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Dear GAP-forum,

I'd like to work with finite-dimensional subspaces of a universal

enveloping algebra, or of a polynomial ring. For example:

U:=UniversalEnvelopingAlgebra(SimpleLieAlgebra("A",1,Rationals));

x:=GeneratorsOfAlgebraWithOne(U);

V:=Subspace(U,[x[1],x[2]]);

After these commands, I'd like to be able to ask for the dimension of

V, and for a basis. But these requests cause errors. Apparently, the

Poincare-Birkhoff-Witt theorem is only used operationally, to

compute a normal form, but GAP does not `know' that PBW-monomials are

independent.

Probably, these problems are not due to the specific setting of UEAs:

they also occur in:

R:=PolynomialRing(Rationals,1);

x:=GeneratorsOfAlgebraWithOne(R);

V:=Subspace(R,[x,x^2]);

Dimension(V);

The problems seem to boil down to GAP's inability to work with infinite

bases. Does anyone have a suggestion how to get around these problems

in an elegant way? Or will they be solved in the next release?

Beste regards,

Jan

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