Obtaining a finite dimensional algebra associated to Lie algebras in QPA

asked 0 years ago

klaaa gravatar image

updated 0 years ago

Let g be a finite dimensional semisimple Lie algebra over a field K (or even more generally a finite dimensional Lie algebra) with basis x1,...,xn. Consider the finite dimensional algebra A(g) (first considered by Smith) given as the quotient of the free polynomial ring K<x1,...,xn,z> in variables x1,...,xn,z with the relations: xizzxi for all i and xixjxjxi[xi,xj]z for all i,j.

Question: Is there an easy way to obtain this algebra for a given Lie algebra g using Lie algebra methods of Sage (such as getting multiplication tables) that is readable for QPA?

Here QPA is a GAP package, so it is technically avaiable in Sage, but I prefer to use GAP in a seperate terminal usually.

Here is an example how the correct output (so that GAP can read it) should look like for the Lie algebra sl2 with basis x=e12,y=e21,h with Lie brackets [x,y]=h,[h,x]=2x,[h,y]=2y:

Q:=Quiver(1,[[1,1,"x"],[1,1,"y"],[1,1,"h"],[1,1,"z"]]);KQ:=PathAlgebra(GF(31),Q);AssignGeneratorVariables(KQ);rel:=[z*x-x*z,z*y-y*z,z*h-h*z,x*y-y*x-h*z,x*h-h*x+2*x*z,y*h-h*y-2*y*z];A:=KQ/rel;Dimension(A);

So the input should be a semisimple Lie algebra and the output a text that is readable in QPA to input the algebra.

Thanks for any help.

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Comments

1

It looks like there are two questions here: (i) getting the algebra; and (ii) presenting it for QPA. Which one concerns you?

Max Alekseyev gravatar imageMax Alekseyev ( 0 years ago )

It is about presenting it for QPA. Getting the algebra inside Sage might be not so useful as GAP and QPA have more commands for dealing with algebras.

klaaa gravatar imageklaaa ( 0 years ago )

Then I do not understand how this aligns with Sage. Why not just stick with GAP for your problem?

Max Alekseyev gravatar imageMax Alekseyev ( 0 years ago )

Sage has the needed data on semisimple Lie algebras I think, for example to get a nice basis with known multiplication table for the bracket.

klaaa gravatar imageklaaa ( 0 years ago )

It might also be useful to have this algebra in Sage, but Im not very experienced with analysing algebras in Sage.

klaaa gravatar imageklaaa ( 0 years ago )