> < ^ Date: Fri, 02 Jun 2000 09:15:45 +0100
> < ^ From: Willem de Graaf <degraaf@math.uu.nl >
> < ^ Subject: Universal Enveloping Algebra

Dear Jan Draisma,

You remarked:

I understand that the category

IsMagmaRingModuloRelations

is not inherited to factor structures, of which I think the universal
enveloping algebra of a Lie algebra is an example. The URL for
finitely presented algebras seems to be under construction.

That is correct; thank you for noting the broken link, we will fix that.

My question: is there nevertheless an analogue of

CoefficientsAndMagmaElements

for u.e.algebras (which should use the PBW-basis)?

The operation `ExtRepOfObj' will give you a list of monomials with
their coefficients. For example:

```gap> L:= SimpleLieAlgebra("A",2,Rationals);;
gap> u:= UniversalEnvelopingAlgebra( L );;
gap> g:= GeneratorsOfAlgebraWithOne( u );
[ [(1)*x.1], [(1)*x.2], [(1)*x.3], [(1)*x.4], [(1)*x.5], [(1)*x.6],
[(1)*x.7], [(1)*x.8] ]
gap> elm:= g[5]*g[4]*g[3];
[(1)*x.1*x.4+(-1)*x.2*x.5+(1)*x.3*x.4*x.5+(-1)*x.3*x.6+(1)*x.8]
gap> ExtRepOfObj( elm );
[ 0, [ [ 1, 1, 4, 1 ], 1, [ 2, 1, 5, 1 ], -1, [ 3, 1, 4, 1, 5, 1 ], 1,
[ 3, 1, 6, 1 ], -1, [ 8, 1 ], 1 ] ]
```

I hope this helps.

Best wishes,

Willem de Graaf

> < [top]