[2]:= bM[2]+c*bM[4]; if not Characteristic( F )= 2 then imgs:= [ ]; c1:= coc21( bM[1],bM[2] ); c2:= coc21( bM[1],bM[3] ); d:= coc21( bM[2],bM[4] )-c2; T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,5] ); SetEntrySCTable( T, 2, 4, [1,5] ); SetEntrySCTable( T, 1, 5, [1,6] ); SetEntrySCTable( T, 3, 4, [1,6] ); N:= LieAlgebraByStructureConstants( F, T ); bM:= Basis( M,bM ); for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) ); Append( cx, cz ); # take care of the coboundary... cx[6]:= cx[6] - c1*cx[3]-c2*cx[5]; if not IsZero( d ) then # map it to 1: cx[1]:= cx[1]/d; cx[3]:= cx[3]/d; cx[4]:= cx[4]/d^2; cx[5]:= cx[5]/d^2; cx[6]:= cx[6]/d^3; # make isom with the Lie alg where d= 0: cx[5]:= cx[5]+( cx[2]+cx[3] )/2; cx[4]:= cx[4]+cx[1]/2; cx[3]:= cx[3]+cx[2]; fi; Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 6, 13 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) ); else c1:= coc21( bM[1],bM[2] ); c2:= coc21( bM[1],bM[3] ); d:= coc21( bM[2],bM[4] )-c2; if IsZero( d ) then imgs:= [ ]; c1:= coc21( bM[1],bM[2] ); c2:= coc21( bM[1],bM[3] ); T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,5] ); SetEntrySCTable( T, 2, 4, [1,5] ); SetEntrySCTable( T, 1, 5, [1,6] ); SetEntrySCTable( T, 3, 4, [1,6] ); N:= LieAlgebraByStructureConstants( F, T ); bM:= Basis( M,bM ); for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) ); Append( cx, cz ); # take care of the coboundary... cx[6]:= cx[6] - c1*cx[3]-c2*cx[5]; Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 6, 13 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) ); else imgs:= [ ]; c1:= coc21( bM[1],bM[2] ); c2:= coc21( bM[1],bM[3] ); T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,5] ); SetEntrySCTable( T, 1, 5, [1,6] ); SetEntrySCTable( T, 2, 4, [1,5,1,6] ); SetEntrySCTable( T, 3, 4, [1,6] ); N:= LieAlgebraByStructureConstants( F, T ); bM:= Basis( M,bM ); for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) ); Append( cx, cz ); # take care of the coboundary... cx[6]:= cx[6] - c1*cx[3]-c2*cx[5]; cx[1]:= cx[1]/d; cx[3]:= cx[3]/d; cx[4]:= cx[4]/d^2; cx[5]:= cx[5]/d^2; cx[6]:= cx[6]/d^3; Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 6, 29 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) ); fi; fi; elif type = [ 5, 6 ] then coc21:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[1]; end; bM:= ShallowCopy( BasisVectors( Basis( M ) ) ); # we change the basis bM so that the cocycle is in "normal form": a:= coc21( bM[1],bM[5] ); b:= coc21( bM[2],bM[3] )-coc21( bM[1],bM[4] ); c:= coc21( bM[2],bM[5] ); if not IsZero( c ) then if not Characteristic( F )= 2 then a21:= -a/c; a42:= -( a^2/c^2+b/c )/2; bM[1]:= bM[1]+a21*bM[2]; bM[2]:= bM[2]+a42*bM[4]; bM[3]:= bM[3]+a42*bM[5]; bM[4]:= bM[4]+a21*bM[5]; c:= coc21( bM[2],bM[5] ); imgs:= [ ]; c1:= coc21( bM[1],bM[2] ); c2:= coc21( bM[1],bM[3] ); c3:= coc21( bM[1],bM[4] ); T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); SetEntrySCTable( T, 1, 4, [1,5] ); SetEntrySCTable( T, 2, 3, [1,5] ); SetEntrySCTable( T, 2, 5, [1,6] ); SetEntrySCTable( T, 3, 4, [-1,6] ); N:= LieAlgebraByStructureConstants( F, T ); bM:= Basis( M, bM ); for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) ); Append( cx, cz ); # take care of the coboundary... cx[6]:= cx[6] - c1*cx[3]-c2*cx[4]-c3*cx[5]; # make c= 1; cx[6]:= cx[6]/c; Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 6, 14 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) ); elif Characteristic( F )= 2 and IsFinite(F) then a21:= a/c; a32:= a/c+( b/c )^( Size( F )/2 ); bM[1]:= bM[1]+a21*bM[2]; bM[2]:= bM[2]+a32*bM[3]; bM[3]:= bM[3]+a32*bM[4]+a21*a32*bM[5]; bM[4]:= bM[4]+( a21+a32 )*bM[5]; c:= coc21( bM[2],bM[5] ); imgs:= [ ]; c1:= coc21( bM[1],bM[2] ); c2:= coc21( bM[1],bM[3] ); c3:= coc21( bM[1],bM[4] ); T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); SetEntrySCTable( T, 1, 4, [1,5] ); SetEntrySCTable( T, 2, 3, [1,5] ); SetEntrySCTable( T, 2, 5, [1,6] ); SetEntrySCTable( T, 3, 4, [1,6] ); N:= LieAlgebraByStructureConstants( F, T ); bM:= Basis( M, bM ); for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) ); Append( cx, cz ); # take care of the coboundary... cx[6]:= cx[6] - c1*cx[3]-c2*cx[4]-c3*cx[5]; # make c= 1; cx[6]:= cx[6]/c; Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 6, 14 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) ); else a21:= a/c; a32:= a/c; bM[1]:= bM[1]+a21*bM[2]; bM[2]:= bM[2]+a32*bM[3]; bM[3]:= bM[3]+a32*bM[4]+a21*a32*bM[5]; bM[4]:= bM[4]+( a21+a32 )*bM[5]; c:= coc21( bM[2],bM[5] ); imgs:= [ ]; c1:= coc21( bM[1],bM[2] ); c2:= coc21( bM[1],bM[3] ); c3:= coc21( bM[1],bM[4] ); T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); SetEntrySCTable( T, 1, 4, [1,5] ); SetEntrySCTable( T, 2, 3, [1,5,b/c,6] ); SetEntrySCTable( T, 2, 5, [1,6] ); SetEntrySCTable( T, 3, 4, [1,6] ); N:= LieAlgebraByStructureConstants( F, T ); bM:= Basis( M, bM ); for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) ); Append( cx, cz ); # take care of the coboundary... cx[6]:= cx[6] - c1*cx[3]-c2*cx[4]-c3*cx[5]; # make c= 1; cx[6]:= cx[6]/c; Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 6, 31, b/c ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) ); fi; else # i.e., c = 0 if not Characteristic( F )= 2 then a21:= b/( 2*a ); bM[1]:= bM[1]+a21*bM[2]; bM[4]:= bM[4]+a21*bM[5]; imgs:= [ ]; c1:= coc21( bM[1],bM[2] ); c2:= coc21( bM[1],bM[3] ); c3:= coc21( bM[1],bM[4] ); T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); SetEntrySCTable( T, 1, 4, [1,5] ); SetEntrySCTable( T, 2, 3, [1,5] ); SetEntrySCTable( T, 1, 5, [1,6] ); SetEntrySCTable( T, 2, 4, [1,6] ); N:= LieAlgebraByStructureConstants( F, T ); bM:= Basis( M, bM ); for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) ); Append( cx, cz ); # take care of the coboundary... cx[6]:= cx[6] - c1*cx[3]-c2*cx[4]-c3*cx[5]; # make a= 1: cx[6]:= cx[6]/a; Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 6, 15 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) ); else imgs:= [ ]; c1:= coc21( bM[1],bM[2] ); c2:= coc21( bM[1],bM[3] ); c3:= coc21( bM[1],bM[4] ); if not IsZero( b ) then T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); SetEntrySCTable( T, 1, 4, [1,5] ); SetEntrySCTable( T, 1, 5, [1,6] ); SetEntrySCTable( T, 2, 3, [1,5,1,6] ); SetEntrySCTable( T, 2, 4, [1,6] ); N:= LieAlgebraByStructureConstants( F, T ); tt:= [ 6, 30 ]; else T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); SetEntrySCTable( T, 1, 4, [1,5] ); SetEntrySCTable( T, 1, 5, [1,6] ); SetEntrySCTable( T, 2, 3, [1,5] ); SetEntrySCTable( T, 2, 4, [1,6] ); N:= LieAlgebraByStructureConstants( F, T ); tt:= [ 6, 15 ]; fi; bM:= Basis( M, bM ); b:= b/a; #( we will divide by a, also changing b ) for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) ); Append( cx, cz ); # take care of the coboundary... cx[6]:= cx[6] - c1*cx[3]-c2*cx[4]-c3*cx[5]; # make a= 1: cx[6]:= cx[6]/a; if not IsZero( b ) then # make it 1: cx[1]:= cx[1]/b; cx[2]:= cx[2]/b^2; cx[3]:= cx[3]/b^3; cx[4]:= cx[4]/b^4; cx[5]:= cx[5]/b^5; cx[6]:= cx[6]/b^6; fi; Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= tt, f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) ); fi; fi; elif type = [ 5, 7 ] then coc21:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[1]; end; bM:= ShallowCopy( BasisVectors( Basis( M ) ) ); # we change the basis bM so that the cocycle is in "normal form": a:= coc21( bM[1],bM[5] ); b:= coc21( bM[2],bM[3] ); c:= coc21( bM[2],bM[5] ); if not IsZero( c ) then a21:= -a/c; bM[1]:= bM[1]+a21*bM[2]; if not Characteristic( F )= 2 then a42:= -( b/c )/2; bM[2]:= bM[2]+a42*bM[4]; bM[3]:= bM[3]+a42*bM[5]; c:= coc21( bM[2],bM[5] ); imgs:= [ ]; c1:= coc21( bM[1],bM[2] ); c2:= coc21( bM[1],bM[3] ); c3:= coc21( bM[1],bM[4] ); T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); SetEntrySCTable( T, 1, 4, [1,5] ); SetEntrySCTable( T, 2, 5, [1,6] ); SetEntrySCTable( T, 3, 4, [-1,6] ); N:= LieAlgebraByStructureConstants( F, T ); bM:= Basis( M, bM ); for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) ); Append( cx, cz ); # take care of the coboundary... cx[6]:= cx[6] - c1*cx[3]-c2*cx[4]-c3*cx[5]; # make c= 1; cx[6]:= cx[6]/c; Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 6, 16 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) ); elif Characteristic( F )= 2 and IsFinite(F) then a32:= ( b/c )^( Size( F )/2 ); bM[2]:= bM[2]+a32*bM[3]; bM[3]:= bM[3]+a32*bM[4]; bM[4]:= bM[4]+a32*bM[5]; c:= coc21( bM[2],bM[5] ); imgs:= [ ]; c1:= coc21( bM[1],bM[2] ); c2:= coc21( bM[1],bM[3] ); c3:= coc21( bM[1],bM[4] ); T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); SetEntrySCTable( T, 1, 4, [1,5] ); SetEntrySCTable( T, 2, 5, [1,6] ); SetEntrySCTable( T, 3, 4, [1,6] ); N:= LieAlgebraByStructureConstants( F, T ); bM:= Basis( M, bM ); for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) ); Append( cx, cz ); # take care of the coboundary... cx[6]:= cx[6] - c1*cx[3]-c2*cx[4]-c3*cx[5]; # make c= 1; cx[6]:= cx[6]/c; Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 6, 16 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) ); else c:= coc21( bM[2],bM[5] ); imgs:= [ ]; c1:= coc21( bM[1],bM[2] ); c2:= coc21( bM[1],bM[3] ); c3:= coc21( bM[1],bM[4] ); T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); SetEntrySCTable( T, 1, 4, [1,5] ); SetEntrySCTable( T, 2, 3, [b/c,6] ); SetEntrySCTable( T, 2, 5, [1,6] ); SetEntrySCTable( T, 3, 4, [1,6] ); N:= LieAlgebraByStructureConstants( F, T ); bM:= Basis( M, bM ); for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) ); Append( cx, cz ); # take care of the coboundary... cx[6]:= cx[6] - c1*cx[3]-c2*cx[4]-c3*cx[5]; # make c= 1; cx[6]:= cx[6]/c; Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 6, 32, b/c ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) ); fi; else imgs:= [ ]; c1:= coc21( bM[1],bM[2] ); c2:= coc21( bM[1],bM[3] ); c3:= coc21( bM[1],bM[4] ); if not IsZero( b ) then T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); SetEntrySCTable( T, 1, 4, [1,5] ); SetEntrySCTable( T, 1, 5, [1,6] ); SetEntrySCTable( T, 2, 3, [1,6] ); N:= LieAlgebraByStructureConstants( F, T ); tt:= [ 6, 17 ]; else T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); SetEntrySCTable( T, 1, 4, [1,5] ); SetEntrySCTable( T, 1, 5, [1,6] ); N:= LieAlgebraByStructureConstants( F, T ); tt:= [ 6, 18 ]; fi; bM:= Basis( M, bM ); b:= b/a; #( we will divide by a, also changing b ) for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) ); Append( cx, cz ); # take care of the coboundary... cx[6]:= cx[6] - c1*cx[3]-c2*cx[4]-c3*cx[5]; # make a= 1: cx[6]:= cx[6]/a; if not IsZero( b ) then # make it 1: cx[1]:= cx[1]/b; cx[2]:= cx[2]/b^2; cx[3]:= cx[3]/b^3; cx[4]:= cx[4]/b^4; cx[5]:= cx[5]/b^5; cx[6]:= cx[6]/b^6; fi; Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= tt, f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) ); fi; elif type = [ 5, 8 ] then coc21:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[1]; end; bM:= ShallowCopy( BasisVectors( Basis( M ) ) ); # we change the basis bM so that the cocycle is in "normal form": a:= coc21( bM[1],bM[4] ); b:= coc21( bM[1],bM[5] ); c:= coc21( bM[2],bM[3] ); d:= coc21( bM[2],bM[4] ); e:= coc21( bM[2],bM[5] ); f:= coc21( bM[3],bM[5] ); if not IsZero( f ) then bM[2]:= bM[2]-e/f*bM[3]; bM[4]:= bM[4]-e/f*bM[5]; elif not IsZero( d ) then bM[3]:= bM[3]-e/d*bM[2]; bM[5]:= bM[5]-e/d*bM[4]; else if not Characteristic( F )= 2 then v:= bM[2]; bM[2]:= bM[2]-bM[3]; bM[3]:= bM[3]+v; v:= bM[4]; bM[4]:= bM[4]-bM[5]; bM[5]:= v+bM[5]; fi; fi; d:= coc21( bM[2],bM[4] ); if not IsZero( d ) then # make d= 1: bM[1]:= bM[1]/d; bM[4]:= bM[4]/d; bM[5]:= bM[5]/d; # and a= c= 0 a:= coc21( bM[1],bM[4] ); c:= coc21( bM[2],bM[3] ); bM[1]:= bM[1]-a*bM[2]; bM[3]:= bM[3]-c*bM[4]; b:= coc21( bM[1],bM[5] ); if not IsZero( b ) then bM[3]:= bM[3]/b; bM[5]:= bM[5]/b; f:= coc21( bM[3],bM[5] ); if not IsZero( f ) then if IsFinite( F ) then if not Characteristic( F )= 2 then xi:= PrimitiveElement( F ); ex:= LogFFE( f, xi ); if IsEvenInt( ex ) then f:= One( F ); eps:= xi^( ex/2 ); else f:= xi; eps:= xi^( ( ex-1 )/2 ); fi; else eps:= f^( Size( F )/2 ); f:= One( F ); fi; elif F = Rationals then u:= LieAlgDBHelper.sq_free( f ); eps:= u[1]; f:= u[2]; else eps:= 1; fi; T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,4] ); SetEntrySCTable( T, 1, 3, [1,5] ); SetEntrySCTable( T, 1, 5, [1,6] ); SetEntrySCTable( T, 2, 4, [1,6] ); SetEntrySCTable( T, 3, 5, [f,6] ); N:= LieAlgebraByStructureConstants( F, T ); bM:= Basis( M, bM ); imgs:= [ ]; c1:= coc21( bM[1],bM[2] ); c2:= coc21( bM[1],bM[3] ); for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) ); Append( cx, cz ); # take care of the coboundary... cx[6]:= cx[6] - c1*cx[4]-c2*cx[5]; # isom cx[2]:= cx[2]*eps; cx[3]:= cx[3]*eps^2; cx[4]:= cx[4]*eps; cx[5]:= cx[5]*eps^2; cx[6]:= cx[6]*eps^2; Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 6, 19, f ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) ); else # i.e., f= 0 T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,4] ); SetEntrySCTable( T, 1, 3, [1,5] ); SetEntrySCTable( T, 1, 5, [1,6] ); SetEntrySCTable( T, 2, 4, [1,6] ); N:= LieAlgebraByStructureConstants( F, T ); bM:= Basis( M, bM ); imgs:= [ ]; c1:= coc21( bM[1],bM[2] ); c2:= coc21( bM[1],bM[3] ); for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) ); Append( cx, cz ); # take care of the coboundary... cx[6]:= cx[6] - c1*cx[4]-c2*cx[5]; Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 6, 20 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) ); fi; else # i.e., b = 0 f:= coc21( bM[3],bM[5] ); if not IsZero( f ) then ff:= ShallowCopy( f ); if IsFinite( F ) then if not Characteristic( F )= 2 then xi:= PrimitiveElement( F ); ex:= LogFFE( f, xi ); if IsEvenInt( ex ) then f:= One( F ); eps:= xi^( ex/2 ); else f:= xi; eps:= xi^( ( ex-1 )/2 ); fi; else eps:= f^( Size( F )/2 ); f:= One( F ); fi; elif F = Rationals then u:= LieAlgDBHelper.sq_free( f ); eps:= u[1]; f:= u[2]; else eps:= 1; fi; T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,4] ); SetEntrySCTable( T, 1, 3, [1,5] ); SetEntrySCTable( T, 1, 5, [1,6] ); SetEntrySCTable( T, 2, 4, [1,6] ); SetEntrySCTable( T, 3, 5, [f,6] ); N:= LieAlgebraByStructureConstants( F, T ); bM:= Basis( M, bM ); imgs:= [ ]; c1:= coc21( bM[1],bM[2] ); c2:= coc21( bM[1],bM[3] ); for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) ); Append( cx, cz ); cx[3]:= cx[3] - cx[1]/ff; # take care of the coboundary... cx[6]:= cx[6] - c1*cx[4]-c2*cx[5]; # isom cx[2]:= cx[2]*eps; cx[3]:= cx[3]*eps^2; cx[4]:= cx[4]*eps; cx[5]:= cx[5]*eps^2; cx[6]:= cx[6]*eps^2; Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 6, 19, f ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) ); fi; fi; else # i.e., d= 0 f:= coc21( bM[3],bM[5] ); if not IsZero( f ) then # make f= 1: bM[1]:= bM[1]/f; bM[4]:= bM[4]/f; bM[5]:= bM[5]/f; # and c= 0, a= 1 a:= coc21( bM[1],bM[4] ); c:= coc21( bM[2],bM[3] ); bM[2]:= bM[2]+c*bM[5]; bM[2]:= bM[2]/a; bM[4]:= bM[4]/a; b:= coc21( bM[1],bM[5] ); T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,4] ); SetEntrySCTable( T, 1, 3, [1,5] ); SetEntrySCTable( T, 1, 5, [1,6] ); SetEntrySCTable( T, 2, 4, [1,6] ); N:= LieAlgebraByStructureConstants( F, T ); bM:= Basis( M, bM ); imgs:= [ ]; c1:= coc21( bM[1],bM[2] ); c2:= coc21( bM[1],bM[3] ); for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) ); Append( cx, cz ); # take care of the coboundary... cx[6]:= cx[6] - c1*cx[4]-c2*cx[5]; if not IsZero( b ) then cx[2]:= cx[2]/b^2; cx[3]:= cx[3]/b; cx[4]:= cx[4]/b^2; cx[5]:= cx[5]/b; cx[6]:= cx[6]/b^2; cx[3]:= cx[3]+cx[1]; fi; cx1:= ShallowCopy( cx ); cx1[2]:= cx[3]; cx1[3]:= cx[2]; cx1[4]:= cx[5]; cx1[5]:= cx[4]; Add( imgs, LinearCombination( Basis( N ), cx1 ) ); od; return rec( type:= [ 6, 20 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) ); else # i.d., d= f= 0 and char 2 # make e= 1: bM[1]:= bM[1]/e; bM[4]:= bM[4]/e; bM[5]:= bM[5]/e; # and a= b= c= 0, a:= coc21( bM[1],bM[4] ); b:= coc21( bM[1],bM[5] ); bM[1]:= bM[1]-b*bM[2]-a*bM[3]; bM[3]:= bM[3]-c*bM[5]; T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,4] ); SetEntrySCTable( T, 1, 3, [1,5] ); SetEntrySCTable( T, 2, 5, [1,6] ); SetEntrySCTable( T, 3, 4, [1,6] ); N:= LieAlgebraByStructureConstants( F, T ); bM:= Basis( M, bM ); imgs:= [ ]; c1:= coc21( bM[1],bM[2] ); c2:= coc21( bM[1],bM[3] ); for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) ); Append( cx, cz ); cx[6]:= cx[6]-c1*cx[4]-c2*cx[5]; Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 6, 33 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) ); fi; fi; elif type = [ 5, 9 ] then coc21:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[1]; end; bM:= ShallowCopy( BasisVectors( Basis( M ) ) ); # we change the basis bM so that the cocycle is in "normal form": a:= coc21( bM[1],bM[4] ); b:= coc21( bM[1],bM[5] ); c:= coc21( bM[2],bM[5] ); if not IsZero( c ) then bM[1]:= bM[1]-b/c*bM[2]; bM[4]:= bM[4]-b/c*bM[5]; elif not IsZero( a ) then bM[2]:= bM[2]-b/a*bM[1]; bM[5]:= bM[5]-b/a*bM[4]; else if not Characteristic( F )= 2 then v:= bM[1]; bM[1]:= bM[1]-bM[2]; bM[2]:= v+bM[2]; bM[3]:= 2*bM[3]; v:= bM[4]; bM[4]:= 2*( bM[4]-bM[5] ); bM[5]:= 2*( v+bM[5] ); fi; fi; b:= coc21( bM[1],bM[5] ); if IsZero( b ) then a:= coc21( bM[1],bM[4] ); c:= coc21( bM[2],bM[5] ); aa:= ShallowCopy( a ); if IsFinite( F ) then if not Characteristic( F )= 2 then xi:= PrimitiveElement( F ); ex:= LogFFE( c/a, xi ); if IsEvenInt( ex ) then c:= One( F )*a; eps:= xi^( ex/2 ); else c:= xi*a; eps:= xi^( ( ex-1 )/2 ); fi; else eps:= ( c/a )^( Size( F )/2 ); c:= One( F )*a; fi; elif F = Rationals then u:= LieAlgDBHelper.sq_free( c/a ); eps:= u[1]; c:= u[2]*a; else eps:= 1; fi; T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); SetEntrySCTable( T, 2, 3, [1,5] ); SetEntrySCTable( T, 1, 4, [1,6] ); SetEntrySCTable( T, 2, 5, [c/a,6] ); N:= LieAlgebraByStructureConstants( F, T ); bM:= Basis( M, bM ); imgs:= [ ]; c1:= coc21( bM[1],bM[2] ); c2:= coc21( bM[1],bM[3] ); c3:= coc21( bM[2],bM[3] ); for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) ); Append( cx, cz ); # take care of the coboundary... cx[6]:= cx[6] - c1*cx[3]-c2*cx[4]-c3*cx[5]; cx[6]:= cx[6]/aa; # isom cx[2]:= cx[2]*eps; cx[3]:= cx[3]*eps; cx[4]:= cx[4]*eps; cx[5]:= cx[5]*eps^2; cx[6]:= cx[6]*eps; Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 6, 21, c/a ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) ); else T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); SetEntrySCTable( T, 1, 5, [1,6] ); SetEntrySCTable( T, 2, 3, [1,5] ); SetEntrySCTable( T, 2, 4, [1,6] ); N:= LieAlgebraByStructureConstants( F, T ); bM:= Basis( M, bM ); imgs:= [ ]; c1:= coc21( bM[1],bM[2] ); c2:= coc21( bM[1],bM[3] ); c3:= coc21( bM[2],bM[3] ); for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) ); Append( cx, cz ); # take care of the coboundary... cx[6]:= cx[6] - c1*cx[3]-c2*cx[4]-c3*cx[5]; cx[6]:= cx[6]/b; Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 6, 34 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) ); fi; elif type = [ 4, 1 ] then coc21:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[1]; end; coc22:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[2]; end; bM:= LieAlgDBHelper.skew_symm_NF( M, coc21 ); if IsZero( coc21( bM[3],bM[4] ) ) then # find a linear combination of coc21, coc22 that is nondegenerate: y:= coc22( bM[1],bM[2] ); a:= coc22( bM[1],bM[3] ); b:= coc22( bM[1],bM[4] ); c:= coc22( bM[2],bM[3] ); d:= coc22( bM[2],bM[4] ); e:= coc22( bM[3],bM[4] ); if IsZero( a*d-b*c ) then x:= 1-y; else x:= -y; fi; if IsZero( x ) then store:= coc21; coc21:= coc22; # so now coc21 has become coc22, need to make coc22 equal to # the old coc21 ( otherwise no longer independent ). coc22:= store; r:= 1; t:= 0; else f1:= coc21; f2:= ShallowCopy( coc22 ); coc21:= function( u,v ) return x*f1( u,v )+f2( u,v ); end; r:= 0; t:= 1; fi; bM:= LieAlgDBHelper.skew_symm_NF( M, coc21 ); u:= x; v:= 1; # i.e., we have made a base change in the centre, corresponding # to theta_1 --> u*theta_1 + v*\theta_2 # theta_2 --> r*theta_1 + t*\theta_2 else u:= 1; v:= 0; r:= 0; t:= 1; fi; y:= coc22( bM[1],bM[2] ); if not IsZero( y ) then f2:= coc22; coc22:= function( u,v ) return -y*coc21( u,v )+f2( u,v ); end; r:= -u*y+r; t:= -v*y+t; # a new base change... fi; a:= coc22( bM[1],bM[3] ); if IsZero( a ) then # try to make it not 0: b:= coc22( bM[1],bM[4] ); c:= coc22( bM[2],bM[3] ); d:= coc22( bM[2],bM[4] ); if not IsZero( b ) then bM[3]:= bM[3]+bM[4]; a:= coc22( bM[1],bM[3] ); elif not IsZero( c ) then bM[1]:= bM[1]+bM[2]; a:= coc22( bM[1],bM[3] ); elif not IsZero( d ) then bM[1]:= bM[1]+bM[2]; bM[3]:= bM[3]+bM[4]; a:= coc22( bM[1],bM[3] ); fi; fi; if not IsZero( a ) then # make it 1: bM[1]:= bM[1]/a; bM[4]:= bM[4]/a; # make b,c -> 0 b:= coc22( bM[1],bM[4] ); c:= coc22( bM[2],bM[3] ); bM[2]:= bM[2]-c*bM[1]; bM[4]:= bM[4]-b*bM[3]; e:= coc22( bM[3],bM[4] ); if IsZero( e ) then b:= coc22( bM[2],bM[4] ); bb:= ShallowCopy( b ); if not IsZero( b ) then if IsFinite( F ) then if not Characteristic( F )= 2 then xi:= PrimitiveElement( F ); ex:= LogFFE( b, xi ); if IsEvenInt( ex ) then b:= One( F ); eps:= xi^( ex/2 ); else b:= xi; eps:= xi^( ( ex-1 )/2 ); fi; else eps:= b^( Size( F )/2 ); b:= Zero( F ); fi; elif F = Rationals then uu:= LieAlgDBHelper.sq_free( b ); eps:= uu[1]; b:= uu[2]; else eps:= 1; fi; fi; T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,5] ); SetEntrySCTable( T, 1, 3, [1,6] ); SetEntrySCTable( T, 2, 4, [b,6] ); SetEntrySCTable( T, 3, 4, [1,5] ); N:= LieAlgebraByStructureConstants( F, T ); bM:= Basis( M, bM ); imgs:= [ ]; q:= coc21( bM[1],bM[2] ); # we have to change that back to 1... for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cz1:= ShallowCopy( cz ); cz1[1]:= u/q*cz[1]+v/q*cz[2]; cz1[2]:= r*cz[1]+t*cz[2]; cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) ); Append( cx, cz1 ); if not IsZero( bb ) then # isom cx[1]:= cx[1]/eps; cx[3]:= cx[3]/eps; cx[5]:= cx[5]/eps; cx[6]:= cx[6]/eps^2; if Characteristic( F )= 2 then # isom cx1:= ShallowCopy( cx ); cx1[1]:= cx[1]+cx[4]; cx1[2]:= cx[2]; cx1[3]:= cx[2]+cx[3]; cx1[4]:= cx[1]; cx1[5]:= cx[6]; cx1[6]:= cx[5]+cx[6]; cx:= ShallowCopy( cx1 ); fi; fi; Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 6, 22, b ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) ); else # make it 1: bM[2]:= bM[2]/e; bM[4]:= bM[4]/e; a:= coc22( bM[2],bM[4] ); if not Characteristic( F )= 2 then if not IsZero( a ) and not IsZero( a+1/4 ) then b:= a+1/4; elif IsZero( a ) then b:= One( F ); elif IsZero( a+1/4 ) then b:= Zero( F ); fi; bb:= ShallowCopy( b ); if not IsZero( b ) then if IsFinite( F ) then xi:= PrimitiveElement( F ); ex:= LogFFE( b, xi ); if IsEvenInt( ex ) then b:= One( F ); eps:= xi^( ex/2 ); else b:= xi; eps:= xi^( ( ex-1 )/2 ); fi; elif F = Rationals then uu:= LieAlgDBHelper.sq_free( b ); eps:= uu[1]; b:= uu[2]; else eps:= 1; fi; fi; T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,5] ); SetEntrySCTable( T, 1, 3, [1,6] ); SetEntrySCTable( T, 2, 4, [b,6] ); SetEntrySCTable( T, 3, 4, [1,5] ); N:= LieAlgebraByStructureConstants( F, T ); bM:= Basis( M, bM ); imgs:= [ ]; q:= coc21( bM[1],bM[2] ); # we have to change that back to 1... for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cz1:= ShallowCopy( cz ); cz1[1]:= u/q*cz[1]+v/q*cz[2]; cz1[2]:= r*cz[1]+t*cz[2]; cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) ); Append( cx, cz1 ); cx1:= ShallowCopy( cx ); if not IsZero( a ) and not IsZero( a+1/4 ) then cx1[1]:= ( 1/( 4*a )+1 )*cx[1]; cx1[2]:= cx[3]; cx1[3]:= a*cx[2]+cx[3]/2; cx1[4]:= cx[4]+1/( 2*a )*cx[1]; cx1[5]:= -cx[5]/2+cx[6]; cx1[6]:= ( a+1/4 )*cx[5]; elif IsZero( a ) then cx1[1]:= cx[4]; cx1[2]:= -cx[2]/4; cx1[3]:= cx[2]/4-cx[3]/2; cx1[4]:= 2*cx[1]-cx[4]; cx1[5]:= -cx[5]/2+cx[6]; cx1[6]:= cx[5]/2; elif IsZero( a+1/4 ) then cx1[1]:= -2*cx[1]+cx[4]; cx1[2]:= cx[2]/2-2*cx[3]; cx1[3]:= cx[2]/2-cx[3]; cx1[4]:= -4*cx[1]+cx[4]; cx1[6]:= -cx[5]+2*cx[6]; fi; cx:= ShallowCopy( cx1 ); if not IsZero( bb ) then # isom cx[1]:= cx[1]/eps; cx[3]:= cx[3]/eps; cx[5]:= cx[5]/eps; cx[6]:= cx[6]/eps^2; fi; Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 6, 22, b ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) ); else b:= coc22( bM[2],bM[4] ); bb:= ShallowCopy( b ); if not IsZero( b ) then if IsFinite( F ) then ex:= First( [0..( Size( F )-1 )],i->Trace( F,PrimitiveElement( F )^i )= One( F ) ); xi:= PrimitiveElement( F )^ex; P:= PolynomialRing( F,["y"] ); y:= P.1; if Trace( F, b )= Zero( F ) then root:= Factors( P,y^2+y+b )[1]; eps:= CoefficientsOfUnivariatePolynomial( root )[1]; b:= Zero( F ); else root:= Factors( P,y^2+y+b+xi )[1]; eps:= CoefficientsOfUnivariatePolynomial( root )[1]; b:= xi; fi; else eps:= 1; fi; fi; T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,5] ); SetEntrySCTable( T, 1, 3, [1,6] ); SetEntrySCTable( T, 2, 4, [b,6] ); SetEntrySCTable( T, 3, 4, [1,5,1,6] ); N:= LieAlgebraByStructureConstants( F, T ); bM:= Basis( M, bM ); imgs:= [ ]; q:= coc21( bM[1],bM[2] ); # we have to change that back to 1... for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cz1:= ShallowCopy( cz ); cz1[1]:= u/q*cz[1]+v/q*cz[2]; cz1[2]:= r*cz[1]+t*cz[2]; cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) ); Append( cx, cz1 ); if not IsZero( bb ) then # isom cx1:= ShallowCopy( cx ); cx1[1]:= cx[1]+cx[4]*eps; cx1[2]:= cx[2]; cx1[3]:= cx[2]*eps+cx[3]; cx1[4]:= cx[4]; cx1[5]:= cx[5]; cx1[6]:= cx[5]*eps+cx[6]; cx:= ShallowCopy( cx1 ); fi; Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 6, 35, b ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) ); fi; fi; else # i.e, a= 0, which means that b= c= d= 0 # make e= 1: e:= coc22( bM[3],bM[4] ); f2:= coc22; coc22:= function( u,v ) return f2( u,v )/e; end; r:= r/e; t:= t/e; # subtract coc22 from coc21: f1:= coc21; coc21:= function( u,v ) return f1( u,v )-coc22( u,v ); end; u:= u-r; v:= v-t; if not Characteristic( F )= 2 then T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,5] ); SetEntrySCTable( T, 1, 3, [1,6] ); SetEntrySCTable( T, 2, 4, [1,6] ); SetEntrySCTable( T, 3, 4, [1,5] ); N:= LieAlgebraByStructureConstants( F, T ); bM:= Basis( M, bM ); imgs:= [ ]; for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cz1:= ShallowCopy( cz ); cz1[1]:= u*cz[1]+v*cz[2]; cz1[2]:= r*cz[1]+t*cz[2]; cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) ); Append( cx, cz1 ); # isom cx1:= ShallowCopy( cx ); cx1[1]:= cx[1]-cx[4]; cx1[2]:= cx[2]/4-cx[3]/4; cx1[3]:= cx[2]/4+cx[3]/4; cx1[4]:= -cx[1]-cx[4]; cx1[5]:= cx[5]/2-cx[6]/2; cx1[6]:= cx[5]/2+cx[6]/2; cx:= ShallowCopy( cx1 ); Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 6, 22, One( F ) ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) ); else T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,5] ); SetEntrySCTable( T, 1, 3, [1,6] ); SetEntrySCTable( T, 3, 4, [1,5,1,6] ); N:= LieAlgebraByStructureConstants( F, T ); bM:= Basis( M, bM ); imgs:= [ ]; for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cz1:= ShallowCopy( cz ); cz1[1]:= u*cz[1]+v*cz[2]; cz1[2]:= r*cz[1]+t*cz[2]; cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) ); Append( cx, cz1 ); # isom cx1:= ShallowCopy( cx ); cx1[1]:= cx[4]; cx1[2]:= cx[2]+cx[3]; cx1[3]:= cx[2]; cx1[4]:= cx[1]+cx[4]; cx1[5]:= cx[5]+cx[6]; cx1[6]:= cx[5]; cx:= ShallowCopy( cx1 ); Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 6, 35, Zero( F ) ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) ); fi; fi; elif type = [ 4, 2 ] then coc21:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[1]; end; coc22:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[2]; end; bM:= ShallowCopy( BasisVectors( Basis( M ) ) ); # coc21 can be moved to one of three normal forms a:= coc21( bM[1],bM[3] ); b:= coc21( bM[1],bM[4] ); c:= coc21( bM[2],bM[3] ); d:= coc21( bM[2],bM[4] ); if IsZero( a ) and not IsZero( c ) then # make a not zero bM[1]:= bM[1]+bM[2]; a:= coc21( bM[1],bM[3] ); b:= coc21( bM[1],bM[4] ); fi; if not IsZero( a ) then bM[1]:= bM[1]/a; bM[2]:= bM[2]*a; c:= coc21( bM[2],bM[3] ); bM[2]:= bM[2]-c*bM[1]; b:= coc21( bM[1],bM[4] ); d:= coc21( bM[2],bM[4] ); if not IsZero( d ) then bM[4]:= bM[4]/d-b/d*bM[3]; else bM[4]:= bM[4]-b*bM[3]; fi; else # here also c= 0. m:= LieAlgDBHelper.comp_mat( b, d, One( F ) ); store:= bM[1]; bM[1]:= m[1][1]*bM[1]+m[1][2]*bM[2]; bM[2]:= m[2][1]*store+m[2][2]*bM[2]; fi; s13:= coc21( bM[1],bM[3] ); s24:= coc21( bM[2],bM[4] ); s14:= coc21( bM[1],bM[4] ); r:= 1; t:= 0; u:= 0; v:= 1; if not IsZero( s13 ) then if not IsZero( s24 ) then a:= coc22( bM[1],bM[3] ); u:= -a; v:= 1; f1:= coc22; f5:= coc21; coc22:= function( u,v ) return -a*f5( u,v )+f1( u,v ); end; b:= coc22( bM[1],bM[4] ); c:= coc22( bM[2],bM[3] ); d:= coc22( bM[2],bM[4] ); if IsZero( c ) then if not IsZero( d ) then bM[3]:= bM[3]/d; bM[2]:= bM[2]/d; bM[1]:= bM[1]-b*bM[2]; bM[4]:= bM[4]+b*bM[3]; cf:= 1/coc21( bM[1],bM[3] ); f2:= coc21; coc21:= function( u,v ) return cf*f2( u,v ); end; r:= r*cf; t:= t*cf; cf1:= 1/coc22( bM[2],bM[4] ); f3:= coc22; coc22:= function( u,v ) return cf1*f3( u,v ); end; u:= u*cf1; v:= v*cf1; f4:= coc21; coc21:= function( u,v ) return f4( u,v )-coc22( u,v ); end; r:= r-u; t:= t-v; c1:= coc21( bM[1],bM[2] ); c2:= coc22( bM[1],bM[2] ); T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,5] ); SetEntrySCTable( T, 2, 4, [1,6] ); N:= LieAlgebraByStructureConstants( F, T ); bM:= Basis( M, bM ); imgs:= [ ]; for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cz1:= ShallowCopy( cz ); cz1[1]:= r*cz[1]+t*cz[2]; cz1[2]:= u*cz[1]+v*cz[2]; cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p,x ) ) ) ); Append( cx, cz1 ); # subtract the cocycle cx[5]:= cx[5]-c1*cx[3]; cx[6]:= cx[6]-c2*cx[3]; Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 6, 27 ], f:= LieAlgDBHelper.liealg_hom( K,N,Basis( K ),imgs ) ); else # d= 0 cf:= 1/coc21( bM[1],bM[3] ); f2:= coc21; coc21:= function( u,v ) return cf*f2( u,v ); end; r:= r*cf; t:= t*cf; cf1:= 1/coc22( bM[1],bM[4] ); f3:= coc22; coc22:= function( u,v ) return cf1*f3( u,v ); end; u:= u*cf1; v:= v*cf1; c1:= coc21( bM[1],bM[2] ); c2:= coc22( bM[1],bM[2] ); T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,5] ); SetEntrySCTable( T, 1, 4, [1,6] ); SetEntrySCTable( T, 2, 4, [1,5] ); N:= LieAlgebraByStructureConstants( F, T ); bM:= Basis( M, bM ); imgs:= [ ]; for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cz1:= ShallowCopy( cz ); cz1[1]:= r*cz[1]+t*cz[2]; cz1[2]:= u*cz[1]+v*cz[2]; cx:= ShallowCopy( Coefficients( bM, Image( tau,Image( p,x ) ) ) ); Append( cx, cz1 ); # subtract the cocycle cx[5]:= cx[5]-c1*cx[3]; cx[6]:= cx[6]-c2*cx[3]; Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 6, 23 ], f:= LieAlgDBHelper.liealg_hom( K,N,Basis( K ),imgs ) ); fi; else # i.e., c not 0 if IsZero( d ) then bM[1]:= bM[1]/c; bM[3]:= bM[3]/c; bM[4]:= bM[4]/( c^2 ); cf:= 1/coc21( bM[1],bM[3] ); f2:= coc21; coc21:= function( u,v ) return cf*f2( u,v ); end; r:= r*cf; t:= t*cf; b:= coc22( bM[1],bM[4] ); c1:= coc21( bM[1],bM[2] ); c2:= coc22( bM[1],bM[2] ); bb:= ShallowCopy( b ); if not IsZero( b ) then if IsFinite( F ) then if not Characteristic( F )= 2 then xi:= PrimitiveElement( F ); ex:= LogFFE( b, xi ); if IsEvenInt( ex ) then b:= One( F ); eps:= xi^( ex/2 ); else b:= xi; eps:= xi^( ( ex-1 )/2 ); fi; else eps:= b^( Size( F )/2 ); b:= Zero( F ); fi; elif F = Rationals then uu:= LieAlgDBHelper.sq_free( b ); eps:= uu[1]; b:= uu[2]; else eps:= 1; fi; fi; T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,5] ); SetEntrySCTable( T, 1, 4, [b,6] ); SetEntrySCTable( T, 2, 3, [1,6] ); SetEntrySCTable( T, 2, 4, [1,5] ); N:= LieAlgebraByStructureConstants( F, T ); bM:= Basis( M, bM ); imgs:= [ ]; for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cz1:= ShallowCopy( cz ); cz1[1]:= r*cz[1]+t*cz[2]; cz1[2]:= u*cz[1]+v*cz[2]; cx:= ShallowCopy( Coefficients( bM, Image( tau,Image( p,x ) ) ) ); Append( cx, cz1 ); # subtract the cocycle cx[5]:= cx[5]-c1*cx[3]; cx[6]:= cx[6]-c2*cx[3]; if not IsZero( bb ) then # isom cx[1]:= cx[1]*eps; cx[3]:= cx[3]*eps; cx[4]:= cx[4]*eps^2; cx[5]:= cx[5]*eps^2; cx[6]:= cx[6]*eps; if Characteristic( F )= 2 then # isom cx1:= ShallowCopy( cx ); cx1[1]:= cx[1]; cx1[2]:= cx[1]+cx[2]; cx1[3]:= cx[3]+cx[4]; cx1[4]:= cx[4]; cx1[5]:= cx[5]; cx1[6]:= cx[5]+cx[6]; cx:= ShallowCopy( cx1 ); fi; fi; Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 6, 24, b ], f:= LieAlgDBHelper.liealg_hom( K,N,Basis( K ),imgs ) ); else # i. e., d not 0 if not Characteristic( F )= 2 then bM[1]:= ( 2*c/d )*bM[1]+bM[2]; bM[2]:= d/( 2*c )*bM[2]; bM[4]:= ( 2*c/d )^2*bM[4]-( 2*c/d )*bM[3]; cf:= 1/coc21( bM[1],bM[3] ); f2:= coc21; coc21:= function( u,v ) return cf*f2( u,v ); end; r:= r*cf; t:= t*cf; cf1:= coc22( bM[2],bM[3] ); f3:= coc22; coc22:= function( u,v ) return ( f3( u,v )-c*coc21( u,v ) )/cf1; end; u:= ( u-c*r )/cf1; v:= ( v-c*t )/cf1; b:= coc22( bM[1],bM[4] ); c1:= coc21( bM[1],bM[2] ); c2:= coc22( bM[1],bM[2] ); bb:= ShallowCopy( b ); if not IsZero( b ) then if IsFinite( F ) then xi:= PrimitiveElement( F ); ex:= LogFFE( b, xi ); if IsEvenInt( ex ) then b:= One( F ); eps:= xi^( ex/2 ); else b:= xi; eps:= xi^( ( ex-1 )/2 ); fi; elif F = Rationals then uu:= LieAlgDBHelper.sq_free( b ); eps:= uu[1]; b:= uu[2]; else eps:= 1; fi; fi; T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,5] ); SetEntrySCTable( T, 1, 4, [b,6] ); SetEntrySCTable( T, 2, 3, [1,6] ); SetEntrySCTable( T, 2, 4, [1,5] ); N:= LieAlgebraByStructureConstants( F, T ); bM:= Basis( M, bM ); imgs:= [ ]; for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cz1:= ShallowCopy( cz ); cz1[1]:= r*cz[1]+t*cz[2]; cz1[2]:= u*cz[1]+v*cz[2]; cx:= ShallowCopy( Coefficients( bM, Image( tau,Image( p,x ) ) ) ); Append( cx, cz1 ); # subtract the cocycle cx[5]:= cx[5]-c1*cx[3]; cx[6]:= cx[6]-c2*cx[3]; if not IsZero( bb ) then # isom cx[1]:= cx[1]*eps; cx[3]:= cx[3]*eps; cx[4]:= cx[4]*eps^2; cx[5]:= cx[5]*eps^2; cx[6]:= cx[6]*eps; fi; Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 6, 24, b ], f:= LieAlgDBHelper.liealg_hom( K,N,Basis( K ),imgs ) ); else bM[1]:= c*bM[1]; bM[2]:= d*bM[2]; bM[3]:= c*d*bM[3]; bM[4]:= c^2*bM[4]; cf:= 1/coc21( bM[1],bM[3] ); f2:= coc21; coc21:= function( u,v ) return cf*f2( u,v ); end; r:= r*cf; t:= t*cf; cf1:= coc22( bM[2],bM[3] ); f3:= coc22; coc22:= function( u,v ) return f3( u,v )/cf1; end; u:= u/cf1; v:= v/cf1; b:= coc22( bM[1],bM[4] ); c1:= coc21( bM[1],bM[2] ); c2:= coc22( bM[1],bM[2] ); bb:= ShallowCopy( b ); if not IsZero( b ) then if IsFinite( F ) then ex:= First( [0..( Size( F )-1 )],i->Trace( F,PrimitiveElement( F )^i )= One( F ) ); xi:= PrimitiveElement( F )^ex; P:= PolynomialRing( F,["y"] ); y:= P.1; if Trace( F,b )= Zero( F ) then root:= Factors( P,y^2+y+b )[1]; eps:= CoefficientsOfUnivariatePolynomial( root )[1]; b:= Zero( F ); else root:= Factors( P,y^2+y+b+xi )[1]; eps:= CoefficientsOfUnivariatePolynomial( root )[1]; b:= xi; fi; else eps:= 1; fi; fi; T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,5] ); SetEntrySCTable( T, 1, 4, [b,6] ); SetEntrySCTable( T, 2, 3, [1,6] ); SetEntrySCTable( T, 2, 4, [1,5,1,6] ); N:= LieAlgebraByStructureConstants( F, T ); bM:= Basis( M, bM ); imgs:= [ ]; for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cz1:= ShallowCopy( cz ); cz1[1]:= r*cz[1]+t*cz[2]; cz1[2]:= u*cz[1]+v*cz[2]; cx:= ShallowCopy( Coefficients( bM, Image( tau,Image( p,x ) ) ) ); Append( cx, cz1 ); # subtract the cocycle cx[5]:= cx[5]-c1*cx[3]; cx[6]:= cx[6]-c2*cx[3]; if not IsZero( bb ) then # isom cx1:= ShallowCopy( cx ); cx1[1]:= cx[1]; cx1[2]:= +cx[1]*eps+cx[2]; cx1[3]:= cx[3]+cx[4]*eps; cx1[4]:= cx[4]; cx1[5]:= cx[5]; cx1[6]:= cx[5]*eps+cx[6]; cx:= ShallowCopy( cx1 ); fi; Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 6, 36, b ], f:= LieAlgDBHelper.liealg_hom( K,N,Basis( K ),imgs ) ); fi; fi; fi; else # theta_1 = Delta_13 b:= coc22( bM[1],bM[4] ); d:= coc22( bM[2],bM[4] ); if not IsZero( d ) then bM[1]:= bM[1]-( b/d )*bM[2]; bM[4]:= bM[4]/d; c:= coc22( bM[2],bM[3] ); if not IsZero( c ) then bM[1]:= bM[1]/c; bM[3]:= bM[3]/c; # normalise coc21 cf:= 1/coc21( bM[1],bM[3] ); f1:= coc21; coc21:= function( u,v ) return cf*f1( u,v ); end; r:= r*cf; t:= t*cf; #make a = 0 by subtracting... a:= coc22( bM[1],bM[3] ); u:= u-a*r; v:= v-a*t; f2:= coc22; coc22:= function( u,v ) return -a*coc21( u,v )+f2( u,v ); end; c1:= coc21( bM[1],bM[2] ); c2:= coc22( bM[1],bM[2] ); if not Characteristic( F )= 2 then T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,5] ); SetEntrySCTable( T, 1, 4, [1,6] ); SetEntrySCTable( T, 2, 3, [1,6] ); SetEntrySCTable( T, 2, 4, [1,5] ); N:= LieAlgebraByStructureConstants( F, T ); bM:= Basis( M, bM ); imgs:= [ ]; for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cz1:= ShallowCopy( cz ); cz1[1]:= r*cz[1]+t*cz[2]; cz1[2]:= u*cz[1]+v*cz[2]; cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) ); Append( cx, cz1 ); # subtract the cocycle cx[5]:= cx[5]-c1*cx[3]; cx[6]:= cx[6]-c2*cx[3]; #isom... cx1:= ShallowCopy( cx ); cx1[1]:= cx[1]+cx[2]; cx1[2]:= -cx[1] +cx[2]; cx1[3]:= 2*cx[3]+cx[4]; cx1[4]:= cx[4]; cx1[5]:= 2*cx[5]+2*cx[6]; cx1[6]:= -2*cx[5]+2*cx[6]; Add( imgs, LinearCombination( Basis( N ), cx1 ) ); od; return rec( type:= [ 6, 24, One( F ) ], f:= LieAlgDBHelper.liealg_hom( K,N,Basis( K ),imgs ) ); else T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,5] ); SetEntrySCTable( T, 2, 3, [1,6] ); SetEntrySCTable( T, 2, 4, [1,5,1,6] ); N:= LieAlgebraByStructureConstants( F, T ); bM:= Basis( M, bM ); imgs:= [ ]; for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cz1:= ShallowCopy( cz ); cz1[1]:= r*cz[1]+t*cz[2]; cz1[2]:= u*cz[1]+v*cz[2]; cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) ); Append( cx, cz1 ); # subtract the cocycle cx[5]:= cx[5]-c1*cx[3]; cx[6]:= cx[6]-c2*cx[3]; #isom... cx1:= ShallowCopy( cx ); cx1[1]:= cx[1]+cx[2]; cx1[2]:= cx[1]; cx1[3]:= cx[3]+cx[4]; cx1[4]:= cx[4]; cx1[5]:= cx[5]+cx[6]; cx1[6]:= cx[5]; Add( imgs, LinearCombination( Basis( N ), cx1 ) ); od; return rec( type:= [ 6, 36, Zero( F ) ], f:= LieAlgDBHelper.liealg_hom( K,N,Basis( K ),imgs ) ); fi; else # c= 0 # normalise coc21 cf:= 1/coc21( bM[1],bM[3] ); f1:= coc21; coc21:= function( u,v ) return cf*f1( u,v ); end; r:= r*cf; t:= t*cf; #make a = 0 by subtracting... a:= coc22( bM[1],bM[3] ); u:= u-a*r; v:= v-a*t; f2:= coc22; coc22:= function( u,v ) return -a*coc21( u,v )+f2( u,v ); end; c1:= coc21( bM[1],bM[2] ); c2:= coc22( bM[1],bM[2] ); T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,5] ); SetEntrySCTable( T, 2, 4, [1,6] ); N:= LieAlgebraByStructureConstants( F, T ); bM:= Basis( M, bM ); imgs:= [ ]; for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cz1:= ShallowCopy( cz ); cz1[1]:= r*cz[1]+t*cz[2]; cz1[2]:= u*cz[1]+v*cz[2]; cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) ); Append( cx, cz1 ); # subtract the cocycle cx[5]:= cx[5]-c1*cx[3]; cx[6]:= cx[6]-c2*cx[3]; Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 6, 27 ], f:= LieAlgDBHelper.liealg_hom( K,N,Basis( K ),imgs ) ); fi; else # d= 0 bM[4]:= bM[4]/b; # now b= 1 c:= coc22( bM[2],bM[3] ); if not IsZero( c ) then bM[1]:= bM[1]/c; bM[3]:= bM[3]/c; bM[4]:= c*bM[4]; # normalise coc21 cf:= 1/coc21( bM[1],bM[3] ); f1:= coc21; coc21:= function( u,v ) return cf*f1( u,v ); end; r:= r*cf; t:= t*cf; #make a = 0 by subtracting... a:= coc22( bM[1],bM[3] ); u:= u-a*r; v:= v-a*t; f2:= coc22; coc22:= function( u,v ) return -a*coc21( u,v )+f2( u,v ); end; c1:= coc21( bM[1],bM[2] ); c2:= coc22( bM[1],bM[2] ); T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,5] ); SetEntrySCTable( T, 2, 3, [1,6] ); SetEntrySCTable( T, 2, 4, [1,5] ); N:= LieAlgebraByStructureConstants( F, T ); bM:= Basis( M, bM ); imgs:= [ ]; for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cz1:= ShallowCopy( cz ); cz1[1]:= r*cz[1]+t*cz[2]; cz1[2]:= u*cz[1]+v*cz[2]; cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) ); Append( cx, cz1 ); # subtract the cocycle cx[5]:= cx[5]-c1*cx[3]; cx[6]:= cx[6]-c2*cx[3]; #isom... cx1:= ShallowCopy( cx ); cx1[1]:= -cx[2]; cx1[2]:= -cx[1]; cx1[3]:= -cx[3]; cx1[4]:= -cx[4]; cx1[5]:= cx[6]; cx1[6]:= cx[5]; Add( imgs, LinearCombination( Basis( N ), cx1 ) ); od; return rec( type:= [ 6, 24, Zero( F ) ], f:= LieAlgDBHelper.liealg_hom( K,N,Basis( K ),imgs ) ); else # c= 0 # normalise coc21 cf:= 1/coc21( bM[1],bM[3] ); f1:= coc21; coc21:= function( u,v ) return cf*f1( u,v ); end; r:= r*cf; t:= t*cf; #make a = 0 by subtracting... a:= coc22( bM[1],bM[3] ); u:= u-a*r; v:= v-a*t; f2:= coc22; coc22:= function( u,v ) return -a*coc21( u,v )+f2( u,v ); end; c1:= coc21( bM[1],bM[2] ); c2:= coc22( bM[1],bM[2] ); T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,5] ); SetEntrySCTable( T, 1, 4, [1,6] ); N:= LieAlgebraByStructureConstants( F, T ); bM:= Basis( M, bM ); imgs:= [ ]; for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cz1:= ShallowCopy( cz ); cz1[1]:= r*cz[1]+t*cz[2]; cz1[2]:= u*cz[1]+v*cz[2]; cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) ); Append( cx, cz1 ); # subtract the cocycle cx[5]:= cx[5]-c1*cx[3]; cx[6]:= cx[6]-c2*cx[3]; Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 6, 25 ], f:= LieAlgDBHelper.liealg_hom( K,N,Basis( K ),imgs ) ); fi; fi; fi; else # i.e., theta_1 = Delta_14 c:= coc22( bM[2],bM[3] ); if not IsZero( c ) then bM[1]:= bM[1]/c; bM[3]:= bM[3]/c; a:= coc22( bM[1],bM[3] ); d:= coc22( bM[2],bM[4] ); bM[1]:= bM[1]-a*bM[2]; bM[4]:= bM[4]-d*bM[3]; # normalise coc21 cf:= 1/coc21( bM[1],bM[4] ); f1:= coc21; coc21:= function( u,v ) return cf*f1( u,v ); end; r:= r*cf; t:= t*cf; #make b = 0 by subtracting... b:= coc22( bM[1],bM[4] ); u:= u-b*r; v:= v-b*t; f2:= coc22; coc22:= function( u,v ) return -b*coc21( u,v )+f2( u,v ); end; c1:= coc21( bM[1],bM[2] ); c2:= coc22( bM[1],bM[2] ); T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,5] ); SetEntrySCTable( T, 2, 4, [1,6] ); N:= LieAlgebraByStructureConstants( F, T ); bM:= Basis( M, bM ); imgs:= [ ]; for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cz1:= ShallowCopy( cz ); cz1[1]:= r*cz[1]+t*cz[2]; cz1[2]:= u*cz[1]+v*cz[2]; cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) ); Append( cx, cz1 ); # subtract the cocycle cx[5]:= cx[5]-c1*cx[3]; cx[6]:= cx[6]-c2*cx[3]; #isom... cx1:= ShallowCopy( cx ); cx1[1]:= -cx[2]; cx1[2]:= cx[1]; cx1[3]:= cx[3]; cx1[4]:= -cx[4]; cx1[5]:= -cx[6]; cx1[6]:= -cx[5]; Add( imgs, LinearCombination( Basis( N ), cx1 ) ); od; return rec( type:= [ 6, 27 ], f:= LieAlgDBHelper.liealg_hom( K,N,Basis( K ),imgs ) ); else #c= 0 a:= coc22( bM[1],bM[3] ); bM[2]:= bM[2]/a; bM[3]:= bM[3]/a; d:= coc22( bM[2],bM[4] ); if not IsZero( d ) then bM[4]:= bM[4]/d; fi; # normalise coc21 cf:= 1/coc21( bM[1],bM[4] ); f1:= coc21; coc21:= function( u,v ) return cf*f1( u,v ); end; r:= r*cf; t:= t*cf; #make b = 0 by subtracting... b:= coc22( bM[1],bM[4] ); u:= u-b*r; v:= v-b*t; f2:= coc22; coc22:= function( u,v ) return -b*coc21( u,v )+f2( u,v ); end; c1:= coc21( bM[1],bM[2] ); c2:= coc22( bM[1],bM[2] ); T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,5] ); SetEntrySCTable( T, 1, 4, [1,6] ); if not IsZero( d ) then SetEntrySCTable( T, 2, 4, [1,5] ); fi; N:= LieAlgebraByStructureConstants( F, T ); bM:= Basis( M, bM ); imgs:= [ ]; for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cz1:= ShallowCopy( cz ); cz1[1]:= r*cz[1]+t*cz[2]; cz1[2]:= u*cz[1]+v*cz[2]; cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) ); Append( cx, cz1 ); # subtract the cocycle cx[5]:= cx[5]-c1*cx[3]; cx[6]:= cx[6]-c2*cx[3]; #isom... cx1:= ShallowCopy( cx ); cx1[5]:= cx[6]; cx1[6]:= cx[5]; Add( imgs, LinearCombination( Basis( N ), cx1 ) ); od; if not IsZero( d ) then return rec( type:= [ 6, 23 ], f:= LieAlgDBHelper.liealg_hom( K,N,Basis( K ),imgs ) ); else return rec( type:= [ 6, 25 ], f:= LieAlgDBHelper.liealg_hom( K,N,Basis( K ),imgs ) ); fi; fi; fi; elif type = [ 4, 3 ] then coc21:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[1]; end; coc22:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[2]; end; bM:= ShallowCopy( BasisVectors( Basis( M ) ) ); a:= coc21( bM[1],bM[4] ); b:= coc21( bM[2],bM[3] ); c:= coc22( bM[1],bM[4] ); d:= coc22( bM[2],bM[3] ); m:= [[a,b],[c,d]]^-1; r:= m[1][1]; t:= m[1][2]; u:= m[2][1]; v:= m[2][2]; cc:= coc21; f1:= coc22; coc21:= function( u,v ) return r*cc( u,v )+t*f1( u,v ); end; coc22:= function( x,y ) return u*cc( x,y )+v*f1( x,y ); end; c1:= coc21( bM[1],bM[2] ); c2:= coc22( bM[1],bM[2] ); c3:= coc21( bM[1],bM[3] ); c4:= coc22( bM[1],bM[3] ); T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,3] ); SetEntrySCTable( T, 1, 3, [1,4] ); SetEntrySCTable( T, 1, 4, [1,5] ); SetEntrySCTable( T, 2, 3, [1,6] ); N:= LieAlgebraByStructureConstants( F, T ); bM:= Basis( M, bM ); imgs:= [ ]; for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cz1:= ShallowCopy( cz ); cz1[1]:= r*cz[1]+t*cz[2]; cz1[2]:= u*cz[1]+v*cz[2]; cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) ); Append( cx, cz1 ); # subtract the cocycle cx[5]:= cx[5]-c1*cx[3]-c3*cx[4]; cx[6]:= cx[6]-c2*cx[3]-c4*cx[4]; Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 6, 28 ], f:= LieAlgDBHelper.liealg_hom( K,N,Basis( K ),imgs ) ); elif type = [ 3, 1 ] then coc21:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[1]; end; coc22:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[2]; end; coc23:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[3]; end; bM:= Basis( M ); m:= [ [], [], [] ]; m[1][1]:= coc21( bM[1],bM[2] ); m[1][2]:= coc21( bM[1],bM[3] ); m[1][3]:= coc21( bM[2],bM[3] ); m[2][1]:= coc22( bM[1],bM[2] ); m[2][2]:= coc22( bM[1],bM[3] ); m[2][3]:= coc22( bM[2],bM[3] ); m[3][1]:= coc23( bM[1],bM[2] ); m[3][2]:= coc23( bM[1],bM[3] ); m[3][3]:= coc23( bM[2],bM[3] ); m:= m^-1; T:= EmptySCTable( 6, Zero( F ), "antisymmetric" ); SetEntrySCTable( T, 1, 2, [1,4] ); SetEntrySCTable( T, 1, 3, [1,5] ); SetEntrySCTable( T, 2, 3, [1,6] ); N:= LieAlgebraByStructureConstants( F, T ); imgs:= [ ]; for x in Basis( K ) do cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) ); cz:= cz*TransposedMat( m ); cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) ); Append( cx, cz ); Add( imgs, LinearCombination( Basis( N ), cx ) ); od; return rec( type:= [ 6, 26 ], f:= LieAlgDBHelper.liealg_hom( K,N,Basis( K ),imgs ) ); fi; end; LieAlgDBHelper.nilp_type:= function( L ) local r, t, a, g, exp, c, K, imgs, pp, s, f, F, name; if (not Dimension(L) in [5,6]) or (not IsLieNilpotent(L)) then Error("This function is only implemented for nilpotent Lie algebras of dim 5,6"); fi; if Dimension(L) = 5 then r:= LieAlgDBHelper.class_dim_5( L ); else r:= LieAlgDBHelper.class_dim_6( L ); fi; F:= LeftActingDomain(L); name:= "N"; Append( name, String(r.type[1]) ); Append( name, "_" ); Append( name, String( r.type[2] ) ); Append( name, "( "); Append( name, LieAlgDBField2String( F ) ); if Length(r.type) = 3 then Append( name, ", " ); Append( name, String( r.type[3] ) ); pp:= [r.type[3]]; else pp:= [ ]; fi; Append( name, " )" ); return rec( name:= [ name, pp ], isom:= r.f ); end; InstallMethod( LieAlgebraIdentification, "for a Lie algebra", true, [ IsLieAlgebra ], 0, function( L ) local n, r; n:= Dimension(L); if n in [2,3,4] then if not IsLieSolvable(L) then Error("<L> has to be a solvable Lie algebra"); fi; r:= LieAlgDBHelper.solv_type(L); elif n in [5,6] then if not IsLieNilpotent(L) then Error("<L> has to be a nilpotent Lie algebra"); fi; r:= LieAlgDBHelper.nilp_type(L); else Error("the dimension has to satisfy 2<= dim <= 6"); fi; return rec( name:= r.name[1], parameters:= r.name[2], isomorphism:= r.isom ); end ); InstallMethod( AllSolvableLieAlgebras, "for a finite field and a positive integer", true, [ IsField and IsFinite, IsPosInt ], 0, function( F, dim ) local parlist, R, fam; if not IsFinite(F) then Error("F has to be a finite field"); fi; if not dim in [1,2,3,4] then Error("solvable Lie algebras of this dimension are not included") if dim = 1 then parlist := [[1,1]]; elif dim = 2 then parlist:= [[2,1],[2,2]]; elif dim = 3 then parlist := EnumeratorByFunctions( NewFamily( IsList ), rec( ElementNumber := function( e, x ) local aa; aa:= [ Zero(F), One(F) ]; if Characteristic( F ) > 2 then Add( aa, PrimitiveRoot(F) ); fi; if x in [1,2] then return [3,x]; elif x in [3..Size( F )+2] then return [3,3,Enumerator( F )[x-2]]; elif Characteristic( F ) = 2 and x in [Size( F )+3..Size( F )+4] then return [3,4,aa[x-Size(F)-2]]; elif Characteristic( F ) <> 2 and x in [Size( F )+3..Size( F )+5] then return [3,4,aa[x-Size(F)-2]]; fi; end, NumberElement := function( e, x ) local list1, list2; list1 := [[3,1],[3,2]]; list2 := [[3,4,Zero(F)],[3,4,One(F)]]; if Characteristic( F ) > 2 then Add( list2, [3,4,PrimitiveRoot(F)]); fi; if x in list1 then return Position( list1, x ); elif x{[1,2]} = [3,3] then return Position( Enumerator( F ), x[3] ) + 2; elif x in list2 then return Position( list2, x ) + Size( F )+2; fi; end, Length := function( e ) if Characteristic( F ) = 2 then return Size( F )+4; else return Size( F )+5; fi; end )); elif dim = 4 then parlist := EnumeratorByFunctions( NewFamily( IsList ), rec( ElementNumber := function( e, x ) local a, T, i, ff, q, l1, l2, l3, l4, l5; q:= Size( F ); a := PrimitiveRoot( F ); l1 := [[4,7, Zero(F), One(F)],[4,7, Zero(F), a],[4,14,One(F)], [4,14,a],[4,7,a^0, Zero(F)],[4,7, a, Zero(F)], [4,7, a^2, Zero(F)]]; l2 := [[4,7, Zero(F), One(F)],[4,11, One(F), Zero(F) ], [4,14, One(F)],[4,7, One(F), Zero(F)]]; l3 := [[4,7, Zero(F), One(F)],[4,7, Zero(F), a],[4,14,One(F)], [4,14,a],[4,7, One(F), Zero(F)]]; l4 := [[4,7, Zero(F), One(F)],[4,11, One(F), Zero(F) ], [4,14, One(F)],[4,7, a^0, Zero(F)], [4,7, a, Zero(F)],[4,7, a^2, Zero(F)]]; l5 := [[4,7, Zero(F), One(F)],[4,7, Zero(F), a],[4,14,One(F)], [4,14,a],[4,7, One(F), Zero(F)]]; if x in [1,2] then return [4,x]; elif x in [3..Size( F )+2] then return [4,3,Enumerator( F )[x-2]]; elif x in [Size( F )+3..Size( F )+4] then return [4,x-Size(F)+1]; elif x in [Size(F)+5..Size(F)^2+Size(F)+4] then return [4,6, Enumerator( F )[Int((x-Size( F )-5)/Size( F ))+1], Enumerator( F )[((x-Size(F)-5) mod Size( F ))+1]]; elif x in [Size(F)^2+Size(F)+5..Size(F)^2+2*Size(F)+4] then return [4,7,Enumerator( F )[x-(Size(F)^2+Size(F)+4)], Enumerator( F )[x-(Size(F)^2+Size(F)+4)]]; elif x = Size(F)^2+2*Size(F)+5 then return [4,8]; elif x = Size(F)^2+2*Size(F)+6 then a:= PrimitiveRoot( F ); T:= Indeterminate(F); for i in [1..q-1] do ff:= Factors( T^2-T-a^i ); if Length(ff) = 1 then return [4,9,a^i]; break; fi; od; elif x = Size(F)^2+2*Size(F)+7 then return [4,12]; elif x in [Size(F)^2+2*Size(F)+8..Size(F)^2+3*Size(F)+7] then return [4,13,Enumerator( F )[x-(Size(F)^2+2*Size(F)+7)]]; elif q mod 6 = 1 and x in [Size(F)^2+3*Size(F)+8..Size(F)^2+3*Size(F)+14] then return l1[x-(Size(F)^2+3*Size(F)+7)]; elif q mod 6 = 2 and x in [Size(F)^2+3*Size(F)+8..Size(F)^2+3*Size(F)+11] then return l2[x-(Size(F)^2+3*Size(F)+7)]; elif q mod 6 = 3 and x in [Size(F)^2+3*Size(F)+8..Size(F)^2+3*Size(F)+12] then return l3[x-(Size(F)^2+3*Size(F)+7)]; elif q mod 6 = 4 and x in [Size(F)^2+3*Size(F)+8..Size(F)^2+3*Size(F)+13] then return l4[x-(Size(F)^2+3*Size(F)+7)]; elif q mod 6 = 5 and x in [Size(F)^2+3*Size(F)+8..Size(F)^2+3*Size(F)+12] then return l5[x-(Size(F)^2+3*Size(F)+7)]; fi; end, NumberElement := function( e, x ) local q, a, l1, l2, l3, l4, l5; q:= Size( F ); a := PrimitiveRoot( F ); l1 := [[4,7, Zero(F), One(F)],[4,7, Zero(F), a],[4,14,One(F)], [4,14,a],[4,7,a^0, Zero(F)],[4,7, a, Zero(F)], [4,7, a^2, Zero(F)]]; l2 := [[4,7, Zero(F), One(F)],[4,11, One(F), Zero(F) ], [4,14, One(F)],[4,7, One(F), Zero(F)]]; l3 := [[4,7, Zero(F), One(F)],[4,7, Zero(F), a],[4,14,One(F)], [4,14,a],[4,7, One(F), Zero(F)]]; l4 := [[4,7, Zero(F), One(F)],[4,11, One(F), Zero(F) ], [4,14, One(F)],[4,7, a^0, Zero(F)], [4,7, a, Zero(F)],[4,7, a^2, Zero(F)]]; l5 := [[4,7, Zero(F), One(F)],[4,7, Zero(F), a],[4,14,One(F)], [4,14,a],[4,7, One(F), Zero(F)]]; if x[2] in [1,2] then return x[2]; elif x[2] = 3 then return Position( Enumerator( F ), x[3] )+2; elif x[2] in [4,5] then return Size( F )+x[2]-1; elif x[2] = 6 then return Size( F )+4+(Position( Enumerator( F ), x[3])-1)* Size( F )+Position( Enumerator( F ), x[4] ); elif x[2] = 7 and x[3]=x[4] then return Size(F)^2+Size(F)+4+Position( Enumerator( F ), x[3] ); elif x[2] = 8 then return Size(F)^2+2*Size(F)+5; elif x[2] = 9 then return Size(F)^2+2*Size(F)+6; elif x[2] = 12 then return Size(F)^2+2*Size(F)+7; elif x[2] = 13 then return Size(F)^2+2*Size(F)+7+ Position( Enumerator( F ), x[3] ); elif q mod 6 = 1 and x in l1 then return Size(F)^2+3*Size(F)+7+Position( l1, x ); elif q mod 6 = 2 and x in l2 then return Size(F)^2+3*Size(F)+7+Position( l2, x ); elif q mod 6 = 3 and x in l3 then return Size(F)^2+3*Size(F)+7+Position( l3, x ); elif q mod 6 = 4 and x in l4 then return Size(F)^2+3*Size(F)+7+Position( l4, x ); elif q mod 6 = 5 and x in l5 then return Size(F)^2+3*Size(F)+7+Position( l5, x ); fi; end, Length := function( e ) local q; q := Size( F ); if q mod 6 = 1 then return q^2+3*q+14; elif q mod 6 = 2 then return q^2+3*q+11; elif q mod 6 = 3 then return q^2+3*q+12; elif q mod 6 = 4 then return q^2+3*q+13; elif q mod 6 = 5 then return q^2+3*q+12; fi; end )); else Error( "not yet implemented" ); fi; R := rec( field := F, dim := dim, type := "Solvable", parlist := parlist ); fam := NewFamily( IsLieAlgDBCollection_Solvable ); R := Objectify( NewType( fam, IsLieAlgDBCollection_Solvable ), R ); return R; end ); InstallMethod( Enumerator, "method for LieAlgDBCollections", [ IsLieAlgDBCollection_Solvable ], function( R ) return EnumeratorByFunctions( NewFamily( CategoryCollections( IsLieAlgebra )), rec( ElementNumber := function( e, n ) local par; par := R!.parlist[n]; return SolvableLieAlgebra( R!.field, par ); end, NumberElement := function( e, x ) return Position( R!.parlist, x!.arg ); end, Length := function( x ) return Length( R!.parlist ); end )); end ); [Verzeichnis aufwärts0.158unsichere VerbindungÜbersetzung europäischer Sprachen durch Browser2026-04-30] |
2026-05-26