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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:=[zx-xz,zy-yz,zh-hz,xy-yx-hz,xh-hx+2xz,yh-hy-2y*z];A:=KQ/rel;Dimension(A);

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);

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);