Past talks - 2025

Abstract: This talk discusses the possible relevance of the Okubonions (i.e., the real Okubo algebra) in quantum chromodynamics (QCD). We start and present the Okubonions within the 8-dimensional real division composition algebras, and then discuss their realization as the traceless cubic simple Jordan algebra over the complex numbers, endowed with a suitable deformation of the Michel-Radicati product. The Okubonions lack a unit element and exhibit the unique feature of sitting in the adjoint representation of their automorphism group SU(3); in this respect, they are fundamentally different from the better-known Octonions. While these latter may represent quarks (and singlets of the QCD SU(3) color gauge group), the Okubonions are conjectured to represent the gluons, i.e. the gauge bosons of the colour group. However, it is remarked that the SU(3) groups pertaining to Okubonions and Octonions are distinct and inequivalent subgroups of Spin(8) that share no common SU(2) subgroup. Main reference: arXiv:2309.17435 [hep-th].

Abstract: An extremely familiar notion in Hopf algebra theory is that of an integral. If G is a compact topological group, then a Haar integral on G is a linear functional from C(G) to R which is translation invariant. A well-known result by Larson and Sweedler shows that integrals on a Hopf algebra can be obtained by applying the Structure Theorem of Hopf modules to the rational part of its linear dual. In this seminar, we will discuss how this construction can be carried out even in the absence of an antipode, offering a novel perspective on integrals which differs profoundly from their classical description as "colinear forms." Our approach leads to a new notion of integrals for bialgebras which does not require, and does not imply, the existence of an antipode. Time permitting, we will seize this opportunity to say a few words about Hopf envelopes of bialgebras, since in many cases of interest, integrals for a bialgebra are in bijection with integrals for its Hopf envelope. The Hopf envelope of a bialgebra B is a certain universal Hopf algebra that we can associate with B and that plays for it the same role that the universal enveloping group plays for a monoid. In categorical terms, the Hopf envelope is the left adjoint to the forgetful functor from Hopf algebras to bialgebras, hence it may be legitimately called the free Hopf algebra generated by B. Its existence is a well-known fact in Hopf algebra theory, but its construction is very technical. Nevertheless, there are a number of cases where we can realize the Hopf envelope of a bialgebra B as a suitable quotient of B itself and we can take advantage of it to study integrals for the corresponding bialgebra. This talk is based on an ongoing project with A. Ardizzoni and C. Menini.

Abstract: In this talk, we establish a connection between evolution algebras of dimension two and Hopf algebras, via the algebraic group of automorphisms of an evolution algebra. This analysis involves the computation of the (tight) p−algebra associated with any 2-dimensional evolution algebra, whenever it exists. Furthermore, if A is perfect and has a faithful tight p-algebra, then this p-algebra is isomorphic to H (the Hopf algebra associated to the evolution algebra).

Abstract: Since their introduction by K. Meyberg in the nonassociative setting, local algebras have played a key role in the study of Jordan systems.  The local inheritance of regularity conditions (such as nondegeneracy, strong primeness or primitivity) is a well-known result that undoubtedly contributed to the development of the structure theory, not only of Jordan algebras, but also of Jordan pairs and triple systems. A rather usual strategy to tackle many Jordan questions is to differentiate  Jordan systems depending on whether their local algebras satisfy or not certain properties. For instance, some of the recent results on localization theory for Jordan algebras have been established taking advantage of the dichotomy between Jordan algebras having, or not,  local algebras satisfying polynomial identities. Analogously, the formulation of Goldie local theory for Jordan algebras is closely related to   Jordan algebras admitting Lesieur-Croisot local algebras. The above considerations lead us to consider, in the Jordan algebra setting, how local algebras of Jordan algebras interact with their algebras of quotients  (in Utumi's sense). This problem is motivated by a previous question originally posed, in the associative setting for (maximal, Martindale, and symmetric) rings of quotients of semiprime rings by Gomez Lozano and Siles Molina, who proved that both constructions commute whenever the element at which the local algebra is defined becomes von Neumann regular in the corresponding ring of quotients. In this talk, we will display the Jordan algebra case of this problem, proving that,  for any nondegenerate Jordan algebra, whenever the element defining the local algebra becomes von Neumann regular in its maximal algebra of quotients, taking local algebras and maximal algebras of quotients are commuting constructions. This is a joint work with Fernando Montaner (University of Zaragoza).

Abstract: I will describe several non-associative versions of Hilbert’s basis theorem, for non-associative Ore extensions and related structures. An interesting asymmetry between the left and right versions of Hilbert’s basis theorem will appear, not present in the associative case. The talk is based on joint work with Per Bäck.

Abstract: In this talk, I will speak on the classification result of irreducible modules for map extended Witt algebras with finite dimensional weight spaces. They turn out to be either modules with uniformly bounded weight spaces or highest weight modules. We further prove that all these modules are single point evaluation modules (n ≥ 2). So they are actually irreducible modules for extended Witt algebras. This is a joint work with S. Eswara Rao, Priyanshu Chakraborty, and Ritesh Kumar Pandey.

Abstract: In this talk, we describe transposed Poisson structures on Witt and Virasoro-type algebras. We compute 1/2-derivations on the deformative Schrodinger-Witt algebra, not-finitely graded Witt algebras, and not-finitely graded Heisenberg-Witt algebras. We classify all transposed Poisson structures on such algebras, as well as deformed generalized Heisenberg-Virasoro and not-finitely graded Heisenberg-Virasoro algebras. Furthermore, we compute the quasi-derivations of the Witt and Virasoro algebras.

Abstract: Let E be the infinite-dimensional Grassmann algebra over a field of characteristic different from 2. In this talk, we investigate the Isomorphism Problem in the context of the natural Z2-grading of E. We show that this grading is completely determined by its graded polynomial identities. Additionally, we explore the connection between Z2-gradings on E and its automorphisms of order two.

Abstract: In this talk, I will introduce the concept of NL bialgebras, an algebraic structure that combines a Lie bialgebra with a Nijenhuis operator on a Lie algebra. The compatibility between these two structures leads to a rich framework for studying deformations and hierarchies of Lie bialgebras. As an important application, I will show how the underlying algebraic framework of the Euler-top system can be represented as a weak NL bialgebra, highlighting the significance of these structures in the context of integrable systems.

Abstract: In this talk, we briefly discuss the recent developments regarding the classification of gradings on simple Lie superalgebras. The interest on Lie superalgebras stems from theoretical physics, while the study of their algebraic properties was fostered by Kac and his classification of the simple finite-dimensional Lie superalgebras. The classification of the gradings on such algebras is an ongoing work, collecting the efforts of many people. We will give an overview of the progress in this direction, and then we will focus on the case of exceptional simple Lie superalgebras.

Abstract: In this talk, we abord the question of the existence of simple nilalgebras within the class of commutative power-associative algebras.  We give some equivalences that relate the existence of such algebras to the non-degenerate bilinear forms or faithful irreducible modules.

Abstract: The orbit method is a fundamental tool to study a finite-dimensional solvable Lie algebra g. It relates the annihilators of irreducible representations of g to the coadjoint orbits of g*. In my talk, I will extend this story to the Witt and Virasoro algebra infinite-dimensional Lie algebras, which are important in physics and representation theory. I will construct an induced module from an element of Vir* and show that its annihilator is a primitive ideal. I will also construct an algebra homomorphism that allows one to relate the orbit method for Vir to that of a finite-dimensional solvable Lie algebra.

Abstract: Starting from Kemer's work, the Grassmann algebra of an infinite-dimensional vector space played a key role in classical PI-Theory. Still, there are new branches of PI-Theory, involving PI-algebras with additional structures, where the Grassmann algebra either is a key ingredient as well, or leads to interesting identities. In particular, the differential identities of the Grassmann algebra under some derivation action will be presented.

Abstract: The Yang-Baxter equation is one of the fundamental equations occurring in statistical mechanics and quantum field theory. I will show that the diagonal mappings are bijections in any non-degenerate set-theoretical solution. This immediately implies that any non-degenerate solution is bijective and affirmatively answers the question stated by Cedo, Jespers, and Verwimp. I also prove that, for a subclass of solutions called permutational, one-sided non-degeneracy is sufficient for the diagonal to be invertible. This is joint work with Premysl Jedlicka (Czech University of Life Sciences).

Abstract: In this talk we are going to introduce the category of twisted relative Rota-Baxter operators in a braided monoidal setting together with a procedure for constructing examples of such structures based on idempotent Hopf algebra morphisms, and also we are going to prove that, under certain conditions, the following results hold: There exists an adjoint pair of functors between the category of Hopf trusses and the category of twisted relative Rota-Baxter operators. The previous adjunction induces a categorical equivalence between the category of Hopf trusses and the subcategory of invertible twisted relative Rota-Baxter operators.

Abstract: Double Poisson brackets, introduced by M. Van den Bergh in 2004, are noncommutative analogs of the usual Poisson brackets in the sense of the Kontsevich-Rosenberg principle: they induce Poisson structures on the space of N-dimensional representations of an associative algebra A  for any N. The problem of deformation quantization of double Poisson brackets was raised by D. Calaque in 2010, and had remained open since then. In the talk, I plan to present a possible answer to the question in the title. Namely, I will discuss a structure on an associative algebra A that induces a star-product under the representation functor and, therefore, according to the Kontsevich-Rosenberg principle, can be viewed as an analog of star-products in noncommutative geometry. If time permits, I will also discuss a way to invert the Kontsevich-Rosenberg principle by introducing the notion of a double algebra over an arbitrary operad. The talk is based on arXiv:2506.00699.

Abstract: When we have an action of sl_2 on a given structure, one may decompose it into its isotypic components. More concretely, if A is a finite-dimensional "algebra" with an operation m: A ⊗ A → A, one may decompose A as a direct sum A = ⊕ V_n ⊗ M_n, where V_n is the irreducible sl_2-module of highest weight n (and dimension n + 1), and M_n is just a "multiplicity" vector space. If the operation m is sl_2-linear, then, a priori, several restrictions for the operation m can be deduced, and algebraic identities (e.g. associativity, Jacobi identity, symmetry, antisymmetry, etc.) of A can be translated into operations and identities on the M_n spaces. In case the only isotypical components that appear are the trivial (V_0) and the adjoint (V_2), then the sl_2-structure is called very short. In case the only isotypical components that appear are the trivial (V_0), the adjoint (V_2), and the defining 2-dimensional representation (V = C^2 = V_1), then the sl_2-structure is called short. The case of very short sl(2) Lie algebras is a classical object studied by Tits and leads to Jordan algebras. There is also a kind of reciprocal knowledge like the TKK-construction (Tits-Kantor-Koecher): given a Jordan algebra, one can assign to it a natural (but not functorial) Lie algebra. The functoriality problem was solved by Allison and Gao; we call it the TAG construction. When the natural representation V also appears, Elduque and collaborators made the "translation" from Lie axioms into an object called a Jordan triple. If the algebra is a super Lie algebra, but very short, then both TKK and TAG constructions were generalized to the super case by Barbier and Shang. In this talk, I will show that the TKK and TAG constructions can be extended to the short super case. That is, one can make a construction beginning from a Jordan super triple (not just a Jordan algebra) and get a Lie superalgebra. In case the Jordan triple is an ordinary one, we get a reciprocal construction to Elduque’s. In case the Jordan triple is just a Jordan algebra, but super, we generalize Shang’s work on super Jordan algebras. In the process of doing this, by adapting to the short case an intrinsic description by Shang of the Jordan algebra associated to a very short Lie algebra, we discover two different possible ternary Jordan structures on the "Jordan data" associated to a short Lie algebra: One was considered previously by Elduque et al. (in the non-super case), The other can be described in a simpler way using the intrinsic Lie-theoretic description, and it turns out that this second one is more suitable for the functorial generalization of the TKK and TAG constructions in the short (and super) setting. Joint work with Gonzalo Emanuel Gutierrez.

Abstract: In this talk, we will delve into the class of Lie algebras defined by quadratic relations, focusing on the explicit computation of their cohomology rings in specific cases. These algebras naturally arise in the broader context of positively graded Lie algebras, where they play a significant structural role. The theory of HNN-extensions plays a crucial role in this context, providing a powerful tool for decomposing quadratic Lie algebras into smaller components. Moreover, we will explore how HNN-extensions can be used to embed finitely presented positively graded Lie algebras into quadratic ones.

Abstract: Let g be a finite-dimensional simple Lie algebra, and consider the loop algebra Lg = g[t, t⁻¹] and the affine Lie algebra ĝ, which is an extension of Lg by a central element c. We investigate two-sided ideals in the universal enveloping algebra U(Lg). It is known that the rings U(ĝ)/(c − λ) are simple for any nonzero scalar λ, but the two-sided structure of U(Lg) = U(ĝ / (c)) is more complicated. We show that U(Lg) does not satisfy the ascending chain condition on two-sided ideals, but that the two-sided ideals still have a nice structure: there is a canonical collection of ideals Iₙ, parameterised by positive integers, so that any two-sided ideal of U(Lg) contains some Iₙ. The ideals Iₙ can be thought of as universal annihilators of classes of finite-dimensional representations of Lg. This is a preliminary report on joint work with Alexey Petukhov.

Abstract: Vertex algebras capture physicists' notion of OPEs in chiral CFTs, in complex dimension one. For various motivations, one would like to have analogs of vertex algebras in higher dimensions. Chiral algebras, in the sense of Beilinson-Drinfeld and Francis-Gaitsgory, provide a promising framework here, because they re-express the vertex algebra axioms (which are rather sui generis, and therefore hard to generalize) as something more recognizable (a chiral algebra is a Lie algebra, of a sort). I will review this, and then go on to introduce a certain concrete model of the unit chiral algebra in higher complex dimensions. We shall see that in going to higher dimensions, one naturally moves from Lie algebras to their homotopy analogs, L-infinity algebras, and from chiral algebras to homotopy chiral algebras in a sense recently introduced by Malikov-Schechtman. The main tool in the talk will be a new model -- the polysimplicial model -- of derived sections of the sheaf of functions on higher configuration spaces. The hope is that this model will prove well-adapted to doing concrete calculations. This is joint work in preparation with Zhengping Gui and Laura Felder.

Abstract: To any braid group B_n ​ there is an associated (Iwahori-)Hecke algebra H_q(n). Over time this algebra has shown to be as intriguing as B_n. For example, H_q(n) possesses a representation for which it is in a Schur–Weyl relation with U_q(sl_d). One possible interpretation of classical braid groups is as a fundamental group of the space of configurations of n distinct points in R^2. Taking this motion group perspective, it is natural to consider configurations of n unit circles S^1. This yields the so-called Loop Braid group. Damiani–Martin–Rowell associated an analogue of the Hecke algebra and made a conjecture on the dimension of this Loop Hecke algebra. In this talk we will firstly briefly introduce the mentioned objects and subsequently tell about how the above Schur–Weyl picture adapts to the Loop setting. In the last part of the talk we will discuss the simple representations and the Jacobson radical.

Abstract: Studying of the geometry of Schubert varieties for simple algebraic finite-dimensional complex groups is a classical topic in algebraic geometry. Tangent cones encode a lot of geometric information about singularity of Schubert varieties. One of the very important tool in investigation properties of tangent cones are Kostant--Kumar polynomials. I will discuss how this topics can be generalized to the case of Kac--Moody groups.

Abstract: Since the works of Patera-Zassenhaus (1989) and Bahturin-Sehgal-Zaicev (2001), the problem of classifying gradings by groups on various algebras has received much attention. There are typically two kinds of classification of gradings on a given algebra A: fine gradings up to equivalence or all G-gradings, for a fixed group G, up to isomorphism. These classifications are related, but it is not straightforward to pass from one to the other. In this talk, based on a recent paper with A. Elduque, we introduce a class of gradings, which we call almost fine, on a finite-dimensional algebra A over an algebraically closed field, such that every G-grading on A is obtained from an almost fine grading in an essentially unique way (which is not the case with fine gradings). For abelian groups, we give a method of obtaining all almost fine gradings if fine gradings are known. If time permits, we will illustrate this approach in the case of simple Lie algebras in characteristic 0: to any abelian group grading with nonzero identity component, we attach a (possibly nonreduced) root system Φ and construct an adapted Φ-grading.

Abstract: Nichols (small shuffle) algebras are a family of graded algebras including the symmetric algebras, the exterior algebras, the positive part of quantized enveloping algebras.  They are defined by generators and relations that depend on a vector space V and a solution of the braid equation on V x V. A subclass of them, which is relevant for the classification program of finite-dimensional Hopf algebras developed by Andruskiewitsch and Schneider, consists of those for which the solution of the braid equation stems from a suitable graded representation of a finite group G. A folklore conjecture states that there are no non-trivial finite-dimensional Nichols algebras in this family if G is a non-abelian simple group. I will report on progress on this conjecture, based on a collaboration with N. Andruskiewitsch, G. García and M. Costantini.

Abstract: Primitive axial algebras of Jordan type half were introduced by Jonathan Hall, Felix Rehren and Sergey Shpectorov in 2015, generalizing Jordan algebras by requiring that their idempotents satisfy the Peirce decomposition. More specifically, primitive axial algebras of Jordan type ½ are commutative non-associative algebras generated by idempotents a such that their multiplication operators Lₐ are diagonalizable with eigenvalues {1,0,½}, such that the fusion laws V₁ = ⟨a⟩, V₀² ⊆ V₀, V₀V_{½} ⊆ V_{½} and V_{½}² ⊆ V₀ ⊕ V₁ hold, where V_λ is the λ-eigenspace of Lₐ. The most well-known examples of this class of algebras are either Jordan algebras or Matsuo algebras, certain non-associative algebras related to 3-transposition groups that Atsushi Matsuo discovered while studying vertex operator algebras. In this talk, we will sketch how one can distinguish these two classes in terms of their automorphism groups. In particular, primitive axial algebras of Jordan type half with large automorphism groups are automatically Jordan while the automorphism groups of non-Jordan Matsuo algebras are usually finite, with one infinite family of exceptions.

Abstract: Since their invention by Fomin-Zelevinsky in 2002, cluster algebras have shown up in an ever growing array of subjects in mathematics (and in physics). In this talk, we will approach their theory starting from elementary examples. More precisely, we will see how the remarkable integrality properties of the Coxeter-Conway friezes and the Somos sequence find a beautiful unification and generalization in Fomin-Zelevinsky's definition of cluster variables and their Laurent phenomenon theorem. Motivated by the periodicity of Coxeter-Conway friezes, we will conclude with a general periodicity theorem, whose proof is based on the interaction between discrete dynamical systems and quiver representations through the combinatorial framework of cluster algebras.

Abstract: It is well known from the PhD thesis of Jim Stasheff that the homotopy theory of associative algebras is encoded by homotopy associative algebras, aka A_infini-algebras, since this latter notion carries infini-morphisms and satisfies a homotopy transfer theorem, for instance. A_infini-algebra structures encode the topological data of a space on the level of cochain complexes. When one wants to encode more data, like the Poincaré duality of manifolds, string topology, or non-commutative derived geometry, then one has to consider further structural operations, like symmetric bitensors or double brackets. The purpose of this talk will be to present the associated new types of homotopy bialgebras, to explain their relationship, and to show that they admit suitable homotopy properties like infini-morphisms and homotopy transfer theorem. To mention them, we will treat pre-Calabi—Yau algebras, homotopy double Poisson bialgebras, and homotopy infinitesimal balanced bialgebras. This is based on a joint work with Johan LERAY available at arXiv:2203.05062.

Abstract: I will present recent results on the existence of SKT and balanced metrics on solvable Lie algebras. The talk is based on joint papers with  Beatrice Brienza,  Asia Mainenti, and Fabio Paradiso.

Abstract: We discuss the class of Jordan operator algebras, including reporting on recent progress on their M-ideals (joint with M. Neal, A. Peralta, S. Su). More generally we consider some nonassociative algebras motivated by Hilbert space operator algebraic theory and group representations.

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.