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\Chapter{Relatively free Algebras}

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

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\Section{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 $a^n = 0$.

\> 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.

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\Section{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 $|F| \in \{2,3,4,5,8,9\}$.

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\Section{Example of accessing the library of Kurosh algebras}

\beginexample
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 ] )
\endexample


