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<html><head><title>[ModIsom] 6 Relatively free Algebras</title></head>
<body text="#000000" bgcolor="#ffffff"> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP005.htm">Previous</a>] [<a href = "theindex.htm">In <h1>6 Relatively free Algebras</h1><p> <P> <H3>Sections</H3> <oL> <li> <A HREF="CHAP006.htm#SECT001">Computing Kurosh Algebras</a> <li> <A HREF="CHAP006.htm#SECT002">A Library of Kurosh Algebras</a> <li> <A HREF="CHAP006.htm#SECT003">Example of accessing the library of Kurosh algebras</a> </ol><p> <p> As described in <a href="biblio.htm#Eic11"><cite>Eic11</cite></a>, the nilpotent quotient algori to determine certain relatively free algebras; that is, algebras that are free within a variety. <p> <p> <h2><a name="SECT001">6.1 Computing Kurosh Algebras</a></h2> <p><p> <a name = "SSEC001.1"></a> <li><code>KuroshAlgebra( d, n, F ) F</code> <p> determines a nilpotent table for the largest associative algebra on <i>d</i> generators over the field <i>F</i> so that every element <i>a</i> of the algebra satisfies <i>a</i><sup><i>n</i></sup> = 0. <p> <a name = "SSEC001.2"></a> <li><code>ExpandExponentLaw( T, n )</code> <p> suppose that <i>T</i> is the nilpotent table of a Kurosh algebra of exponent <i>n</i> defined over a prime field. This function determines polynomials describing the corresponding Kurosh algebras over all fields with the same characteristic as the prime field. <p> <p> <h2><a name="SECT002">6.2 A Library of Kurosh Algebras</a></h2> <p><p> The package contains a library of Kurosh algebras. This can be accessed as follows. <p> <a name = "SSEC002.1"></a> <li><code>KuroshAlgebraByLib(d, n, F) F</code> <p> At current, the library contains the Kurosh algebras for <i>n</i>=2, (<i>d</i>,<i>n</i>) = (2,3), (<i>d</i>,<i>n</i>) = (3,3) and <i>F</i> = <b>Q</b> or |<i>F</i>| ∈ {2,3,4}, (<i>d</i>,<i>n</i>) = (4,3) and <i>F</i> = <b>Q</b> or |<i>F</i>| ∈ {2,3,4}, (<i>d</i>,<i>n</i>) = (2,4) and <i>F</i> = <b>Q</b> or |<i>F</i>| ∈ {2,3,4,9}, (<i>d</i>,<i>n</i>) = (2,5) and <i>F</i> = <b>Q</b> or |<i>F</i>| ∈ {2,3,4,5,8,9}. <p> <p> <h2><a name="SECT003">6.3 Example of accessing the library of Kurosh algebras</a></h2> <p><p> <pre> gap> KuroshAlgebra(2,2,Rationals); ... some printout .. rec( bas := [ [ 1, 0, 0, 0 ], [ 0, 1, 1, 0 ], [ 0, 0, 0, 1 ], [ 0, 1, 0, 0 ] ] , com := false, dim := 3, fld := Rationals, rnk := 2, tab := [ [ [ 0, 0, 0 ], [ 0, 0, -1 ], [ 0, 0, 0 ] ], [ [ 0, 0, 1 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ] ], wds := [ ,, [ 2, 1 ] ], wgs := [ 1, 1, 2 ] ) </pre> <p> <p> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP005.htm">Previous</a>] [<a href = "theindex.htm">In <P> <address>ModIsom manual<br>September 2024 </address></body></html> ¤ Dauer der Verarbeitung: 0.12 Sekunden (vorverarbeitet) ¤*© Formatika GbR, Deutschland |
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2026-03-28