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 $$\alpha\ge 2$$ 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 )