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Minimal varieties of polynomial growth

Abstract

In this research project we plan to study questions related to the polynomial identities (and their generalizations) satisfied by an algebra A over a field. In general in characteristic zero the study of the identities is essentially based on the representation theory of the symmetric group and the general linear group and it involves the combinatorics of partitions, tableaux and Young diagrams. The relation between the T-ideals and the varieties of algebras is well understood, and it is useful to translate notions from one to another category. An important characteristic of a T-ideal is its sequence of codimensions. A celebrated result of Regev states that this sequence grows at most exponentially for any associative PI algebra (while the codimensions of the free associative algebra grow as a factorial). Later on Giambruno and Zaicev proved that actually the codimension sequence of a PI algebra grows as the exponent of an integer. Thus they define the exponent of a PI algebra as the limit of the n-th root of the n-th codimension. Their main result is that this exponent always exists (as long as the algebra satisfies nontrivial PI), and is always an integer. There are descriptions, due to Kemer, of the PI algebras with polynomial growth of the codimensions. It follows from Kemer's results that there is no associative algebra with intermediate growth of the codimensios: it is either polynomially bounded or it is exponential. In this project we aim at classifying all minimal varieties with biquadratic codimensions growth. Moreover we will be studying analogous questions concerning group graded identities and also concerning identities with involution. The notion of codimension in these cases is well defined, and a lot of research has been done. We aim at describing all minimal varieties of polynomial growth in these two situations as well. (AU)

Scientific publications
(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)
LA MATTINA, DANIELA. On algebras of polynomial codimension growth. SAO PAULO JOURNAL OF MATHEMATICAL SCIENCES, v. 10, n. 2, p. 312-320, DEC 2016. Web of Science Citations: 0.
KOSHLUKOV, PLAMEN; LA MATTINA, DANIELA. Graded algebras with polynomial growth of their codimensions. Journal of Algebra, v. 434, p. 115-137, JUL 15 2015. Web of Science Citations: 9.

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