Identities of unitary finite-dimensional algebras

Abstract

We deal with growth functions of sequences of codimensions of identities in finite-dimensional algebras with unity over a field of characteristic zero. For three-dimensional algebras, it is proved that the codimension sequence grows asymptotically as a n , where a is 1, 2, or 3. For arbitrary finite-dimensional algebras, it is shown that the codimension growth either is polynomial or is not slower than 2 n . We give an example of a finite-dimensional algebra with growth rate an with fractional exponent $ a = \frac{3}{{\sqrt[3]{4}}} + 1 $ .