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# 6 Relatively free Algebras

### Sections

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| ∈ {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}.

## 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 2020