Ciao Mimmo,

You asked about restricted enveloping algebras. In fact, some time
ago I have written a small package containing a function for creating
these algebras. It is called `penvelope', and can be found on my
homepage, www.math.uu.nl/people/graaf/penv.html.

An example:

gap> RequirePackage("penvelope");
                  PEnvelope!            true
gap> L:= SimpleLieAlgebra("A",2,GF(3) );
<Lie algebra of dimension 8 over GF(3)>
gap> u:= PEnvelopingAlgebra( L );
PEnvAlgebra( <Lie algebra of dimension 8 over GF(3)> )
gap> g:= GeneratorsOfAlgebra(u);
[ (Z(3)^0)*1, (Z(3)^0)*x1, (Z(3)^0)*x2, (Z(3)^0)*x3, (Z(3)^0)*x4,
  (Z(3)^0)*x5, (Z(3)^0)*x6, (Z(3)^0)*x7, (Z(3)^0)*x8 ]
gap> g[3]^3;
<zero>
gap> g[8]^3;
(Z(3)^0)*x7
gap> g[8]^4*g[5]^2*g[2];
(Z(3)^0)*x1*x4^2+(Z(3))*x1*x4^2*x7+(Z(3)^0)*x1*x4^2*x7^2+(Z(3))*x4+(Z(3)^0)*x
4*x7^2

If you have questions about it, please ask.

All the best,

Willem

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