Growth of algebras

Summary

The growth of a finitely generated associative algebra over a field measures how fast a dimension increases with $n$. This is the dimension of the span of all products of at most $n$ generators. Up to a natural equivalence, this rate does not depend on the chosen generators. Algebras then split into algebras of polynomial growth, of exponential growth and of intermediate growth. The notion parallels the growth rate of a group, and its numerical summary is the Gelfand–Kirillov dimension.