Tag - Associative rings

Alice Dell’Arciprete: Scopes equivalence for blocks of Ariki-Koike algebras

19th July, 2022

We consider representations of the Ariki-Koike algebra, a q-deformation of the group algebra of the complex reflection group Cr ≀ Sn. The representations of this algebra are naturally indexed by multipartitions of n. We examine blocks of the Ariki-Koike algebra, in an attempt to generalise the combinatorial representation theory of the Iwahori-Hecke algebra. In particular, we prove a sufficient condition such that restriction of modules leads to a natural correspondence between the multipartitions of n whose Specht modules belong to a block B and those of n-δi(B) whose Specht modules belong to the block B', obtained from B applying a Scopes equivalence.

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.

Ivan Ezequiel Angiono: Finite-dimensional pointed Hopf algebras over central extensions of abelian groups

7th July, 2022

One of the most studied kinds of finite-dimensional Hopf algebras is the family of pointed ones: it means that the coradical is the algebra of the group-like elements. When the group is abelian, all such examples are known following the so-called Lifting Method by Andruskiewitsch-Schneider and include deformations of small quantum groups, their super analogues and some exceptional examples of Nichols algebras. When the group is not abelian, the classification is not known yet. Even more, the first step of the Lifting Method (the computation of all finite-dimensional Nichols algebras) has not been completed: the classification has been performed by Heckenberger-Vendramin when the elements in degree one form a non-simple Yetter-Drinfeld module, and consist of low rank exceptions and large rank families.

In this talk we will present finite-dimensional Hopf algebras whose coradical is the group algebra of a central extension of an abelian group. They fall into families associated with a simple Lie algebra together with a Dynkin diagram automorphism.

We will show conversely that every finite-dimensional pointed Hopf algebra over a non-abelian group with a non-simple infinitesimal braiding is of this form for large rank families. The proof follows the steps of the Lifting Method. Indeed we prove that the large rank families are cocycle twists of Nichols algebras constructed by Lentner as foldings of Nichols algebras of Cartan type over abelian groups by outer automorphisms. This enables us to give uniform Lie-theoretic descriptions of the large rank families, prove generation in degree one and construct liftings.

We also show that every lifting is a cocycle deformation of the corresponding coradically graded Hopf algebra using an explicit presentation by generators and relations of the Nichols algebra.

Rob Muth: Superalgebra deformations of web categories

5th July, 2022

For a superalgebra A, and even subalgebra a, one may define an associated diagrammatic monoidal supercategory Web(A,a), which generalizes a number of symmetric web category constructions. In this talk, I will define and discuss Web(A,a)), focusing on two interesting applications: Firstly, Web(A,a) is equipped with an asymptotically faithful functor to the category of 𝔤𝔩n(A)-modules generated by symmetric powers of the natural module, and may be used to establish Howe dualities between 𝔤𝔩n(A) and 𝔤𝔩m(A) in some cases. Secondly, Web(A,a) yields a diagrammatic presentation for the ‘Schurification' TAa(n,d). For various choices of A/a, these Schurifications have proven connections to RoCK blocks of Hecke algebras, and conjectural connections to RoCK blocks of Schur algebras and Sergeev superalgebras.

Misha Dokuchaev: Strong equivalence of graded algebras

24th June, 2022

We introduce the notion of a strong equivalence between graded algebras and prove that any partially-strongly-graded algebra by a group G is strongly-graded-equivalent to the skew group algebra by a product partial action of G. We show that strongly-graded-equivalence preserves strong gradings and is nicely related to Morita equivalence of product partial actions. Furthermore, we show that strongly-graded-equivalent partially-strongly-graded algebras with orthogonal local units are stably isomorphic as graded algebras.

Vladimir Dotsenko: New examples of Nielsen-Schreier varieties of algebras

10th June, 2022

A variety of algebras is said to be a Nielsen-Schreier variety if every subalgebra of every free algebra is free. Using methods of the operad theory, we propose an effective combinatorial criterion for that property in the case of algebras over a field of zero characteristic. Using this criterion, we show, in particular, that the variety of all pre-Lie algebras (also known as right-symmetric algebras) is Nielsen-Schreier, and that, quite surprisingly, there are already infinitely many Nielsen-Schreier varieties of algebras with one binary operation and identities of degree three.

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.

José Serrano: Algebraic sets, ideals of points and the Hilbert’s Nullstellensatz theorem for skew PBW extensions

19th May, 2022

In this talk we define the algebraic sets and the ideal of points for bijective skew PBW extensions with coefficients in left Noetherian domains. Some properties of affine algebraic sets of commutative algebraic geometry will be extended, in particular, a Zariski topology will be constructed. Assuming additionally that the extension is quasi-commutative with polynomial center and the ring of coefficients is an algebraically closed field, we will prove an adapted version of Hilbert's Nullstellensatz theorem that covers the classical one. The Gröbner bases of skew PBW extensions will be used for defining the algebraic sets and for proving the main theorem. Many key algebras and rings coming from mathematical physics and non-commutative algebraic geometry are skew PBW extensions.

Yuri Bahturin: Group Gradings and Actions of Pointed Hopf Algebras

12th May, 2022

Pointed Hopf algebras are a wide class of Hopf algebras, including group algebras and enveloping algebras of Lie algebras. In this talk, based on a recent work with Susan Montgomery, we study actions of pointed Hopf algebras on simple algebras. These actions are known to be inner, as in the case of Skolem-Noether theorem. We try to give explicit descriptions, whenever possible, and consider Taft algebras, their Drinfeld doubles and some quantum groups.

Eduardo Marcos: Homogeneous triples for algebras with two relations

14th April, 2022

We define the category of homogeneous triples, which is equivalent to the category of graded algebras, with a fixed semisimple degree zero part. We apply the results to algebras whose defining ideal has two generators, and give a partial classification.

We thank Fapesp, grant 2018/23690-6, for the support.