Let A be a finite-dimensional algebra over a field, and S be its generating set. The smallest positive integer k such that all products of the elements from S of degree not exceeding k span A is called the length of S. The maximal length of S among all sets S that generate A is called the length of A. The length function is an important invariant widely used to study finite-dimensional algebras since 1959. Due to its numerous applications, it was thoroughly investigated as a purely algebraic problem. It is straightforward to check that the length of an associative finite-dimensional algebra is less than its dimension d, and can be equal to d-1 only for commutative algebras.

We show that the length of a d-dimensional non-associative algebra is bounded by 2^{d-2}, and this bound is sharp. The investigations of the length function in the non-associative case is closely related with the combinatorial properties of addition chains, i.e., the sequences of natural numbers in which each term is a sum of two previous terms. These sequences are known since ancient times and are useful in the number of applications. In particular, Fibonacci sequence is a classical example of an addition chain without doublings. We prove that the length of a quadratic algebra of dimension n is bounded by the Fibonacci number F_{n-1}, and this bound is sharp.

The precise length evaluation is a difficult problem even in the associative case. For example, the length of the full matrix algebra is unknown. It was conjectured by Paz in 1984 to be a linear function of the matrix size; this conjecture is still open. We investigate different algebraic properties of the length function for associative and non-associative algebras and estimate length for different classes of non-associative algebras.

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