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<html><head><title>[ModIsom] 5 Nilpotent Quotients</title></head>
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<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>Eic11</cite></a> for
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>
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<address>ModIsom manual<br>September 2024
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