Speaker: Antonio Ioppolo, University of Milano Bicocca.
Date: 23 sep 2022, 10h.
Place: Room 407, Bloco H, Campus Gragoatá, UFF.
Abstract: An identity is a symbolic expression involving operations and variables which is always satisfied when the variables are replaced in a given algebraic structure. This talk aims to give the motivation behind the study of these objects, the flavor of some important results and future directions.
I will start with a motivating example leading into the basic notions of the theory of polynomial identities in algebras. Then I will present the celebrated theorem of Amitsur and Levitzk (1950) stating that a certain standard polynomial is an identity for the algebra of square matrices. This initial combinatorial method proved to be limited until Regev introduced in 1972 a growth function measuring the size of identities. This new analytic approach, combined with techniques from ring theory, combinatorics and representation theory of groups, forms one of the current points of view of the theory (see the recent book [1]).
Next I will give an idea of the latest developments in this area. In particular, I will introduce the so-called algebras with trace and we shall see how in this setting also a computational approach turns out to be important for the development of the theory. Along the way, I will present some results concerning the generating identities of important algebras with trace and the characterization of trace algebras with interesting properties of the Regev growth function (see [2, 3, 4, 5]).
References
[1] E. Aljadeff, A. Giambruno, C. Procesi, A. Regev, Rings with polynomial identities and finite dimensional representations of algebras, American Mathematical Society Colloquium Publications 66, Providence R.I., 2020, 630 pp.
[2] A. Giambruno, A. Ioppolo, D. La Mattina, Trace codimensions of algebras and their exponential growth, accepted in Israel J. Math.
[3] A. Ioppolo, P. Koshlukov, D. La Mattina, Trace identities and almost polynomial growth, J. Pure Appl. Algebra 225 (2021), no. 2, Article ID. 106501.
[4] A. Ioppolo, P. Koshlukov, D. La Mattina, Trace identities on diagonal matrix algebras, Proceedings of the INdAM Workshop Polynomial identities in algebras, in Polynomial Identities in Algebras, Springer INdAM Series 44, Editors O. M. Di Vincenzo, A. Giambruno, Springer, 2021.
[5] A. Ioppolo, P. Koshlukov, D. La Mattina, Matrix algebras with degenerate traces and trace identities, J. Algebra 592 (2022), 36–63.