Past talks - 2024

Abstract: The starting point of this talk is the fact that the class of evolution algebras over a fixed field is closed under tensor product. We prove that, under certain conditions, the tensor product is an evolution algebra if and only if every factor is an evolution algebra. Another issue arises about the inheritance of properties from the tensor product to the factors and conversely. For instance, nondegeneracy, irreducibility, perfectness and simplicity are investigated. The four-dimensional case is illustrative and useful to contrast conjectures, so we achieve a complete classification of four-dimensional perfect evolution algebras emerging as tensor product of two-dimensional ones. We find that there are four-dimensional evolution algebras that are the tensor product of two non evolution algebras. This is a joint work together with Yolanda Cabrera Casado, Dolores Martín Barquero, and Cándido Martín González.

Abstract: I will present joint work with Sue Sierra where we proved the ACC for radical Poisson ideals of the symmetric algebra of a Dicksonian Lie algebra. Part of the talk will be devoted to explaining what Dicksonian means (and give a variety of examples), and then discuss the method of proof of the basis theorem. We will observe why our result applies to graded-simple Lie algebras.

Abstract: In order to define suitable noncommutative Poisson structures, M. Van den Bergh introduced double Poisson algebras and double quasi-Poisson algebras. Furthermore, N. Iyudu and M. Kontsevich found an insightful correspondence between double Poisson algebras and pre-Calabi-Yau algebras; certain cyclic A∞-algebras which can be seen as noncommutative versions of shifted Poisson manifolds. In this talk I will present an extension of the Iyudu-Kontsevich correspondence to the differential graded setting. I will also explain how double quasi-Poisson algebras give rise to pre-Calabi-Yau algebras. This is a joint work with E. Herscovich (EPFL).

Abstract: Describing the conjugacy classes and/or irreducible characters of the unitriangular group over a finite field is known to be an impossibly difficult problem. Superclasses and supercharacters have been introduced (under the names of "basic varieties" and "basic characters") as an attempt to approximate conjugacy classes and irreducible characters using a cruder version of Kirillov's method of coadjoint orbits. In the past thirty years, these notions have been recognised in several areas (seemingly unrelated to representation theory): exponential sums in number theory, random walks in probability and statistics, association schemes in algebraic combinatorics... In this talk, we will describe and illustrate the main ideas and recent developments of the standard supercharacter theory of adjoint groups of radical rings. We will explore the close relation to Schur rings, and extend a well-known factorisation of supercharacters of unitriangular groups which explains the alternative definition as basic characters.

Abstract: By an ultra classical result, the tensor product of a simple representation of gl(n,C) and its defining representation decomposes as a direct sum of simple representations without multiplicities. This means that for each highest weight, the space of highest weight vectors is one dimensional. We will give an explicit construction of these highest weight vectors, and show that they arise from the action of certain elements in the enveloping algebra of gl(n,c)+gl(n,C) on the tensor product. These elements are independent of the simple representation we started with, and in fact produce highest weight vectors in several other contexts. (Joint with Joanna Meinel from Bonn University)

Abstract: Let F be a field of characteristic zero, L a Lie algebra over F, and A an L-algebra - that is, an associative algebra over F with an action of L induced by derivations. This action of L on A can be extended to an action of its universal enveloping algebra U(L), leading to the concept of L-identities or differential identities of A: polynomials in variables x^u := u(x), where u \in U(L), that vanish under all substitutions of elements from A. Differential identities were first introduced by Kharchenko in 1978, and, in later years, subsequent work by Gordienko and Kochetov has spurred a renewed interest in both their structure and quantitative properties. In this talk, I will present recent results on the differential identities of matrix L-algebras, with a particular focus on their classification and growth behavior.

Abstract: In this talk we expose ongoing joint work with I Paniello on systems of quotients (in a sense partially extending the localization theory of Jordan algebras, which in turn is inspired by the localization theory of associative algebras). Localization theory in associative algebras originated in the purpose of extending the construction of fields of quotients of integral domains, and therefore in the purpose of defining  ring extensions in which a selected set of elements become invertible. As it is well known in associative theory  that led to Goldie's theorems, and these in turn to more general localization theories for which the denominators of the fraction-like elements of the extensions are (one-sided) ideals taken in a class of filters (Gabriel filters). These ideas have been partially extended to Jordan algebras by several authors (starting with Zelmanov's version of Goldie theory in the Jordan setting, and its extension by Fernandez López-García Rus and Montaner) and Paniello and Montaner (among others) definition of algebras of quotients of Jordan algebras. Following the development of Jordan theory, a natural direction for extending these results is considering the context of Jordan pairs. This is the objective of the research presented here. Since obviously a Jordan pair cannot have invertible elements unless it is an algebra, and in this case we are back in the already developed theory, the kind of quotients that would make a significative (proper) extension of the case of algebras should be based in a different notion of quotient.  An approach that seems to be promising is considering the Jordan extension of Fountain and Gould notion of local order, as has been adapted to Jordan algebras by the work of Fernández López, and more recently by Montaner and Paniello with the notion of local order, in which the bridge between algebras and pairs is established by local algebras following the ideas of D'Amour and McCrimmon. In the talk this idea is exposed, together with the state of the research, and the open problems that it raises.

Abstract: In a now classic paper, Belavin and Drinfeld categorized solutions to the classical Yang-Baxter equation (CYBE), an equation crucial to the theory of integrable systems, into three classes: elliptic, trigonometric and rational. It is possible to reproduce this result by geometrizing solutions of the CYBE and then applying algebro-geometric methods. In this talk, we will explain how this approach can be used to categorize Lie bialgebra structures on power series Lie algebras, as well as non-associative generalizations of these structures: D-bialgebra structures on more general power series algebras.

Abstract: In this talk, I will discuss a general method to renormalise singular stochastic partial differential equations (SPDEs) using the theory of regularity structures. It turns out that, to derive the renormalised equation, one can employ a convenient multi-pre-Lie algebra. The pre-Lie products in this algebra are reminiscent of the pre-Lie product on the Grossman-Larson algebra of trees, but come with several important twists. For the renormalisation of SPDEs, the important feature of this multi-pre-Lie algebra is that it is free in a certain sense. Based on joint work with Yvain Bruned, Ajay Chandra, and Martin Hairer.

Abstract: One important problem in the vertex algebra theory is to associate certain vertex algebra-like objects, the quantum vertex algebras, to various classes of quantum groups, such as quantum affine algebras or double Yangians. In this talk, I will discuss this problem in the context of Etingof--Kazhdan's quantum affine vertex algebra V^c(gl_N)  associated with the trigonometric R-matrix of type A. The main focus will be on the explicit description of the center of V^c(gl_N) at the critical level c=-N and, furthermore, on the connection between certain classes of V^c(gl_N)-modules and representation theories of the quantum affine algebra of type A and the orthogonal twisted h-Yangian. The talk is in part based on the joint works with Alexander Molev and Lucia Bagnoli.

Abstract: We investigate a special class of solutions of the set-theoretic Yang-Baxter equation, called Frobenius-Separability (FS) type solutions. In particular, we show that the category of solutions of the set-theoretic Yang-Baxter equation of Frobenius-Separability (FS) type is equivalent to the category of pointed Kimura semigroups. As applications, all involutive, idempotent, nondegenerate, surjective, finite order, unitary, or indecomposable solutions of FS type are classified. For instance, if |X| = n, then the number of isomorphism classes of all such solutions on X that are (a) left non-degenerate, (b) bijective, (c) unitary or (d) indecomposable and left-nondegenerate is: (a) the Davis number d(n), (b) \sum_{m|n} p(m), where p(m) is the Euler partition number, (c) τ(n) + \sum_{d|n}[d/2], where τ is the number of divisors of n, or (d) the Harary number. The automorphism groups of such solutions can also be recovered as automorphism groups Aut(f) of sets X equipped with a single endo-function f ⁣: X→X. We describe all groups of the form Aut(f) as iterations of direct and (possibly infinite) wreath products of cyclic or full symmetric groups, characterize the abelian ones as products of cyclic groups, and produce examples of symmetry groups of FS solutions not of the form Aut(f). Based on joint work with A. Chirvasitu and G. Militaru.

Abstract: Recall that a set-theoretic solution of the Yang-Baxter equation is a tuple (X,r), where X is a non-empty set and r: X х X → X  х X a bijective map such that (r х id_X ) (id_X х r) (r х id_X) = (id_X х r) (r х id_X ) (id_X х r), where one denotes r(x,y)=(l_x(y), r_y(x)). Attention is often restricted to so-called non-degenerate solutions, i.e. l_x and r_y are bijective. We will call these solutions for short in the remainder of this abstract. To understand more general objects, it is an important technique to study 'minimal' objects and glue them together. For solutions both indecomposable and simple solutions fit the bill for being a minimal object. In this talk, we will report on recent work with I. Colazzo, E. Jespers and L. Kubat on simple solutions. In particular, we will discuss an extension of a result of M. Castelli that allows to identify whether a solution is simple, without having to know or calculate all smaller solutions. This method employs so-called skew braces, which were constructed to provide more examples of solutions, but also govern many properties of general solutions. In the latter part of the talk, we discuss the extension of a method to construct new indecomposable or simple solutions from old ones via cabling, originally introduced by V. Lebed, S. Ramirez, and L. Vendramin to unify the known results on indecomposability of solutions.

Abstract: A 2-plectic form ω on a Lie algebra is a 3-form on the algebra such that it is closed and non-degenerate in the sense that, for every nonzero x, the bilinear form ω(x, ·, ·) is not identically zero. We will study the existence of 2-plectic structures on the so-called quadratic Lie algebras, which are Lie algebras admitting an ad-invariant pseudo-Euclidean product. It is well-known that every centerless quadratic Lie algebra admits a 2- plectic form but not many quadratic examples with nontrivial center are known. We give several constructions to obtain large families of 2-plectic quadratic Lie algebras with nontrivial center, many of them among the class of nilpotent Lie algebras. We give some sufficient conditions to assure that certain extensions of 2-plectic quadratic Lie algebras result to be 2-plectic as well. For instance, we show that oscillator algebras can be naturally endowed with 2-plectic structures. We prove that every quadratic and symplectic Lie algebra with dimension greater than 4 also admits a 2-plectic form. Further, conditions to assure that one may find a 2-plectic which is exact on certain quadratic Lie algebras are obtained.

Abstract: Leibniz algebras were introduced by Blo(c)h in the 1960’s and rediscovered by Loday in the 1990’s as non-anticommutative analogues of Lie algebras. Many results for Lie algebras have been proven to hold for Leibniz algebras, but there are also several results that are not true in this more general context. In my talk, I will investigate the structure of semi-simple Leibniz algebras. In particular, I will prove a simplicity criterion for (left) hemi-semidirect products of a Lie algebra g and a (left) g-module. For example, in characteristic zero every finite-dimensional simple Leibniz algebra is such a hemi-semidirect product. But this also holds for some infinite-dimensional Leibniz algebras or sometimes in non-zero characteristics. More generally, the structure of finite- dimensional semi-simple Leibniz algebras in characteristic zero can be reduced to the well-known structure of finite-dimensional semi-simple Lie algebras and their finite-dimensional irreducible modules. If time permits, I will apply these structure results to derive some properties of finite-dimensional semi-simple Leibniz algebras in characteristic zero and other Leibniz algebras that are hemi-semidirect products.

Abstract: Post-Lie algebras appeared in 2007 in algebraic combinatorics, and independently in 2008 in the study of numerical schemes on homogeneous spaces. Gavrilov's K-map is a particular Hopf algebra isomorphism, which can be naturally described in the context of free post-Lie algebras. Post-groups, which are to post-Lie algebras what groups are to Lie algebras, were defined in 2023 by C. Bai, L. Guo, Y. Sheng and R. Tang. Although skew-braces and braided groups are older equivalent notions, their reformulation as post-groups brings crucial new information on their structure. After giving an account of the above-mentioned structures, I shall introduce free post-groups, and describe a group isomorphism which can be seen as an analogon of Gavrilov's K-map for post-groups. Based on joint work with M. J. H. Al-Kaabi and K. Ebrahimi-Fard.

Abstract: The skew brace was devised by Guanieri and Vendramin in 2017, building on Rump's brace. Since then, the skew brace has been central to the study of solutions to the Yang-Baxter equation, with connections to many other areas of mathematics including Hopf-Galois theory. We introduce the skew bracoid, a generalisation of the skew brace which can arise as a partial quotient thereof. We explore the connection between skew bracoids and Hopf-Galois theory, as well as the more recent connection to solutions of the Yang-Baxter equation.

Abstract: Although not every 1-connected solvable Lie group G admits a simply transitive action via affine maps on R^n, it is known that such an action exists if one replaces R^n by a suitable nilpotent Lie group H, depending on G. However, not much is known about which pairs of Lie groups (G,H) admit such an action, where ideally you only need information about the Lie algebras corresponding to G and H. In recent work with Marcos Origlia, we show that every simply transitive action induces a post-Lie algebra structure on the corresponding Lie algebras. Moreover, if H has nilpotency class 2 we characterize the post-Lie algebra structures coming from such an action by giving a new definition of completeness, extending the known cases where G is nilpotent or H is abelian.

Abstract: In 1985, Novikov and Balinskii introduced what became known as Novikov algebras in an attempt to construct generalizations of Witt Lie algebra. To their disappointment, Zelmanov showed that the only simple finite-dimensional Novikov algebra is one-dimensional (and corresponds to Witt algebra). The picture is much more interesting in the super case, where there are many more generalizations of Witt algebra, called superconformal Lie algebras. In 1988 Kac and Van de Leur gave a conjectural list of simple superconformal Lie algebras. Their list was amended with a Cheng-Kac superalgebra, which was constructed several years later. However, Novikov superalgebras are not flexible enough to describe all simple superconformal Lie algebras. In this talk, we shall present the class of quasi-Poisson algebras. Quasi-Poisson algebras have two products: it is a commutative associative (super)algebra, a Lie (super)algebra, and has an additional unary operation, subject to certain axioms. All known simple superconformal Lie algebras arise from finite-dimensional simple quasi-Poisson superalgebras. In this talk, we shall present basic constructions, describe the examples of quasi-Poisson superalgebras, and mention some results about their representations.

Abstract: 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 Schroeder 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 primitives elements.

Abstract: To each solution of the Yang-Baxter equation one may associate a quadratic algebra over a field, called the YB-algebra, encoding certain information about the solution. It is known that YB-algebras of finite non-degenerate solutions are (two-sided) Noetherian, PI and of finite Gelfand-Kirillov dimension. If the solution is additionally involutive then the corresponding YB-algebra shares many other properties with polynomial algebras in commuting variables (e.g., it is a Cohen-Macaulay domain of finite global dimension). The aim of this talk is to explain the intriguing relationship between ring-theoretical and homological properties of YB-algebras and properties of the corresponding solutions of the Yang-Baxter equation. The main focus is on when such algebras are Noetherian, (semi)prime and representable. The talk is based on a joint work with I. Colazzo, E. Jespers and A. Van Antwerpen.

Abstract: Let K be a field and X a connected partially ordered set. In this talk we show that the finitary incidence algebra FI(X, K) of X over K has an involution of the second kind if and only if X has an involution and K has an automorphism of order 2. We also present a characterization of the involutions of the second kind on FI(X, K). We conclude by giving necessary and sufficient conditions for two involutions of the second kind on FI(X, K) to be equivalent in the case where characteristic of K is different from 2 and every multiplicative automorphism of FI(X, K) is inner.

Abstract: In this talk we define and study Quillen-Barr-Beck cohomology for the category of restricted Lie algebras. We prove that the first Quillen-Barr-Beck’s cohomology classifies general abelian extensions of restricted Lie algebras. Moreover, using Duskin-Glenn’s torsors cohomology theory, we prove a classification theorem for the second Quillen-Barr-Beck cohomology group in terms of 2-fold extensions of restricted Lie algebras. Finally, we give an interpretation of Cegarra-Aznar’s exact sequence for torsor cohomology.

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.