Past talks - 2025

Abstract: A double bialgebra is a family (A, m, Δ, δ) such that both (A, m, Δ) and (A, m, δ) are bialgebras, with the extra condition that seeing δ as a right coaction on itself, m and Δ are right comodule morphisms over (A, m, δ). A classical example is given by the polynomial algebra ℂ[X], with its two classical coproducts. In this talk, we will present a double bialgebra structure on the symmetric algebra generated by noncrossing partitions. The first coproduct is given by the separations of the blocks of the partitions, with respect to the entanglement, and the second one by fusions of blocks. This structure implies that there exists a unique polynomial invariant on noncrossing partitions which respects both coproducts: we will give some elements on this invariant, and applications to the antipode of noncrossing partitions.

Abstract: Nichols algebras appear in several areas of mathematics, from Hopf algebras and quantum groups to Schubert calculus and conformal field theories. In this talk, I will review the main problems related to finite-dimensional Nichols algebras over groups and discuss a very recent classification theorem written in collaboration with Andruskiewitsch and Heckenberger.

Abstract: Associative central simple algebras are a classical subject, related to many areas of study including Galois cohomology and algebraic geometry. An associative central simple algebra is a form of matrices because a maximal étale subalgebra acts on the algebra faithfully by left and right multiplication. In an attempt to extract and isolate the full potential of this point of view, we study nonassociative algebras whose nucleus contains an étale subalgebra bi-acting faithfully on the algebra. We show that these algebras, termed semiassociative, are forms of a nonassociative analogue of matrix algebras. Finally, we consider the monoid composed of semiassociative algebras modulo the nonassociative matrix algebras, and discuss its connection to the classical Brauer group. Based on joint work with Darrell Haile, Eliyahu Matzri, Edan Rein, and Uzi Vishne.

Abstract: Recently, Li, Sheng, and Tang introduced post-Hopf algebras and relative Rota—Baxter operators (on cocommutative Hopf algebras), providing an adjunction between the respective categories under the assumption that the structures involved are cocommutative. We introduce Yetter— Drinfeld post-Hopf algebras, which become usual post-Hopf algebras in the cocommutative setting. In analogy with the correspondence between cocommutative post-Hopf algebras and cocommutative Hopf braces, the category of Yetter—Drinfeld post-Hopf algebras is isomorphic to the category of Yetter—Drinfeld braces, introduced by the author in a joint work with D. Ferri. The latter structures are equivalent to matched pairs of actions on Hopf algebras and generalise both Hopf braces and Majid’s transmutation. We also prove that the category of Yetter—Drinfeld post-Hopf algebras is equivalent to a subcategory of Yetter—Drinfeld relative Rota—Baxter operators (that generalise bijective relative Rota—Baxter operators on cocommutative Hopf algebras). Once the surjectivity of the latter operators is removed, the equivalence is replaced by an adjunction and one recovers, in the cocommutative case, the result of Li, Sheng, and Tang. The talk is partially based on a joint work with D. Ferri.

Abstract: This talk is an introduction to the Tits–Kantor–Koecher Lie algebras associated with Jordan triples. In particular, we will obtain representations of these algebras as matrix Lie algebras. The necessary background will be provided, as to render the talk self-contained.

Abstract: Let S be an arbitrary reductive algebraic group. Let's call a homomorphism Φ:S → Aut(g) an S-structure on the Lie algebra g. S-structures were previously investigated by various authors, including E.B. Vinberg. The talk deals with SL_2-structures. Let's call the SL_2-structure short if the representation Φ of the group SL_2 decomposes into irreducible representations of dimensions 1, 2, and 3. If we consider irreducible representations of dimensions only 1 and 3, we get the well-known Tits-Kantor-Koeher construction, which establishes a one-to-one correspondence between simple Jordan algebras and simple Lie algebras of a certain type. Similarly to the Tits–Kantor–Koeher theorem, in the case of short SL_2-structures, there is a one-to-one correspondence between simple Lie algebras with such a structure and the so-called simple symplectic Lie-Jordan structures. Let g be a Lie algebra with SL_2-structure and the map ρ:g→gl(U) be a linear representation of g. The homomorphism Φ:S →GL(U) is called a SL_2-structure on the Lie g-module U if  Φ(s)ρ(ξ)u =ρ(Φ(s)ξ)Ψ(s)u. This construction has interesting applications to the representation theory of Jordan algebras, which will be discussed during the talk. We will also present a complete classification of irreducible short g-modules for simple Lie algebras.

Abstract: It is well known that in the semi-abelian category Grp of groups, split extensions, or equivalently internal actions, are represented by automorphisms. This means that the category Grp is action representable and the actor of a group X is the group Aut(X). The notion of action representable category has proven to be quite restrictive: for instance, if a non-abelian variety of non-associative algebras, over an infinite field of characteristic different from two, is action representable, then it is the category of Lie algebras. More recently G. Janelidze introduced the notion of weakly action representable category, which includes a wider class of categories. In this talk we show that for an algebraically coherent variety of algebras and an object X of it, it is always possible to construct a partial algebra E(X), called external weak actor of X, which allows us to describe internal actions on X. Moreover, we show that the existence of a weak representation is connected to the amalgamation property, and we give an application of the construction of the external weak actor in the context of varieties of unitary algebras. This is joint work with J. Brox (Universidad de Valladolid), Xabier García Martínez (Universidade de Vigo), Tim Van der Linden and Corentin Vienne (Université catholique de Louvain).

Abstract: We discuss narrow in the sense of Shalev and Zelmanov positively graded Lie (super)algebras. They appear in different problems of geometry, topology and math physics. We will pay attention to the classification results as well as to the applications.

Abstract: Let k be a field, let G be a smooth affine k-group of finite type, and V a finite-dimensional G-module. We say V is rigid if the socle series and radical series coincide for the action of G on each indecomposable summand of V; say V is geometrically rigid (resp. absolutely rigid) if V is rigid after base change of G and V to an algebraic closure of k (resp. any field extension of k). We show that all simple G-modules are geometrically rigid, though they are not in general absolutely rigid. More precisely, we show that if V is a simple G-module, then there is a finite purely inseparable extension k_V /k naturally attached to V such that V_{k_V} is absolutely rigid as a G_{k_V} -module. The proof for connected G turns on an investigation of algebras of the form K \otimes_k E where K and E are field extensions of k; we give an example of such an algebra which is not rigid as a module over itself. We establish the existence of the purely inseparable field extension k_V /k through an analogous version for Artinian algebras.