Past talks - 2024

Abstract: Associated to a Lie algebra g and a g-module M is a standard complex C*(g,M) computing the cohomology of g with coefficients in M; this classical construction goes back to Chevalley and Eilenberg of the late 1940s. Shortly afterwards, it was realized that this cohomology is an example of a derived functor in the category of g-modules. The Lie algebra g can be replaced by a differential graded Lie algebra and M – with a dg g-module  with the  same conclusion. Later, a deep connection with Koszul duality was uncovered in the works of Quillen (late 1960s) and then Hinich (late 1990s). In this talk I will discuss the cohomology of (dg) Lie coalgebras with coefficients in dg comodules. The treatment is a lot more delicate, underscoring how different Lie algebras and Lie coalgebras are (and similarly their modules and comodules). A definitive answer can be obtained for so-called conilpotent Lie coalgebras (though not necessarily conilpotent comodules). If time permits, I will also discuss some topological applications.

Abstract: We generalize the concept of truncation of operators to JB*-triples and study some general properties of bijections preserving the truncation of triple products in both directions between general JB*-triples. In our main result, we show that a (non-necessarily linear nor continuous) bijection between atomic JBW*-triples preserving the truncation of triple products in both directions (and such that the restriction to each rank-one Cartan factor is a continuous mapping)  is an isometric real linear triple isomorphism.

Abstract: The problem of the existence of a finite basis of identities for a variety of associative algebras over a field of characteristic zero was formulated by Specht in 1950. We say that a variety of algebras has the Specht property if any of its subvariety has a finite basis of identities. In 1988, A. Kemer proved that the variety of associative algebras over a field of characteristic zero has the Specht property. Specht’s problem has been studied for many well-known varieties of algebras, such as Lie algebras, alternative algebras, right-alternative algebras, and Novikov algebras. An algebra is called right-symmetric if it satisfies the identity (a, b, c) = (a, c, b) where (a, b, c) = (ab)c − a(bc) is the associator of a, b, c. The talk is devoted to the Specht problem for the variety of right-symmetric algebras. It is proved that the variety of right-symmetric algebras over an arbitrary field does not satisfy the Specht property. The talk is based on the results of joint work with U. Umirbaev.

Abstract: In non commutative geometry, a quantum principal bundle over an affine base is recovered through a deformation of the algebra of its global sections: the property of being a principal bundle is encoded by the notion of Hopf Galois extension, while the local triviality is expressed by the cleft property.  We examine the case of a projective base X in the special case X=G/P, where G is a complex semisimple group and P a parabolic subgroup. The quantization of G will then be interpreted as the quantum principal bundle on the quantum base space X, obtained via a quantum section.

Abstract: Using the theory of algebraic operads, we give a combinatorial description of free pre-Lie algebras (also known as left-symmetric algebras) with rooted trees. A numerical coincidence hints a similar description for algebras with several pre-Lie products sharing the Lie bracket using rooted Greg trees which are rooted trees with black and white vertices such that black vertices have at least two children. We then show that those Greg trees can be used to give a description of the free Lie algebras.

Abstract: Corestriction is an important technique in the theory of central-simple associative algebras over a field. Given a finite étale extension K/F, e.g. a Galois extension, corestriction associates a central-simple associative F-algebra with every central-simple associative K-algebra. In this talk, I will give an introduction to corestriction over fields, applicable to nonassociative algebras. Towards the end of my talk, I will indicate why it is of interest to generalize corestruction to  schemes and sketch how this can be done (joint work Philippe Gille and Cameron Ruether).

Abstract: The classification of finite-dimensional semisimple Lie algebras in characteristic 0 represents one of the significant achievements in algebra during the first half of the 20th century. This classification was developed by Killing and Cartan. According to the Killing–Cartan classification, the isomorphism classes of simple Lie algebras over an algebraically closed field of characteristic zero correspond one-to-one with irreducible root systems. In the infinite-dimensional case, the situation is more complicated, and the so-called algebras of Cartan type appear. It is somewhat surprising that graded identities for Lie algebras have been relatively few results to that extent. In this presentation, we will discuss some of the results obtained thus far and introduce an algorithm capable of generating a basis for all graded identities in Lie algebras with Cartan gradings. Specifically, over any infinite field, we will apply this algorithm to establish a basis for all graded identities of U_1, the Lie algebra of derivations of the algebra of Laurent polynomials  K[t,t^{-1}], and demonstrate that they do not admit any finite basis. The findings discussed in this presentation are joint works with P. Koshlukov (UNICAMP).

Abstract: In this talk, we will present multi-Novikov algebras, a generalisation of Novikov algebras with several binary operations indexed by a given set, and show that the multi-indices recently introduced in the context of singular stochastic partial differential equations can be interpreted as free multi-Novikov algebras. This is parallel to the fact that decorated rooted trees arising in the context of regularity structures are related to free multi-pre-Lie algebras. This is a joint work with Vladimir Dotsenko.

Abstract: The theory of the parametric set-theoretic Yang-Baxter equation is established from a purely algebraic point of view.  We introduce generalizations of the familiar shelves and racks named parametric (p)-shelves and racks. These objects satisfy a "parametric self-distributivity" condition and lead to solutions of the Yang-Baxter equation. Novel, non-reversible solutions are  obtained from p-shelve/rack solutions by a suitable parametric twist, whereas all reversible set-theoretic solutions are reduced to the identity map via a parametric twist. The universal algebras associated to both p-rack and generic parametric set-theoretic solutions are next presented and the corresponding universal R-matrices are derived.  By introducing the concept of a parametric coproduct we prove the existence of a parametric co-associativity. We show that the parametric coproduct is an algebra homomorphsim and the universal R-matrices intertwine with the algebra coproducts.

Abstract: We first explore the definition of an affine space which makes no reference to the underlying vector space and then formulate the notion of a Lie bracket and hence a Lie algebra on an affine space in this framework. Since an affine space has neither distinguished elements nor additive structure, the concepts of antisymmetry and Jacobi identity need to be modified. We provide suitable modifications and illustrate them by a number of examples. The talk is based in part on joint works with James Papworth and Krzysztof Radziszewski.

Abstract: In earlier work, we studied the structure of primitive axial algebras of Jordan type (PAJ's), not necessarily commutative, in terms of their primitive axes. In this paper we weaken primitivity and permit several pairs of (left and right) eigenvalues satisfying a more general fusion rule, bringing in interesting new examples such as the band semigroup algebras and various noncommutative examples. Also, we broaden our investigation to the case of 2-generated algebras for which only one axis satisfies the fusion rules. As an example we describe precisely the 2-dimensional axial algebras and the 3-dimensional and 4-dimensional weakly primitive axial algebras of Jordan type (weak PAJ's), and we see, in contrast to the case for PAJ's, that there are higher dimensional weak PAJ's generated by two axes. We also prove a theorem that enables us to reduce weak PAJ's to uniform components.

Abstract: For any associative algebra A, the left regular representation is an embedding of A into its linear endomorphism algebra End(A). In this talk, I shall explain how this elementary observation can be generalised to a (less elementary) structure result for general non-associative algebras. The describes the category of unital, not necessarily associative, algebras in terms of associative algebras with certain distinguished subspaces.

Abstract: I will report on joint work in progress with Samuel Lopes and Isaac Oppong where we aim to compute the derivations of quantum nilpotent algebras, a class on noncommutative algebras which includes in particular the positive part of quantised enveloping algebras and quantum Schubert cells.

Abstract: The Płonka sum is one of the most significant composition methods in Universal Algebra introduced by Jerzy Płonka in 1967. In particular, Clifford semigroups have turned out to be the first instances of Płonka sums of groups. In this talk, we illustrate a method for constructing set-theoretical solutions of the Yang-Baxter equation that is inspired by the notion of the Płonka sums. Moreover, we will show how to obtain solutions of this type by considering dual weak braces, algebraic structures recently studied and described in a joint work with Francesco Catino and Marzia Mazzotta.

Abstract: In this presentation, I will introduce the audience to ternary algebras called heaps and trusses. Specifically, I will familiarize the audience with modules over trusses, highlighting differences with modules over rings. The main point will be to show the close relationship between modules over trusses and affine spaces over rings. I will illustrate that modules over trusses occupy a position between modules over rings and affine spaces over rings.

Abstract: Derivations on group algebras are linear operators. They satisfy the Leibniz rule. Another examples are Fox derivatives, which satisfy a different (but very similar) identity. We will give a construction which generalises all such identities and the corresponding operator families. The main element of such a construction is an action groupoid and the space of characters on it. The second step of the construction are characters on special graphs (action diagrams) which are equivalent to classical Cayley graphs for the case of left multiplication action. I will show the way to interpret inner derivations as a special case of trivial on loops characters. And we will consider a more general ideal of quasi-inner derivations. These results are based on the author's results, and the main approach was proposed in collaboration with A. Mischchenko.

Abstract: We will discuss representations of the Smith algebra which are free of finite rank over a subalgebra which plays a role analogous to that of the (enveloping algebra of the) Cartan subalgebra of the simple Lie algebra sl_2. In the case of rank 1 we obtain a full description of the isomorphism classes, a simplicity criterion, and a combinatorial algorithm to produce all composition series and the multiplicities of the simple factors. This is joint work with V. Futorny (SUSTech & USP) and E. Mendonça (Lyon & USP).

Abstract: The L'vov-Kaplansky conjecture claims that the image of a multilinear polynomial on the full matrix algebra is a vector space. Positive results concerning the conjecture are known only for small cases (polynomials of small degree or matrices of small size). Besides presenting the main results on the L'vov-Kaplasnky conjecture, in this talk we also will discuss some of its variations such as images of multilinear polynomials on some subalgebras of the full matrix algebra with additional structure (gradings, involutions, graded involutions).

Abstract: The aim of this work is to study the relation between S-expansions and other transformations of Lie algebras. In particular, we prove that contractions, deformations and central extensions of Lie algebras are preserved by S-expansions. We also provide several examples and give conditions so transformations of reduced subalgebras of S-expanded algebras are preserved by the S-expansion procedure. This is a joint work with Javier Rosales-Gómez.

Abstract: Given a 2-generated primitive axial algebra of Monster Type, it has been shown that it has an axet which is regular or skew. With all the known examples being regular, it was proposed if any axial algebra were skew and if so, can they be classified. We will begin by defining axial algebras and axets, before producing examples of axial algebras with skew axets. We will finish by stating the complete classification of these skew axial algebras and mention how it was proven.

Abstract: We show that pseudo-composition algebras and train algebras of rank 3 generated by idempotents are characterized as axial algebras with fusion laws derived from the Peirce decompositions of idempotents in these classes of algebras. The corresponding axial algebras are called PC(η)-axial algebras, where η is an element of the ground field. As a first step towards their classification, we describe 2− and 3-generated subalgebras of such algebras.

Abstract: In 2019, Alsaody and Gille showed that, for octonion algebras over unital commutative rings, there is an equivalence between isotopes and isometric quadratic forms. This leads us to a question: can this equivalence be generalized to octonion algebras over a (not necessarily affine) scheme? We give the basic definitions of octonion algebras over schemes. We show that an isotope of an octonion algebra C over a scheme is isomorphic to a twist by an Aut(C)–torsor. We conclude by giving an affirmative answer to our question.

Abstract: A pseudo-Euclidean left-symmetric algebra (A, .,< , >) is a real left-symmetric algebra (A,.) endowed with a non-degenerate symmetric bilinear  form < , > such that left multiplications by any element of A are skew-symmetric with respect to < , >, . We recall that a pseudo-Euclidean Lie algebra (g, [ , ], < , >) is flat if and only if (g, ., ,< , >)  its underlying vector space endowed with the Levi- Civita product associated with < , > is a pseudo-Euclidean left-symmetric algebra. In this talk, We will give an inductive classification of  pseudo-Euclidean left-symmetric algebras (A, .,< , >) such that commutators of allelements of A are contained in the left annihilator of (A, .), these algebras will be called pseudo-Euclidean left-symmetric L−algebras of any signature.. To do this, we will develop double extension processes that allow us to have inductive descriptions of all pseudo-Euclidean left-symmetric L−algebras and of all its pseudo-Euclidean modules.

Abstract: We study Lie algebraic properties of subalgebras of the Witt algebra and the one-sided Witt algebra: we compute derivations, one-dimensional extensions, and automorphisms of these subalgebras. In particular, all these properties are inherited from the full Witt algebra (e.g. derivations of subalgebras are simply restrictions of derivations of the Witt algebra). We also prove that any isomorphism between subalgebras of finite codimension extends to an automorphism of the Witt algebra. We explain this "rigid" behavior by proving a universal property satisfied by the Witt algebra as a completely non-split extension of any of its subalgebras of finite codimension. This is a purely Lie algebraic property which I will introduce in the talk.

Abstract: In this talk, I will introduce a bialgebra theory for the Novikov algebra, namely the Novikov bialgebra, which is characterized by the fact that its affinization (by a quadratic right Novikov algebra) gives an infinite-dimensional Lie bialgebra. A Novikov bialgebra is also characterized as a Manin triple of Novikov algebras. The notion of Novikov Yang-Baxter equation is introduced, whose skewsymmetric solutions can be used to produce Novikov bialgebras and hence Lie bialgebras. These solutions also give rise to skewsymmetric solutions of the classical Yang-Baxter equation in the infinite-dimensional Lie algebras from the Novikov algebras. Moreover, a similar connection between Novikov bialgebras and Lie conformal bialgebras will be introduced. This talk is based on joint works with Chengming Bai and Li Guo.

Abstract: This is joint work with Dietrich Burde (University of Vienna, Austria). Intrigued by computations of Richardson, our goal is to compute the adjoint cohomology spaces of Lie algebras which are the semi-direct product of a simple Lie algebra s and an s-module. We present some theorems and conjectures in these cohomologies.

Abstract: How to recover an algebra structure if the algebra does NOT satisfy any reasonable identity? How to characterize its idempotents, their spectrum, or fusion laws? In my talk, I will discuss what can be thought of as "nonassociative algebra in large", imitating a well-known concept of "geometry in large". In other words, the properties of nonassociative algebras which crucially depend on a complete set of idempotents. The latter is very related to the concept of generic algebras. I will explain some recent results in this direction and some unsolved problems. 

Abstract: We report on ongoing research about a module-theoretic construction which, when successful, yields natural extensions of infinite dimensional modules over arbitrary algebras. Whether the construction works or not depends on the basis that one chooses to carry on such a construction. Bases that work are said to be amenable. A natural example on which one may focus is when the module is the algebra itself. For instance, a great deal of the work done so far has focused on infinite dimensional algebra of polynomials on a single variable. We will see that amenability and related notions serve to classify the distinct bases according to interesting complementary properties having to do with the types of relations induced on them by the properties of their change-of-basis matrices.