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6 Relatively free Algebras

Sections

  1. Computing Kurosh Algebras
  2. A Library of Kurosh Algebras
  3. 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.

6.1 Computing Kurosh Algebras

  • 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 an = 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.

    6.2 A Library of Kurosh Algebras

    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| 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}.

    6.3 Example of accessing the library of Kurosh algebras

    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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    ModIsom manual
    January 2016