User:Harmonicmap/draft/Gelfand-Kirillov Dimension

Definition

$\operatorname {dim} _{GK}(R)=\limsup _{n\to \infty }(\log _{n}g_{V}(n))~$

Examples

• Every finitely generated PI-algebra has a finite GK-dimension. ( Berele )
• $\operatorname {dim} _{GK}(k[x_{1},\ldots ,x_{n}])=n~$
• $\operatorname {dim} _{GK}(k\langle x_{1},\ldots ,x_{n}\rangle )=\infty ~$ for $n>1$
• For any real $\alpha \geq 2$ there exists an algebra with two generators and with Gelfand-Kirillov dimension equal to $\alpha$.
• For any real $\alpha \in (1,2)$, there exists a Lie algebra with two generators and with Gelfand-Kirillov dimension equal to $\alpha$. ( Petrogradsky )