Combinatorial methods to investigate the length of algebras

Link to the recording

An important combinatorial characteristic of finite-dimensional algebras is their length.  The length of a finite system of generators for a finite-dimensional
algebra over a field is the least positive integer $k$, such that the products of generators of length not exceeding $k$ span this algebra as a vector space.
The maximum length over all systems of generators of an algebra is called the length of this algebra.

Length function is an important invariant widely utilized to investigate finite dimensional algebras since 1959. 

It is actively used in various applications, for example, in mechanics of isotropic continua or in quantum sciences.
However, even the length of the matrix algebra is unknown. It was conjectured by Azaria Paz in 1984 that the length of the full matrix algebra is a linear function
on the size of matrices, and this conjecture is still open. It turns out that the investigation of the length function requires a number of tools from algebra and combinatorics. In particular, the length of nonassociative algebras is closely related with the integer sequences known in combinatorics as addition chains. These are the sequences where each term is a sum of two previous terms. The most well-known example of an addition chain is the Fibonacci sequence.  In the talk we obtain optimal upper bounds for the length of several classes of non-associative algebras using  addition chains and prove some other results on the length of algebras.

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