As described in [Eic11], the nilpotent quotient algorithm can also be used to determine certain relatively free algebras; that is, algebras that are free within a variety.
‣ KuroshAlgebra( d, n, F ) | ( function ) |
determines a nilpotent table for the largest associative algebra on \(d\) generators over the field \(F\) so that every element \(a\) of the algebra satisfies \(a^n = 0\).
‣ ExpandExponentLaw( T, n ) | ( function ) |
suppose that \(T\) is the nilpotent table of a Kurosh algebra of exponent \(n\) 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.
The package contains a library of Kurosh algebras. This can be accessed as follows.
‣ KuroshAlgebraByLib( d, n, F ) | ( function ) |
At present, the library contains the Kurosh algebras for \(n=2\), \((d,n) = (2,3)\), \((d,n) = (3,3)\) and \(F = ℚ\) or \(|F| \in \{2,3,4\}\), \((d,n) = (4,3)\) and \(F = ℚ\) or \(|F| \in \{2,3,4\}\), \((d,n) = (2,4)\) and \(F = ℚ\) or \(|F| \in \{2,3,4,9\}\), \((d,n) = (2,5)\) and \(F = ℚ\) or \(|F| \in \{2,3,4,5,8,9\}\).
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 ] )
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