Tag - PI-algebras

Diogo Diniz: Gradings on block-triangular matrix algebras

17th November, 2022

Upper triangular, and more generally, block-triangular matrices, are rather important in linear algebra, and also in ring theory, namely in the theory of PI algebras. The group gradings on such algebras have been studied extensively during the last decades. In 2007 A. Valenti and M. Zaicev conjectured that every grading on these algebras is obtained from an elementary grading on a block-triangular matrix algebra and a division grading on a matrix algebra. In this talk we present recent results on this problem.

Viviane da Silva: On the minimal varieties of PI *-superalgebras and the factorability of their T-ideals

3rd November, 2022

In this talk, we deal with varieties of PI-superalgebras with graded involution of finite basic rank over a field of characteristic zero and we present some recent results concerning the minimality of these varieties (of fixed *-graded exponent) and the factorability of their *-graded polynomial identities.

This video was produced by the Universidade de São Paulo, as part of the LieJor Online Seminar: Algebras, Representations, and Applications.

Allan Berele: Poincaré Series of the Trace Rings of Generic Matrices

22nd September, 2022

We first give some background on the Poincaré series of the algebra of generic matrices and its associated trace ring, and then focus on some recent work, including a conjecture for the denominator of the one variable series for the trace rings. Time permitting we will also say a bit about traces of direct sums.

Irina Sviridova: Hook theorem for identities and its generalizations

15th July, 2022

The hook theorem is one of the key result of the classical theory of polynomial identities of algebras in the case of a field of characteristic zero. This well known result is fundamental for applications of the technique of the classic representation theory of the symmetric group to study identities. It has essential connections with many important facts of PI-theory, and implies many important and interesting consequences. In particular, it is one of the basic results for Kemer's positive solution of the Specht problem. Also it is the base to construct the growth theory for varieties of associative algebras over a field of of characteristic zero.

In the last years, one of the most popular directions of the theory of polynomial identities is to consider algebras with some additional structures (such as gradings, involutions, actions by automorphisms, etc.), and to study identities of such algebras with the additional signature.

We will discuss the versions of the hook theorem for various types of such identities with complementary structures. In particular, we will represent some version of the hook theorem for identities with some types of actions. This result generalizes the analogous results known before, for example, for graded identities or identities with involution. We also will discuss some possible consequences and applications of this theorem.

Efim Zelmanov: Automorphism groups and Lie algebras of vector fields on affine varieties

26th May, 2022

Let V be an affine algebraic variety over a commutative ring K and let A be the K-algebra of regular (polynomial) functions on V.

The group of automorphisms of V, namely Aut(A), is, generally speaking, not linear. We will discuss the following two questions: which properties of linear groups extend to Aut(A), and which properties of finite-dimensional Lie algebras extend to the Lie algebra Der(A) of vector fields on V?

In particular, we will focus on analogues of classical theorems of Selberg, Burnside, and Schur for Aut(A) and an analogue of the Engel theorem for Der(A). In order to achive natural degree of generality and to include some interesting non-commutative cases we prove the theorems for PI-algebras.

Plamen Koshlukov: Gradings on upper triangular matrices

22nd February, 2022

The upper triangular matrix algebras are important in Linear Algebra, and represent a powerful tool in Ring Theory. They also appear in the theory of PI algebras.

In addition to the usual associative product, one can consider the Lie bracket and also the symmetric (Jordan) product on the upper triangular matrices.

We discuss the group gradings on the upper triangular matrices viewed as an associative, Lie and Jordan algebra, respectively. Valenti and Zaicev proved that the associative gradings are, in a sense, given by gradings on the matrix units. Di Vincenzo, Valenti and Koshlukov classified such gradings. Later on, Yukihide and Koshlukov, described the Lie and the Jordan gradings. In this talk we recall some of these results as well as a new development in a rather general setting, obtained by Yukihide and Koshlukov.

Eli Aljadeff: Generic Azumaya G-graded algebras

26th November, 2020

The algebra of generic nxn-matrices and its localizations (e.g. the generic division algebra) has attracted much attention among researchers in different areas as PI theory, Brauer theory and algebraic geometry. We construct the corresponding generic objects for an arbitrary finite dimensional G-graded simple algebra where G is a finite group. In particular we construct a generic G-graded Azumaya algebra which represents all forms in the sense of descent theory of a finite dimensional G-graded simple algebra.

Milen Yakimov: Quantum cluster algebras at roots of unity and discriminants

3rd October, 2020

Cluster algebras were invented by Fomin and Zelevinsky twenty years ago. Since then they have played an important role in a number of settings in combinatorics, geometry, representation theory and topology. We will introduce a notion of root of unity quantum cluster algebras which are PI algebras, and will show that they have large canonical central subalgebras isomorphic to the original cluster algebras. These are far reaching generalizations of the De Concini-Kac-Procesi central subalgebras that appear in the study of the irreducible representations of big quantum groups. We will describe a general theorem computing the discriminants of these algebras. In a special situation it yields a formula for the discriminants of the quantum unipotent cells at roots of unity associated to all symmetrizable Kac-Moody algebras.