Gelfand-Kirillov dimension and Jordan algebras

TL;DR

The paper investigates the Gelfand-Kirillov dimension in associative and Jordan algebras, providing a counterexample that challenges a known inequality in the non-associative Jordan algebra context.

Contribution

It demonstrates that the inequality relating Gelfand-Kirillov dimensions does not always hold for Jordan algebras, unlike in associative algebras, with explicit counterexamples.

Findings

Counterexample for Jordan algebras showing inequality failure

Existence of specific Z_2-grading on Jordan matrix algebras

Challenges previous assumptions in non-associative algebra theory

Abstract

Let A be any associative algebra graded by a finite abelian group G, then if we denote by GKdim_k(A) and GKdim^G_k (A) the Gelfand-Kirillov dimension of its relatively free algebra and its relatively free G-graded algebra in k variables respectively, then GKdim_k(A)\leq GKdim^G_k (A). We show a counterexample of the previous result for Jordan algebras (hence non-associative). In particular, there exists a $Z_2$-grading on $UJ_n$, the Jordan algebra of $n\times n$ upper triangular matrices, n equal to 2 or 3, such that the previous inequality does not hold.