Tag - Associative rings

Jieru Zhu: Double centralizer properties for the Drinfeld double of the Taft algebras

10th November, 2020

The Drinfeld double of the Taft algebra, Dn, whose ground field contains nth roots of unity, has a known list of 2-dimensional irreducible modules. For each of such module V, we show that there is a well-defined action of the Temperley-Lieb algebra TLk on the k-fold tensor product of V, and this action commutes with that of Dn. When V is self-dual and when k ≤ 2(n−1), we further establish a isomorphism between the centralizer algebra of Dn on V⊗k, and TLk. Our inductive argument uses a rank function on the TL diagrams, which is compatible with the nesting function introduced by Russell-Tymoczko.

Rob Muth: Specht modules and cuspidal ribbon tableaux

27th October, 2020

Representation theory of Khovanov-Lauda-Rouquier (KLR) algebras in affine type A can be studied through the lens of Specht modules, associated with the cellular structure of cyclotomic KLR algebras, or through the lens of cuspidal modules, associated with categorified PBW bases for the quantum group of affine type A. Cuspidal ribbons provide a sort of combinatorial bridge between these approaches. I will describe some recent results on cuspidal ribbon tableaux, and some implications in the world of KLR representation theory, such as bounds on labels of simple factors of Specht modules, and the presentation of cuspidal modules.

Mikhail Zaicev: Polynomial identities: anomalies of codimension growth

23rd October, 2020

We consider numerical invariants associated with polynomial identities of algebras over a field of characteristic zero. Given an algebra A, one can construct a sequence of non-negative integers cn(A), n = 1, 2, ..., called the codimensions of A, which is an important numerical characteristic of identical relations of A. In present talk we discuss asymptotic behaviour of the codimension sequence in different classes of algebras.

Alberto Elduque: Graded-simple algebras and twisted loop algebras

8th October, 2020

The loop algebra construction by Allison, Berman, Faulkner, and Pianzola (2008), describes graded-central-simple algebras with "split centroid" in terms of central simple algebras graded by a quotient of the original grading group. Particular versions of this result were considered by several authors.

In this talk it will be shown how the restriction on the centroid can be removed, at the expense of allowing some deformations (cocycle twists) of the loop algebras.

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.

Chris Bowman: Tautological p-Kazhdan-Lusztig Theory for cyclotomic Hecke algebras

15th September, 2020

We discuss a new explicit isomorphism between (truncations of) quiver Hecke algebras and Elias-Williamson’s diagrammatic endomorphism algebras of Bott-Samelson bimodules. This allows us to deduce that the decomposition numbers of these algebras (including as examples the symmetric groups and generalised blob algebras) are tautologically equal to the associated p-Kazhdan-Lusztig polynomials, provided that the characteristic is greater than the Coxeter number. This allows us to give an elementary and explicit proof of the main theorem of Riche-Williamson’s recent monograph and extend their categorical equivalence to cyclotomic Hecke algebras, thus solving Libedinsky-Plaza’s categorical blob conjecture.

Andrea Solotar: A cup-cap duality in Koszul calculus

21st August, 2020

In this talk I will introduce a cup-cap duality in the Koszul calculus of N-homogeneous algebras. As an application of this duality, it follows that the graded symmetry of the Koszul cap product is a consequence of the graded commutativity of the Koszul cup product. I will also comment on a conceptual approach to this problem that may lead to a proof of the graded commutativity, based on derived categories in the framework of DG-algebras and DG-bimodules.

Mikhail Chebotar: On polynomials over nil rings

31st July, 2020

will discuss some recent results related to polynomials over nil rings. In particular, we will present solutions of problems posed by Beidar, Puczylowski and Smoktunowicz, Greenfeld, Smoktunowicz and Ziembowski.

Matej Brešar: Images of non-commutative polynomials

23rd July, 2020

Let f = f(X1, ..., Xm) be a non-commutative polynomial with coefficients in a field F. We will discuss various questions concerning the image of f in an F-algebra A, which is defined to be the set f(A) = {f(a1, ..., am) | a1, ..., am ∈ A}. A special emphasis will be on the Waring type problem, asking about the existence of a positive integer N (independent of f, provided that f is neither an identity nor a central polynomial of A) such that every element, or at least every commutator, in A is a linear combination of N elements from f(A). We are primarily interested in the case where A = Mn(F), but some other algebras will also be considered.