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DTSTAMP:20260417T111735
DTSTART;TZID=America/Detroit:20260508T143000
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SUMMARY:Workshop / Seminar:Growth of the length function for finite-dimensional algebras (Combinatorics Seminar)
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.
UID:145710-21897721@events.umich.edu
URL:https://events.umich.edu/event/145710
CLASS:PUBLIC
STATUS:CONFIRMED
CATEGORIES:Mathematics
LOCATION:East Hall - 3866 East Hall
CONTACT:
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