Seminars in Hopf Algebras

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Pierre Catoire: The free tridendriform algebra, Schröder trees and Hopf algebras

12th August, 2024

The notions of dendriform algebras, respectively tridendriform, describe the action of some elements of the symmetric groups called shuffle, respectively quasi-shuffle over the set of words whose letters are elements of an alphabet, respectively of a monoid. A link between dendriform and tridendriform algebras will be made. Those words algebras satisfy some properties but they are not free. This means that they satisfy extra properties like commutativity. In this talk, we will describe the free tridendriform algebra. It will be described with planar trees (not necessarily binary) called Schröder trees. We will describe the tridendriform structure over those trees in a non-recursive way. Then, we will build a coproduct on this algebra that will make it a (3, 2)-dendriform bialgebra graded by the number of leaves. Once it will be build, we will study this Hopf algebra: duality, quotient spaces, dimensions, study of the primitive elements.

Julia Plavnik: Remarks on the tensor product property for support varieties for finite tensor categories

22nd April, 2024

For non-semisimple tensor categories satisfying some finiteness conditions, support varieties are meaningful geometric invariants of objects. Their theory began in the work of Quillen and Carlson on finite group representations. In more recent years, the theory of support varieties was generalized in many directions, including representations of finite-dimensional Hopf algebras and self-injective algebras, and objects in finite tensor categories and triangulated categories, among others.

In this talk, we will start by introducing the definition of support varieties for finite tensor categories and some of their basic properties. We will also present some conditions under which the tensor product property holds for support varieties, and we will present some applications to certain Hopf algebras. We will also discuss a construction of non-semisimple finite tensor categories with finitely generated cohomology for which the tensor product property does not hold for support varieties.

Kang Lu: A Drinfeld presentation of twisted Yangians via degeneration

2nd March, 2024

We formulate a new family of algebras, twisted Yangians (of split type) in current generators and relations, via degeneration of Drinfeld presentations of affine iquantum groups (associated with split Satake diagrams). These new algebras admit PBW type bases and are shown to be a deformation of twisted current algebras. For type AI, it matches with the Drinfeld presentation of twisted Yangian obtained via Gauss decomposition. We conjecture that our twisted Yangians are isomorphic to twisted Yangians constructed in RTT presentation.

Senne Trappeniers: The interplay between skew braces, the Yang-Baxter equation and Hopf-Galois structures

27th November, 2023

In 2007, Wolfgang Rump introduced algebraic objects called braces, these generalize Jacobson radical rings and are related to involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation (YBE). These objects were subsequently generalized to skew braces by Leandro Guarnieri and Leandro Vendramin in 2017, and a similar relation was shown to hold for non-degenerate set-theoretic solutions of the YBE which are not necessarily involutive. In this talk, we will describe this interplay between skew braces and the YBE. We will also discuss their relation to Hopf-Galois structures and see how this extends the classical Galois theory in an elegant way.

Kang Lu: A Drinfeld presentation of twisted Yangians

12th May, 2023

In this talk, I will discuss the Drinfeld’s new presentation for (Olshanski’s) twisted Yangians of type AI that is open for 30 years. This presentation comes from the Gauss decomposition and turns out to be compatible with the one obtained from degeneration of affine iQuantum Groups. If time permits, I will also discuss our work on twisted Yangians of other types.

Milen Yakimov: On the spectrum and support theory of a finite tensor category

4th April, 2023

Finite tensor categories are important generalizations of the categories of finite-dimensional modules of finite-dimensional Hopf algebras. There are two support theories for them, the cohomological one and one based on the noncommutative Balmer spectrum of the corresponding stable module category. We will describe general results linking the two types of support via a new notion of categorical center of the cohomology ring of a finite tensor category and will state a conjecture giving the exact relation. The construction and results will be illustrated with various examples.

Tekin Karadağ: Lie Structure on Hochschild and Hopf Algebra Cohomologies II

26th September, 2022

Murray Gerstenhaber constructed a graded Lie structure (Gerstenhaber bracket) on Hochschild cohomology, which makes Hochschild cohomology a Lie algebra. However, it is not easy to calculate bracket structure with the original definition. There is an alternative technique to compute Gerstenhaber bracket on Hochschild cohomology, introduced by Chris Negron and Sarah Witherspoon. It is also known that Hopf algebra cohomology has a bracket and the bracket is trivial when a Hopf algebra is quasi-triangular. We use a similar technique to the technique given by Negron and Witherspoon to calculate the Lie structure on Hochschild cohomology of the Taft algebra Tp for any integer p>2 which is a nonquasi-triangular Hopf algebra. Then, we find the corresponding bracket on Hopf algebra cohomology of Tp. We show that the bracket is indeed zero on Hopf algebra cohomology of Tp, as in all known quasi-triangular Hopf algebras. This example is the first known bracket computation for a nonquasi triangular algebra.

Tekin Karadağ: Lie Structure on Hochschild and Hopf Algebra Cohomologies I

19th September, 2022

Murray Gerstenhaber constructed Lie structure (Gerstenhaber bracket) on Hochschild cohomology, which makes Hochschild cohomology a graded Lie algebra. Later, it is shown that Hopf algebra cohomology also has a Lie structure. We will introduce a general formula for the bracket on Hopf algebra cohomology of any Hopf algebra with bijective antipode on the bar resolution that is reminiscent of Gerstenhaber’s original formula for Hochschild cohomology.

Jason Bell: Recent results on the Dixmier-Moeglin equivalence

12th August, 2022

Dixmier and Moeglin showed that if L is a finite-dimensional complex Lie algebra then the primitive ideals of the enveloping algebra U(L) are the prime ideals of Spec(U(L)) that are locally closed in the Zariski topology. In addition, they proved that a prime ideal P of U(L) is primitive if and only if the Goldie ring of quotients of U(L)/P has the property that its centre is just the base field of the complex numbers. Algebras that share this characterization of primitive ideals are said to satisfy the Dixmier-Moeglin equivalence. We give an overview of this property and mention some recent work on proving this equivalence holds for certain classes of twisted homogenous coordinate rings and classes of Hopf algebras of small Gelfand-Kirillov dimension.

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.