Polynomial Identities in Algebras

This volume contains the talks given at the INDAM workshop entitled "Polynomial identites in algebras", held in Rome in September 2019. The purpose of the book is to present the current state of the art in the theory of PI-algebras. The review of the classical results in the last few years has pointed out new perspectives for the development of the theory. In particular, the contributions emphasize on the computational and combinatorial aspects of the theory, its connection with invariant theory, representation theory, growth problems. It is addressed to researchers in the field.

Y. Bahturin, Some thoughts on the current state on the theory of identical relations in Lie algebras.- D. Bessades et al., Minimal degree of identities of matrix algebras with additional structures.- F. S. Benanti and A. Valenti, On the asymptotics of Capelli polynomials.- A. Berele, Regev’s conjecture  for algebras with Hopf actions.- G. Blachar et al., l-Weak identities and central polynomials for matrices.- L. Carini, Computing multiplicities in the sign trace cocharacters of M12 (F).- V. De Filippis et al., b-Generalized skew derivations on multilinear polynomials in prime rings.- T. C. de Mello and F. Y. Yasumura, Relatively free algebras of finite rank.- V. Drensky, Graded algebras, algebraic functions, planar trees and elliptic integrals.- A. Giambruno and M. Zaicev, Central polynomials of algebras and their growth.- A. Ioppolo et al., Trace identities on diagonal matrix algebras.- P. Koshlukov and D. Macedo, Codimension growth for weak polynomial identities, and non-integrality of the PI exponent.- D. La Mattina, On codimensions of algebras with involution.- R. La Scala and D. Piontkovski, Context-free languages and associative algebras with algebraic Hilbert series.- S. P. Mishchenko and A. Valenti, On almost nilpotent varieties of linear algebras.- V. Nardozza, (d,e)-Differential identities of UTm(F).- V. Petrogradsky, Identities in group rings, enveloping algebras and Poisson algebras.- C. Polcino Milies, Notes on the history of identities on group (and loop) algebras.- C. Procesi, Cayley Hamilton algebras.- C. Rizzo, Growth of differential identities.- S. Siciliano, Derived lengths of symmetric Poisson algebras.- E. Spinelli, Group and polynomial identities in group rings.

Onofrio Mario Di Vincenzo has been a full professor of Algebra since 2000. He currently works at the University of Basilicata where he has been the head of the department for the last 8 years. His research interests focus on the polynomial identities of algebras, in particular in the presence of gradings or involutions acting on them.

Antonio Giambruno received his PhD in Mathematics from the University of Chicago in 1977 under the supervision of Israel N. Herstein. He became a professor at the department of mathematics of the University of Palermo in 1980. He has been a visiting researcher at numerous institutions, including University of Southern California, MSRI, University of Alberta, and University of San Paulo. His research focuses on noncommutative algebra, especially on polynomial identities and their connection with representation theory and asymptotic methods.