GK-dimension of the Lie algebra of generic $2\times 2$ matrices

TL;DR

This paper provides a new proof for the Gelfand-Kirillov dimension of the Lie algebra generated by generic 2x2 matrices, connecting classical results with the structure of the algebra's commutator ideal.

Contribution

It offers an alternative proof for the GK-dimension of the algebra generated by generic 2x2 matrices using classical invariant theory and module structure analysis.

Findings

Confirmed that GK-dimension of the algebra is 3(m-1)

Connected the commutator ideal to the center of the generated algebra

Provided a new proof method based on classical results

Abstract

Recently Machado and Koshlukov have computed the Gelfand-Kirillov dimension of the relatively free algebra $F_m=F_m(\text{var}(sl_2(K)))$ of rank $m$ in the variety of algebras generated by the three-dimensional simple Lie algebra $sl_2(K)$ over an infinite field $K$ of characteristic different from 2. They have shown that $\text{GKdim}(F_m)=3(m-1)$. The algebra $F_m$ is isomorphic to the Lie algebra generated by $m$ generic $2\times 2$ matrices. Now we give a new proof for $\text{GKdim}(F_m)$ using classical results of Procesi and Razmyslov combined with the observation that the commutator ideal of $F_m$ is a module of the center of the associative algebra generated by $m$ generic traceless $2\times 2$ matrices.