# Obtaining a finite dimensional algebra associated to Lie algebras in QPA

Let g be a finite dimensional semisimple Lie algebra over a field K (or even more generally a finite dimensional Lie algebra) with basis $x_1,...,x_n$. Consider the finite dimensional algebra A(g) (first considered by Smith) given as the quotient of the free polynomial ring $K<x_1,...,x_n,z>$ in variables $x_1,...,x_n,z$ with the relations: $x_i z-z x_i$ for all $i$ and $x_i x_j -x_j x_i - [x_i, x_j] z$ 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 $sl_2$ with basis $x=e_{12}, y=e_{21}, h$ 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.

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

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.

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

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.

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