Gelfand–Kirillov dimension

Remarks

If $A$ is finitely generated and $V$ is a finite-dimensional generating subspace with $1 \in V$, then the supremum is attained at $V$: $\operatorname{GKdim} A = \limsup_n \log d_V(n) / \log n$, where $d_V$ is the growth function of $A$. So $\operatorname{GKdim} A$ depends only on the growth of $A$.

Example: let $A = k[x_1, \ldots, x_m]$ be a polynomial ring and $V = k + k x_1 + \cdots + k x_m$. The space $V^n$ is spanned by the monomials of degree at most $n$, so $\dim V^n = \binom{n + m}{m}$, which is a polynomial of degree $m$ in $n$. Hence $\operatorname{GKdim} A = m$.

Example: if $A$ is finite-dimensional, then $\dim V^n \le \dim A$ for all $n$, so $\operatorname{GKdim} A = 0$.

Non-example of finiteness: for the free algebra $A = k\langle x, y \rangle$ and $V = k + kx + ky$, one has $\dim V^n = 2^{n+1} - 1$. Then $\log \dim V^n / \log n \to \infty$, so $\operatorname{GKdim} A = \infty$.

The base of the logarithm does not matter, because only the ratio of two logarithms appears.