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<div class="ChapSects"><a href="chap4.html#X7E8A43A484CE0BA8">4 <span class="Heading">Path Algebras</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7DFB63A97E67C0A1">4.1 <span class="Heading">Introduction</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X848A225A84A15B1E">4.2 <span class="Heading">Constructing Path Algebras</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7CA1C87B8202C2E9">4.2-1 PathAlgebra</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X85A3A8767E7C11AD">4.3 <span class="Heading">Categories and Properties of Path Algebras</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8255FDF78315E1B3">4.3-1 IsPathAlgebra</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7DE2F2A48492041A">4.4 <span class="Heading">Attributes and Operations for Path Algebras</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7DF51D297E0E6A8B">4.4-1 AssociatedMonomialAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X83FBA499856580B0">4.4-2 QuiverOfPathAlgebra</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7CEF60107CE4616B">4.5 <span class="Heading">Operations on Path Algebra Elements</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X841C00E87E19528E">4.5-1 ElementOfPathAlgebra</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7F0D555379C97A6E">4.6 <span class="Heading">Constructing Quotients of Path Algebras</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X814203E281F3272E">4.6-1 AssignGeneratorVariables</a></span>
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on ideals</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X82ACACDD7D8E9B25">4.7-1 Ideal</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X83E8D45B82356D8E"><code>4.7-6 \in</code></a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X82A1683E7A402E73">4.8 <span class="Heading">Categories and properties of ideals</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7F40193B877D76BC">4.8-1 IsAdmissibleIdeal</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7D2896F27C976231">4.10-1 GroebnerBasisOfIdeal</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7A3CA333873389AD">4.11 <span class="Heading">Categories and Properties of Quotients of Path Algebras</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7ACDD33087F98B88">4.11-1 IsAdmissibleQuotientOfPathAlgebra</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X85F2FFFD78355788">4.12-1 IsValidString</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X85B82A4086AA53D6">4.12-2 StringsLessThan</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X85D0616C82375B5C">4.12-7 BridgeQuiver</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X82DBBB737885C73B">4.12-8 LocalARQuiver</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X86647D317A961513">4.13 <span class="Heading">Attributes and Operations (for Quotients) of Path
Algebras</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X84E3FEF587CB66C3">4.13-1 CartanMatrix</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7F8084A67A3BE874">4.13-2 Centre/Center</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X835A161E8524797A">4.13-3 ComplexityOfAlgebra</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7F6F526C86052150">4.13-5 CoxeterPolynomial</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X818AF5A979F8E539">4.13-11 NakayamaAutomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7C068FE379FBCE18">4.13-12 NakayamaPermutation</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X782EFE477EC0C1C6">4.13-14 RadicalSeriesOfAlgebra</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7BD7DB497917893C">4.14 <span class="Heading">Attributes and Operations on Elements
of Quotients of Path Algebra</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X87495684791B5742">4.14-1 IsElementOfQuotientOfPathAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X80B32F667BF6AFD8">4.14-2 Coefficients</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7F2527747A3D0D6D">4.14-5 ElementOfQuotientOfPathAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8088721187BA8D82">4.14-6 OriginalPathAlgebra</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7B209E0A7DD93C08">4.15 <span class="Heading">Predefined classes and classes
of (quotients of) path algebras</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X820AE0208636B9AA">4.15-1 BrauerConfigurationAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7E63A4F37856A075">4.15-2 CanonicalAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X83498D3D856CC08A">4.15-3 KroneckerAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7C678A08836F77CC">4.15-4 NakayamaAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8369BB398212101C">4.15-5 AdmissibleSequenceGenerator</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X78C675F2836D1B18">4.15-6 PosetAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7F877C5F839A3AA9">4.15-7 PosetOfPosetAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7EF1AE62790D7486">4.15-8 TruncatedPathAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X87E17E137E2B0FC4">4.15-9 IsSpecialBiserialQuiver</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X840BAB827C62AA4C">4.16 <span class="Heading">Opposite algebras</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X85794BE082B632B9">4.16-1 OppositePath</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X842527EC7F90C8C5">4.17 <span class="Heading">Tensor products of path algebras</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X835BBBE18104654A">4.17-1 QuiverProduct</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8455692C7E282C6C">4.17-5 ProjectFromProductQuiver</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7A9026937BDDFA6C">4.17-6 TensorProductOfAlgebras</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7EE8921D787C8377">4.17-7 TensorAlgebrasInclusion</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7B31F4F680135E72">4.17-8 SimpleTensor</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7F0EBF88866A537D">4.17-9 TensorProductDecomposition</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X820195C47E2BE7E0">4.17-10 EnvelopingAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8210F6627AB95229">4.17-11 EnvelopingAlgebraHomomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7AE13B567B5F72FC">4.17-12 IsEnvelopingAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X80D827747ACD76FA">4.17-13 AlgebraAsModuleOverEnvelopingAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7A4C262287D74AB0">4.17-14 DualOfAlgebraAsModuleOverEnvelopingAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8101415F7FFB34CF">4.17-15 TrivialExtensionOfQuiverAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X81D98182822E8911">4.17-16 TrivialExtensionOfQuiverAlgebraProjection</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X79B8B91F8097BB80">4.18 <span class="Heading">Operations on quiver algebras</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7EE5A11883B86971">4.18-1 QuiverAlgebraOfAmodAeA</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7B2D7385829F5EC6">4.18-2 QuiverAlgebraOfeAe</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X8561BCB6835D561F">4.19 <span class="Heading">Finite dimensional algebras over finite fields</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X810A29FB7E6EA24D">4.19-1 AlgebraAsQuiverAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7E9078077EE8B51B">4.19-2 BlocksOfAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X84B423137F933795">4.19-3 IsBasicAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7D30E9C878221B42">4.19-4 IsElementaryAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7B35109B8176FE56">4.19-5 PreprojectiveAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X80C0C6C37C4A2ABD">4.19-6 PrimitiveIdempotents</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7DDBF6F47A2E021C">4.20 <span class="Heading">Algebras</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X821B7B047871B42D">4.20-1 LiftingCompleteSetOfOrthogonalIdempotents</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X83041BDF78BF3CCA">4.20-2 LiftingIdempotent</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X850B9F12806FF76B">4.21 <span class="Heading">Saving and reading quotients of path algebras to and from a
file</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X82A638C77FA75549">4.21-1 ReadAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7E60DDCE848CB739">4.21-2 SaveAlgebra</a></span>
</div></div>
</div>
<h3>4 <span class="Heading">Path Algebras</span></h3>
<p><a id="X7DFB63A97E67C0A1" name="X7DFB63A97E67C0A1"></a></p>
<h4>4.1 <span class="Heading">Introduction</span></h4>
<p>A path algebra is an algebra constructed from a field <span class="Math">F</span> (see Chapter 56 and 57 in the <strong class="pkg">GAP</strong> manual for information about fields) and a quiver <span class="Math">Q</span>. The path algebra <span class="Math">FQ</span> contains all finite linear combinations of paths of <span class="Math">Q</span>. This chapter describes the functions in <strong class="pkg">QPA</strong> that deal with path algebras and quotients of path algebras. Path algebras are algebras, so see Chapter 60: Algebras in the <strong class="pkg">GAP</strong> manual for functionality such as generators, basis functions, and mappings.</p>
<p>The only supported ordering of elements in a path algebra is length left lexicographic ordering. See <a href="chap3.html#X78BBB63B828EB9FB"><span class="RefLink">3.4</span></a> for more information.</p>
<p><a id="X848A225A84A15B1E" name="X848A225A84A15B1E"></a></p>
<h4>4.2 <span class="Heading">Constructing Path Algebras</span></h4>
<p><a id="X7CA1C87B8202C2E9" name="X7CA1C87B8202C2E9"></a></p>
<h5>4.2-1 PathAlgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PathAlgebra</code>( <var class="Arg">F</var>, <var class="Arg">Q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Arguments: <var class="Arg">F</var> -- a field, <var class="Arg">Q</var> -- a quiver.<br /></p>
<p>Returns: the path algebra <var class="Arg">FQ</var> of <var class="Arg">Q</var> over the field <var class="Arg">F</var>.</p>
<p>For construction of fields, see the <strong class="pkg">GAP</strong> documentation. The elements of the path algebra <var class="Arg">FQ</var> will be ordered by left length-lexicographic ordering.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Q := Quiver( ["u","v"] , [ ["u","u","a"], ["u","v","b"], </span>
<span class="GAPprompt">></span> <span class="GAPinput">["v","u","c"], ["v","v","d"] ] );</span>
<quiver with 2 vertices and 4 arrows>
<span class="GAPprompt">gap></span> <span class="GAPinput">F := Rationals;</span>
Rationals
<span class="GAPprompt">gap></span> <span class="GAPinput">FQ := PathAlgebra(F,Q);</span>
<Rationals[<quiver with 2 vertices and 4 arrows>]>
</pre></div>
<p><a id="X85A3A8767E7C11AD" name="X85A3A8767E7C11AD"></a></p>
<h4>4.3 <span class="Heading">Categories and Properties of Path Algebras</span></h4>
<p><a id="X8255FDF78315E1B3" name="X8255FDF78315E1B3"></a></p>
<h5>4.3-1 IsPathAlgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPathAlgebra</code>( <var class="Arg">object</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Arguments: <var class="Arg">object</var> -- any object in <strong class="pkg">GAP</strong>. <br /></p>
<p>Returns: true whenever <var class="Arg">object</var> is a path algebra.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPathAlgebra(FQ);</span>
true
</pre></div>
<p><a id="X7DE2F2A48492041A" name="X7DE2F2A48492041A"></a></p>
<h4>4.4 <span class="Heading">Attributes and Operations for Path Algebras</span></h4>
<p><a id="X7DF51D297E0E6A8B" name="X7DF51D297E0E6A8B"></a></p>
<h5>4.4-1 AssociatedMonomialAlgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AssociatedMonomialAlgebra</code>( <var class="Arg">A</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Arguments: <var class="Arg">A</var> -- a quiver algebra.<br /></p>
<p>Returns: the associated monomial algebra of <var class="Arg">A</var> with respect to the Groebner basis the path algebra is endoved with.</p>
<p><a id="X83FBA499856580B0" name="X83FBA499856580B0"></a></p>
<h5>4.4-2 QuiverOfPathAlgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ QuiverOfPathAlgebra</code>( <var class="Arg">FQ</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Arguments: <var class="Arg">FQ</var> -- a path algebra. <br /></p>
<p>Returns: the quiver from which <var class="Arg">FQ</var> was constructed.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">QuiverOfPathAlgebra(FQ);</span>
<quiver with 2 vertices and 4 arrows>
</pre></div>
<p><a id="X8279084B828E5FD7" name="X8279084B828E5FD7"></a></p>
<h5>4.4-3 OrderingOfAlgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OrderingOfAlgebra</code>( <var class="Arg">FQ</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Arguments: <var class="Arg">FQ</var> -- a path algebra.<br /></p>
<p>Returns: the ordering of the quiver of the path algebra.</p>
<p><em>Note:</em> As of the current version of <strong class="pkg">QPA</strong>, only left length lexicographic ordering is supported.</p>
<p><a id="X86CDD46F7F05ADE9" name="X86CDD46F7F05ADE9"></a></p>
<h5>4.4-4 .</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ .</code>( <var class="Arg">FQ</var>, <var class="Arg">generator</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">FQ</var> -- a path algebra, <var class="Arg">generator</var> -- a vertex or an arrow in the quiver <var class="Arg">Q</var>. <br /></p>
<p>Returns: the <var class="Arg">generator</var> as an element of the path algebra.</p>
<p>Other elements of the path algebra can be constructed as linear combinations of the generators. For further operations on elements, see below.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">FQ.a;</span>
(1)*a
<span class="GAPprompt">gap></span> <span class="GAPinput">FQ.v;</span>
(1)*v
<span class="GAPprompt">gap></span> <span class="GAPinput">elem := 2*FQ.a - 3*FQ.v;</span>
(-3)*v+(2)*a
</pre></div>
<p><a id="X7CEF60107CE4616B" name="X7CEF60107CE4616B"></a></p>
<h4>4.5 <span class="Heading">Operations on Path Algebra Elements</span></h4>
<p><a id="X841C00E87E19528E" name="X841C00E87E19528E"></a></p>
<h5>4.5-1 ElementOfPathAlgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ElementOfPathAlgebra</code>( <var class="Arg">PA</var>, <var class="Arg">path</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">PA</var> -- a path algebra, <var class="Arg">path</var> -- a path in the quiver from which <var class="Arg">PA</var> was constructed.<br /></p>
<p>Returns: The embedding of <var class="Arg">path</var> into the path algebra <var class="Arg">PA</var>, or it returns false if <var class="Arg">path</var> is not an element of the quiver from which <var class="Arg">PA</var> was constructed.</p>
<p><a id="X7E3FAB1F803E26FF" name="X7E3FAB1F803E26FF"></a></p>
<h5>4.5-2 <</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ <</code>( <var class="Arg">a</var>, <var class="Arg">b</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">a</var> and <var class="Arg">b</var> -- two elements of the same path algebra.<br /></p>
<p>Returns: True whenever <var class="Arg">a</var> is smaller than <var class="Arg">b</var>, according to the ordering of the path algebra.</p>
<p><a id="X853C8B0B8665BFBB" name="X853C8B0B8665BFBB"></a></p>
<h5>4.5-3 IsLeftUniform</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsLeftUniform</code>( <var class="Arg">element</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">element</var> -- an element of the path algebra.<br /></p>
<p>Returns: true if each monomial in <var class="Arg">element</var> has the same source vertex, false otherwise.</p>
<p><a id="X7C06BE7483992634" name="X7C06BE7483992634"></a></p>
<h5>4.5-4 IsRightUniform</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsRightUniform</code>( <var class="Arg">element</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">element</var> -- an element of the path algebra.<br /></p>
<p>Returns: true if each monomial in <var class="Arg">element</var> has the same target vertex, false otherwise.</p>
<p><a id="X8735FBE180797557" name="X8735FBE180797557"></a></p>
<h5>4.5-5 IsUniform</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsUniform</code>( <var class="Arg">element</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">element</var> -- an element of the path algebra. <br /></p>
<p>Returns: true whenever <var class="Arg">element</var> is both left and right uniform.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsLeftUniform(elem);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsRightUniform(elem);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsUniform(elem);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">another := FQ.a*FQ.b + FQ.b*FQ.d*FQ.c*FQ.b*FQ.d;</span>
(1)*a*b+(1)*b*d*c*b*d
<span class="GAPprompt">gap></span> <span class="GAPinput">IsLeftUniform(another);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsRightUniform(another);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsUniform(another);</span>
true
</pre></div>
<p><a id="X84C98E687A3A84D8" name="X84C98E687A3A84D8"></a></p>
<h5>4.5-6 LeadingTerm</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeadingTerm</code>( <var class="Arg">element</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Tip</code>( <var class="Arg">element</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">element</var> -- an element of the path algebra. <br /></p>
<p>Returns: the term in <var class="Arg">element</var> whose monomial is largest among those monomials that have nonzero coefficients (known as the "tip" of <var class="Arg">element</var>).</p>
<p><em>Note: </em> The two operations are equivalent.</p>
<p><a id="X80710E9B7D8340BD" name="X80710E9B7D8340BD"></a></p>
<h5>4.5-7 LeadingCoefficient</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeadingCoefficient</code>( <var class="Arg">element</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TipCoefficient</code>( <var class="Arg">element</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">element</var> -- an element of the path algebra. <br /></p>
<p>Returns: the coefficient of the tip of <var class="Arg">element</var> (which is an element of the field).</p>
<p><em>Note: </em> The two operations are equivalent.</p>
<p><a id="X7B3EAE41795598A5" name="X7B3EAE41795598A5"></a></p>
<h5>4.5-8 LeadingMonomial</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeadingMonomial</code>( <var class="Arg">element</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TipMonomial</code>( <var class="Arg">element</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">element</var> -- an element of the path algebra. <br /></p>
<p>Returns: the monomial of the tip of <var class="Arg">element</var> (which is an element of the underlying quiver, not of the path algebra).</p>
<p><em>Note: </em> The two operations are equivalent.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">elem := FQ.a*FQ.b*FQ.c + FQ.b*FQ.d*FQ.c+FQ.d*FQ.d;</span>
(1)*d^2+(1)*a*b*c+(1)*b*d*c
<span class="GAPprompt">gap></span> <span class="GAPinput">LeadingTerm(elem);</span>
(1)*b*d*c
<span class="GAPprompt">gap></span> <span class="GAPinput">LeadingCoefficient(elem);</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">mon := LeadingMonomial(elem);</span>
b*d*c
<span class="GAPprompt">gap></span> <span class="GAPinput">mon in FQ;</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">mon in Q;</span>
true
</pre></div>
<p><a id="X8172B40181E1B7D2" name="X8172B40181E1B7D2"></a></p>
<h5>4.5-9 MakeUniformOnRight</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MakeUniformOnRight</code>( <var class="Arg">elems</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">elems</var> -- a list of elements in a path algebra.<br /></p>
<p>Returns: a list of right uniform elements generated by each element of <var class="Arg">elems</var>.</p>
<p><a id="X796249A682818750" name="X796249A682818750"></a></p>
<h5>4.5-10 MappedExpression</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MappedExpression</code>( <var class="Arg">expr</var>, <var class="Arg">gens1</var>, <var class="Arg">gens2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">expr</var> -- element of a path algebra, <var class="Arg">gens1</var> and <var class="Arg">gens2</var> -- equal-length lists of generators for subalgebras.<br /></p>
<p>Returns: <var class="Arg">expr</var> as an element of the subalgebra generated by <var class="Arg">gens2</var>.</p>
<p>The element <var class="Arg">expr</var> must be in the subalgebra generated by <var class="Arg">gens1</var>. The lists define a mapping of each generator in <var class="Arg">gens1</var> to the corresponding generator in <var class="Arg">gens2</var>. The value returned is the evaluation of the mapping at <var class="Arg">expr</var>.</p>
<p><a id="X7D6DDDF178B0F2D9" name="X7D6DDDF178B0F2D9"></a></p>
<h5>4.5-11 SupportOfQuiverAlgebraElement</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SupportOfQuiverAlgebraElement</code>( <var class="Arg">algebra</var>, <var class="Arg">element</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftSupportOfQuiverAlgebraElement</code>( <var class="Arg">algebra</var>, <var class="Arg">element</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RIghtSupportOfQuiverAlgebraElement</code>( <var class="Arg">algebra</var>, <var class="Arg">element</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">algebra</var> -- a QuiverAlgebra, <var class="Arg">element</var> -- an element of the QuiverAlgebra<br /></p>
<p>Returns: the second version returns a list of the index of the vertices such that the product from the left is non-zero, the third version returns a list of the index of the vertices such that the product from the right is non-zero, and the first version returns the both of the previous lists of indices of vertices.</p>
<p><a id="X849AC0F67A131929" name="X849AC0F67A131929"></a></p>
<h5>4.5-12 VertexPosition</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ VertexPosition</code>( <var class="Arg">element</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">element</var> -- an element of the path algebra on the form <span class="Math">k*v</span>, where <span class="Math">v</span> is a vertex of the underlying quiver and <span class="Math">k</span> is an element of the field.<br /></p>
<p>Returns: the position of the vertex <var class="Arg">v</var> in the list of vertices of the quiver.</p>
<p><a id="X86795D8E7ED73048" name="X86795D8E7ED73048"></a></p>
<h5>4.5-13 RelationsOfAlgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RelationsOfAlgebra</code>( <var class="Arg">A</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Arguments: <var class="Arg">A</var> -- a quiver algebra. <br /></p>
<p>Returns: a set of generators for the ideal in the path algebra <span class="SimpleMath">kQ</span> from which the algebra <var class="Arg">FQ</var> was constructed. If <var class="Arg">A</var> is a path algebra, then an empty list is returned.</p>
<p><a id="X7F0D555379C97A6E" name="X7F0D555379C97A6E"></a></p>
<h4>4.6 <span class="Heading">Constructing Quotients of Path Algebras</span></h4>
<p>In the introduction we saw already one way of constructing a quotient of a path algebra. In addition to this there are at least two other ways of constructing a quotient of a path algebra; one with factoring out an ideal and one where a Groebner basis is attached to the quotient. We discuss these two next.</p>
<p>For several functions in <strong class="pkg">QPA</strong> to function properly one needs to have a Groebner basis attached to the quotient one wants to construct, or equivalently a Groebner basis for the ideal one is factoring out. For this to work the ideal must admit a finite Groebner basis. However, to our knowledge there is no algorithm for determining if an ideal has a finite Groebner basis. On the other hand, it is known that if the factor algebra is finite dimensional, then the ideal has a finite Groebner basis (independent of the ordering of the elements, see <a href="chapBib.html#biBGreen">[Gre00]</a> ). In addition to having a finite Groebner basis, several functions also need that the factoring ideal is admissible. A quotient of a path algebra by an admissible ideal belongs to the category <code class="func">IsAdmissibleQuotientOfPathAlgebra</code> (<a href="chap4.html#X7ACDD33087F98B88"><span class="RefLink">4.11-1</span></a>). The method used in the introduction constructs a quotient in this category. However, there are situations where it is interesting to analyze quotients of path algebras by a non-admissible ideal, so we provide also additional methods.</p>
<p>In the example below, we construct a factor of a path algebra purely with commands in <strong class="pkg">GAP</strong> (cf. also Chapter 60: Algebras in the <strong class="pkg">GAP</strong> manual on how to construct an ideal and a quotient of an algebra). Functions which use Groebner bases like <code class="func">IsFiniteDimensional</code> (<a href="chap4.html#X802DB9FB824B0167"><span class="RefLink">4.11-3</span></a>), <code class="func">Dimension</code> (<a href="chap4.html#X7E6926C6850E7C4E"><span class="RefLink">4.13-6</span></a>), <code class="func">IsSpecialBiserialAlgebra</code> (<a href="chap4.html#X7D7AC1D07A9607DF"><span class="RefLink">4.11-19</span></a>) or a membership test <code class="func">\in</code> (<a href="chap4.html#X83E8D45B82356D8E"><span class="RefLink">4.7-6</span></a>) will work properly (they simply compute the Groebner basis if it is necessary). But some "older" functions (like <code class="func">IndecProjectiveModules</code> (<a href="chap6.html#X85EDCFE27F66093F"><span class="RefLink">6.5-5</span></a>)) can fail or give an incorrect answer! This way of constructing a quotient of a path algebra can be useful e.g. if we know that computing a Groebner basis will take a long time and we do not need this because we want to deal only with modules.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Q := Quiver( 1, [ [1,1,"a"], [1,1,"b"] ] );</span>
<quiver with 1 vertices and 2 arrows>
<span class="GAPprompt">gap></span> <span class="GAPinput">kQ := PathAlgebra(Rationals, Q);</span>
<Rationals[<quiver with 1 vertices and 2 arrows>]>
<span class="GAPprompt">gap></span> <span class="GAPinput">gens := GeneratorsOfAlgebra(kQ);</span>
[ (1)*v1, (1)*a, (1)*b ]
<span class="GAPprompt">gap></span> <span class="GAPinput">a := gens[2];</span>
(1)*a
<span class="GAPprompt">gap></span> <span class="GAPinput">b := gens[3];</span>
(1)*b
<span class="GAPprompt">gap></span> <span class="GAPinput">relations := [a^2,a*b-b*a, b*b];</span>
[ (1)*a^2, (1)*a*b+(-1)*b*a, (1)*b^2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">I := Ideal(kQ,relations);</span>
<two-sided ideal in <Rationals[<quiver with 1 vertices and 2 arrows>]>
, (3 generators)>
<span class="GAPprompt">gap></span> <span class="GAPinput">A := kQ/I;</span>
<Rationals[<quiver with 1 vertices and 2 arrows>]/
<two-sided ideal in <Rationals[<quiver with 1 vertices and 2 arrows>]>
, (3 generators)>>
<span class="GAPprompt">gap></span> <span class="GAPinput">IndecProjectiveModules(A);</span>
Compute a Groebner basis of the ideal you are factoring out with before \
you form the quotient algebra, or you have entered an algebra which \
is not finite dimensional.
fail
</pre></div>
<p>To resolve this matter, we need to compute the Gröbner basis of the ideal generated by the relations in <span class="Math">kQ</span> (yes, it seems like we are going in circles here. Remember, then, that an ideal in the "mathematical sense" may exist independently of the a corresponding <code class="code">Ideal</code> object in <strong class="pkg">GAP</strong>. Also, Gröbner bases in <strong class="pkg">QPA</strong> are handled by the <strong class="pkg">GBNP</strong> package, with constructor methods not dependent on <code class="code">Ideal</code> objects. After creating the ideal <span class="Math">I</span>, we need to perform yet another Gröbner basis operation which just set a respective attribute for <span class="Math">I</span>, see <code class="func">GroebnerBasis</code> (<a href="chap5.html#X7A43611E876B7560"><span class="RefLink">5.1-2</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">gb := GBNPGroebnerBasis(relations,kQ);</span>
[ (1)*a^2, (-1)*a*b+(1)*b*a, (1)*b^2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">I := Ideal(kQ,gb); </span>
<two-sided ideal in <Rationals[<quiver with 1 vertices and 2 arrows>]>,
(3 generators)>
<span class="GAPprompt">gap></span> <span class="GAPinput">GroebnerBasis(I,gb); </span>
<complete two-sided Groebner basis containing 3 elements>
<span class="GAPprompt">gap></span> <span class="GAPinput">IndecProjectiveModules(A); </span>
fail
<span class="GAPprompt">gap></span> <span class="GAPinput">A := kQ/I; </span>
<Rationals[<quiver with 1 vertices and 2 arrows>]/
<two-sided ideal in <Rationals[<quiver with 1 vertices and 2 arrows>]>,
(3 generators)>>
<span class="GAPprompt">gap></span> <span class="GAPinput">IndecProjectiveModules(A);</span>
[ <[ 4 ]> ]
</pre></div>
<p>Note that the instruction <code class="code">A := kQ/relations;</code> used in Introduction is exactly an abbreviation for a sequence of instructions with Groebner basis as in above example.</p> | | |
