Finite-dimensional algebras and their length

The length of a finite system of generators for a finite-dimensional (not necessarily associative) algebra over a field is the least positive integer k such that the products of length not exceeding k span this algebra as a vector space. The maximum length for the systems of generators of an algebra is called the length of this algebra. Length function is an important invariant widely used to study finite dimensional algebras since 1959. The length evaluation can be a difficult problem, for example, the length of the full matrix algebra is unknown and it was conjectured by Paz in 1984 to be a linear function on the size of matrices. We investigate different algebraic properties of the length function for associative and non-associative algebras and in particular present our recent results on the upper bounds for the length.

The talk is based on the series of joint works with Dmitry Kudryavtsev and Olga Markova.