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<html><head><title>[ModIsom] 5 Nilpotent Quotients</title></head> <body text="#000000" bgcolor="#ffffff"> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP004.htm">Previous</a>] [<a href ="CHAP006.htm">Nex <h1>5 Nilpotent Quotients</h1><p> <P> <H3>Sections</H3> <oL> <li> <A HREF="CHAP005.htm#SECT001">Computing nilpotent quotients</a> <li> <A HREF="CHAP005.htm#SECT002">Example of nilpotent quotient computation</a> </ol><p> <p> This chapter contains a description of the nilpotent quotient algorithm for associative finitely presented algebras. We refer to <a href="biblio.htm#Eic11"><cite>Eic background on the algorithms used in this Chapter. <p> <p> <h2><a name="SECT001">5.1 Computing nilpotent quotients</a></h2> <p><p> Let <i>A</i> be a finitely presented algebra in the GAP sense. The following function can be used to determine the class-<i>c</i> nilpotent quotient of <i>A</i>. The quotient is described by a nilpotent table. <p> <a name = "SSEC001.1"></a> <li><code>NilpotentQuotientOfFpAlgebra( A, c ) F</code> <p> The output of this function is a nilpotent table with some additional entries. In particular, there is the additional entry <i>img</i> which describes the images of the generators of <i>A</i> in the nilpotent table. <p> <p> <h2><a name="SECT002">5.2 Example of nilpotent quotient computation</a></h2> <p><p> <pre> gap> F := FreeAssociativeAlgebra(GF(2), 2);; gap> g := GeneratorsOfAlgebra(F);; gap> r := [g[1]^2, g[2]^2];; gap> A := F/r;; gap> NilpotentQuotientOfFpAlgebra(A,3); rec( def := [ 1, 2 ], dim := 8, fld := GF(2), img := [ <a GF2 vector of length 8>, <a GF2 vector of length 8> ], mat := [ [ ], [ ] ], rnk := 2, tab := [ [<a GF2 vector of length 8>, <a GF2 vector of length 8>, [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ], [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ] ], [ <a GF2 vector of length 8>, <a GF2 vector of length 8>, [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ] ], [ [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ] ], [ [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ] ]], wds := [ ,, [ 2, 1 ], [ 1, 2 ], [ 1, 3 ], [ 2, 4 ], [ 2, 5 ], [ 1, 6 ] ], wgs := [ 1, 1, 2, 2, 3, 3, 4, 4 ] ) </pre> <p> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP004.htm">Previous</a>] [<a href ="CHAP006.htm">Nex <P> <address>ModIsom manual<br>September 2024 </address></body></html>
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