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        "event_title":"Growth of the length function for finite-dimensional algebras (Combinatorics Seminar)",
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        "combined_title":"Growth of the length function for finite-dimensional algebras (Combinatorics Seminar): Alexander Guterman (Bar-Ilan University)",
        "event_subtitle":"Alexander Guterman (Bar-Ilan University)",
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        "description":"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. \n\nWe 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. \n\nThe 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.   \n\nThe talk is based on a series of joint works with Dmitry Kudryavtsev, Olga Markova and Svetlana Zhilina.",
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