- Computing Kurosh Algebras
- A Library of Kurosh Algebras
- Example of accessing the library of Kurosh algebras

As described in Eic11, the nilpotent quotient algorithm also allows to determine certain relatively free algebras; that is, algebras that are free within a variety.

`KuroshAlgebra( d, n, F ) F`

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 )`

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) F`

At current, the library contains the Kurosh algebras for
*n*=2,
(*d*,*n*) = (2,3),
(*d*,*n*) = (3,3) and *F* = **Q** or |*F*| ∈ {2,3,4},
(*d*,*n*) = (4,3) and *F* = **Q** or |*F*| ∈ {2,3,4},
(*d*,*n*) = (2,4) and *F* = **Q** or |*F*| ∈ {2,3,4,9},
(*d*,*n*) = (2,5) and *F* = **Q** or |*F*| ∈ {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 ] )

ModIsom manual

January 2020