Past talks - 2023

Abstract: Recently, the theory of semiassociative algebras and their Brauer monoid was introduced by Blachar, Haile,  Matri,  Rein, and  Vishne as a canonical generalization of the theory of associative central simple algebras and their Brauer group: together with the tensor product semiassociative algebras over a field form a monoid that contains the classical Brauer group as its unique maximal subgroup. We present classes of semiassociative algebras that are canonical generalizations of classes of certain central simple algebras and explore their behaviour in the Brauer monoid. Time permitting, we also discuss some - hopefully interesting - particularities of this newly defined Brauer monoid.

Abstract: We prove that a transposed Poisson algebra is simple if and only if its associated Lie bracket is simple. Consequently, any simple finite-dimensional transposed Poisson algebra over an algebraically closed field of characteristic zero is trivial. Similar results are obtained for transposed Poisson superalgebras. An example of a non-trivial simple finite-dimensional transposed Poisson algebra is constructed by studying the transposed Poisson structures on the modular Witt algebra. Furthermore, we show that the Kantor double of a transposed Poisson algebra is a Jordan superalgebra, that is, we prove that transposed Poisson algebras are Jordan brackets.  Additionally, a simplicity criterion for the Kantor double of a transposed Poisson algebra is obtained. 

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

Abstract: For every two-dimensional non-associative algebra A we describe generators for the algebra I(A) of polynomial invariants of several copies of A. We also discuss Artin's conjecture on invariants, which claims that I(A) is generated by the traces of operators of left and right multiplication over the algebra A. This is a joint work with María Alejandra Alvarez. 

Abstract: This is an overview of joint work with Alexandr N. Zubkov. We discuss linkage principles and blocks for general linear, ortho-symplectic, and periplectic supergroups over fields of positive characteristics. In the end, we describe the strong linkage principle and blocks for the queer supergroup Q(2).

Abstract: The length function of a non-associative algebra describes the guaranteed number of multiplications which will be sufficient to generate the whole algebra with its arbitrary generating set. In this talk we present a new method for length computation based on the sequence of differences between the dimensions of a certain sequence of subspaces. It allows us to compute the length of an Okubo algebra A over an arbitrary field. Namely, if A contains either nonzero idempotents or zero divisors, then its length equals four, and otherwise its length equals three. We also show that, in the latter case, A is generated by any two elements which do not belong to the same two-dimensional subalgebra. The talk is based on a joint work with Alexander Guterman.

Abstract: Solvability and nilpotence arise naturally from the commutator theory in congruence modular varieties. In the presence of associativity, the resulting concepts agree with the classical concepts of group theory. But the two kinds of solvability differ in loops (= not necessarily associative groups) and it is a difficult question to determine the boundary where the two theories coincide. I will review the general theory and report on recent results, particularly in Moufang loops. For instance, we will prove the Odd Order Theorem for Moufang loops for the stronger notion of solvability. This is joint work with Ales Drapal and David Stanovsky.

Abstract: It is well-known that strongly homotopy structures can be transferred over chain homotopy equivalences. Using the uniqueness results of Markl & Rogers we show that the transfers could be organized into a discrete Grothendieck bifibration. An immediate aplication is e.g. functoriality up to isotopy.

Abstract: The talk is a survey of our recent results on the homotopy theory of operated algebras such as Rota-Baxter associative (or Lie) algebras and differential associative (or Lie) algebras etc. We make explicit the Kozul dual homotopy cooperads and the minimal models of the operads governing these operated algebras. As a consequence the L-infinity structures on the deformation complexes are described as well.

Abstract: Lie algebras having bases of a special form (nice and beautiful bases) are considered. For nice bases, it is proved that in any nilpotent Lie algebra their number (up to equivalence) is finite. For some Lie algebras of low dimension, it is shown that, when passing from a complex Lie algebra to its realification, the property to have a beautiful basis is lost. Also nilpotent Lie algebras of dimensions less than 8 are considered.

Abstract: Given a directed graph, one can associate two algebraic entities: the Leavitt path algebra and the talented monoid. The Graded Classification conjecture states that the talented monoid could be a graded invariant for the Leavitt path algebra, i.e., isomorphism in the talented monoids reflects as graded equivalence in the category of graded modules over the Leavitt path algebra of the corresponding directed graphs. In this talk, we shall see confirmations of this invariance in the ideal structure of the talented monoid with the so-called Gelfand-Kirillov Dimension of the Leavitt path algebra. The last part of the talk is an affirmation of the Graded classification conjecture in the finite-dimensional case. This is a compilation of joint works with Roozbeh Hazrat, Wolfgang Bock, and Jocelyn P. Vilela.

Abstract: We present the recent results on Jordan quadruple systems. We show the Peirce decomposition for a Jordan quadruple system with respect to a quadripotent. We extend the notions of the orthogonality, primitivity, and minimality of tripotents in a Jordan triple system to that of quadripotents in a Jordan quadruple system. We show the relation between minimal and primitive quadripotents in a Jordan quadruple system. We also discuss the results on complemented subsystems of Jordan quadruple systems.

Abstract: We study the roots and critical points (i.e., points at which the formal derivative vanishes) of standard polynomials over Cayley-Dickson algebras. In the anisotropic real case, we prove that the critical points live inside the convex hull of the roots of the polynomial. The talk is based on joint work with Alexander Guterman, Solomon Vishkautsan and Svetlana Zhilina.

Abstract: We describe the defining identities of a variety of binary perm algebras which is a subvariety of the variety of alternative algebras. Moreover, we construct a basis of the free binary perm algebra. In addition, we describe the subalgebras of binary perm algebras under commutator which has a connection with Malcev algebras.

Abstract: A transposed Poisson algebra is a triple (L,⋅,[⋅,⋅]) consisting of a vector space L with two bilinear operations ⋅ and [⋅,⋅], such that (L,⋅) is a commutative associative algebra; (L,[⋅,⋅]) is a Lie algebra; the "transposed" Leibniz law holds: 2z⋅[x,y]=[z⋅x,y]+[x,z⋅y] for all x,y,z∈L. A transposed Poisson algebra structure on a Lie algebra (L,[⋅,⋅]) is a (commutative associative) multiplication ⋅ on L such that (L,⋅,[⋅,⋅]) is a transposed Poisson algebra. I will give an overview of my recent results in collaboration with Ivan Kaygorodov (Universidade da Beira Interior) on classification of transposed Poisson structures on several classes of Lie algebras.

Abstract: Let K<X> denote the free associative algebra generated by a finite set X with n elements over a field K of characteristic 0. Let Ip denote the two-sided associative ideal in K<X> generated by all commutators of length p, where p is an arbitrary positive integer greater than 1. The group GL(n) acts in a natural way on the quotient K<X>/Ip and the GL(n)-module structure of K<X>/Ip is known for p=2,3,4,5. In this talk, we give some results on the GL(n)-module structure of K<X>/Ip for any p. More precisely, we give a bound on the values of the highest weights of irreducible GL(n)-modules which appear in the decomposition of K<X>/Ip. We discuss also applications of these results related to the algebras of G-invariants in K<X>/Ip, where G is one of the classical GL(n)-subgroups SL(n), O(n), SO(n), or Sp(2k) (for n=2k).

Abstract: I will present some interesting computations concerning polynomial and rational invariants of nilpotent Lie algebras. I will say more about standard filiform Lie algebras which appear to have the highest level of complication among the small-dimensional algebras. I will outline an implementable algorithm for the computation of generators of the field of rational invariants.

Abstract: I will present the first notions concerning heaps and trusses. Heaps were introduced for the first time by H. Prüfer (1924) and R. Baer (1929). A heap is a pair (H, [−−−]) consisting of a set H and a ternary operation [−−−] : H × H × H → H, (x, y, z)  → [x, y, z], such that, for all v, w, x, y, z∈H,  [v, w, [x, y, z]] = [[v, w, x, ], y, z], [x, x, y] = y, and [y, x, x]= y. Truss is a much more recent algebraic structure (T. Brzeziński, 2019). A truss is a heap with a further associative binary operation, denoted by juxtaposition, which distributes over [−−−], that is, for all w, x, y, z∈T, w[x, y, z] = [wx, wy, wz], [x, y, z]w = [xw, yw, zw], and [x, y, z] =[z, y, x].

Abstract: Stable equivalences occur frequently in the representation theory of finite-dimensional algebras; however, these equivalences are poorly understood. An interesting class of stable equivalences is obtained by ‘gluing’ two idempotents. More precisely, let A be a finite-dimensional algebra with a simple projective module and a simple injective module. Assume that B is a subalgebra of A having the same Jacobson radical. Then B is constructed by identifying the two idempotents belonging to the simple projective module and to the simple injective module, respectively. In this talk we will compare the first Hochschild cohomology groups of finite-dimensional monomial algebras under gluing two arbitrary idempotents (hence not necessarily inducing a stable equivalence). As a corollary, we will show that stable equivalences obtained by gluing two idempotents provide `some functoriality' to the first Hochschild cohomology, that is, HH^1(A) is isomorphic to a quotient of HH^1(B).

Abstract: Let F be a finitely generated free algebra in a variety of algebras over a field of characteristic zero. A polynomial in F is called symmetric, if it is preserved under any permutation of the generators. The set S(F) of symmetric polynomials is a subalgebra of F. In this talk, we examine the algebras S(F), where F is the free metabelian associative, Lie, Leibniz, Poisson algebra or the free algebra generated by generic traceless matrices or the free algebra in the variety generated by Grassmann algebras.

Abstract: The whole structure given by the Hochschild cohomology and homology of an associative algebra A together with the cup and cap products, the Gerstenhaber bracket and the Connes differential is called the Tamarkin-Tsygan calculus. It is invariant under derived equivalence and if we can compute all these invariants provides a lot of information. The calculation of the whole Tamarkin-Tsygan calculus is very difficult and generally not even possible for particular algebras. However, there exist some calculations for individual algebras. The problem is, in general, that the minimal projective bimodule resolutions are difficult to find and even if one is able to compute such a resolution, it might be so complicated that the computation of the Tamarkin-Tsygan calculus is not within reach. For monomial algebras the minimal projective bimodule resolution is known and in the case of quadratic monomial algebras it is simple enough, to embark on the extensive calculations of the Tamarkin Tsygan calculus. Yet even for quadratic monomial algebras, the combinatorial level of the calculations is such that it is too complicated to calculate the whole calculus. On the other hand for gentle algebras, the additional constraints on their structure are such that the calculations become possible. We will focus on the concrete aspects of these calculations.

Abstract: In my talk I would like to discuss my joint articles with S. Sierra about the primitive ideals of universal enveloping U(W) and the symmetric algebra S(W) of Witt Lie algebra W and similar Lie algebras (including Virasoro Lie algebra). The key theorem in this setting is that every nontrivial quotient by a two-sided ideal of U(W) or S(W) has finite Gelfand-Kirillov dimension. Together with S. Sierra we enhanced this statement to the description of primitive Poisson ideals of S(W) in terms of certain points on the complex plane plus a few parameters attached to these points. In the end I will try to explain how all these concepts works for the ideals whose quotient has Gelfand-Kirillov dimension 2.

Abstract: In this talk we will present recent results on the category of finite-dimensional modules for map superalgebras. Firstly, we will show a new description of certain irreducible modules. Secondly, we will use this new description to extract homological properties of the category of finite-dimensional modules for map superalgebras, most importantly, its block decomposition.

Abstract: Diagram categories are a special kind of tensor categories that can be represented using diagrams. In this talk I will give an introduction to categories represented using Brauer diagrams. In particular I will explain the relation with the Brauer algebra and how the categorical framework can be applied to representation theory of the corresponding algebra.

Abstract: We say that an element x in a ring R is nilpotent last-regular if it is nilpotent of certain index n+1 and its last nonzero power x^n is regular von Neumann, i.e., there exists another element y∈R such that x^nyx^n=x^n. This type of elements naturally arise when studying certain inner derivations in the Lie algebra Skew(R,*) of a ring R with involution * whose indices of nilpotence differ when considering them acting as derivations on Skew(R,*) and on the whole R. When moving to the symmetric Martindale ring of quotients Q^s_m(R) of R we still obtain inner derivations with the same indices of nilpotence on Q^s_m(R) and on the skew-symmetric elements Skew(Q^s_m(R),*) of Q^s_m(R), but with the extra condition of being generated by a nilpotent last-regular element. This condition strongly determines the structure of Q^s_m(R) and of Skew(Q^s_m(R),*). We will review the Jordan canonical form of nilpotent last-regular elements and show how to get gradings in associative algebras (with and without involution) when they have such elements.

Abstract: The title matches that of a series of papers by various authors beginning in 1997, whose goal was the study and classification of such algebras over fields of positive characteristic. The original motivation came from group theory: the Leedham-Green and Newman coclass conjectures on pro-p groups from 1980 had all become theorems relatively recently, and subsequent results of Shalev and Zelmanov had raised interest in what one could say about Lie algebras of finite coclass. In positive characteristic, the simplest case of coclass one (i.e., 'Lie algebras of maximal class', also called 'filiform' in some quarters) appeared challenging even under the strong assumptions of those Lie algebras being infinite-dimensional and graded over the positive integers. I will review motivations and results of those studies, including some classifications obtained by Caranti, Newman, Vaughan-Lee. Then I will describe some generalizations recently established with three of my former PhD students.

Abstract: The problem of determining centralizers in the enveloping algebras of Lie algebras is considered from both the algebraic and analytical perspectives. Applications of the procedure, such as the decomposition problem of the enveloping algebra of a simple Lie algebra, the labeling problem and the construction of orthonormal bases of states are considered.

Abstract: Let L be a graded Lie algebra by integers with k-th homogenous space L_k where k are integers. An L-module V is called a smooth module if any vector in V can be annihilated by L_k for all sufficiently large k. Smooth modules for affine Kac-Moody algebras were introduced and studied by Kazhdan and Lusztig in 1993. I will show why this class of modules should be studied and what results are known now. An easy characterization for simple smooth modules for some Lie algebras will be provided.

Abstract: Nonzero real vectors of an affine Lie superalgebra act on a simple module either locally nilpotently or injectively. This helps us to divide simple finite weight modules over a twisted affine Lie superalgebra L into two subclasses called hybrid and tight. We will talk about the characterization as well as the classification problem of modules in each subclass. In this regard, the classification of bases of the root system of L is crucial. We will discuss how we can classify the bases and how we can use the obtained classification to study simple finite weight modules over L.

Abstract: We say that a Lie ring R is called a unique addition Lie ring, or briefly a UA-Lie ring, if any commutator-preserving bijection on R preserves the addition as well. We prove that any semisimple Lie algebra and any its parabolic subalgebra is a UA-Lie ring. Also we describe wide classes of solvable UA-Lie rings.

Abstract: Finite W-algebras were introduced by Premet in full generality, and they quickly became quite famous for their many applications in the representation theory of complex semisimple Lie algebras, especially the classification of primitive ideals. However, these algebras first appeared in the representation theory of Lie algebras associated to reductive groups in positive characteristic. In this talk I will survey the history of finite W-algebras in modular representation theory, and explain some of the contributions I have made to the field. The main applications in this talk will be the construction and classification of​ ''small'' modules of Lie algebras.

Abstract: To an n-dimensional representation of a finite dimensional Lie algebra one can naturally associate an algebra of equivariant polynomial maps from the space of m-tuples of elements of the Lie algebra into the space of n-by-n matrices. In the talk, we mainly deal with the special case of irreducible representations of the simple 3-dimensional complex Lie algebra, and discuss results on the generators of the corresponding associative algebra of concomitants as well as results on the quantitative behaviour of the identities of these representations.

Abstract: Rota-Baxter operators on Lie algebras were first studied by Belavin, Drinfeld and Semenov-Tian-Shansky as operator forms of the classical Yang-Baxter equation. Integrating the Rota-Baxter operators on Lie algebras, we introduce the notion of Rota-Baxter operators on Lie groups and more generally on groups. Then the factorization theorem can be achieved directly on groups. We introduce the notion of post-Lie groups, whose differentiations are post-Lie algebras. A Rota-Baxter operator on a group naturally induces a post-group. Post-groups are also closely related to operads, braces, Lie-Butcher groups and various structures.

Abstract: The so-called Lvov-Kaplansky Conjecture states that the image of a multilinear polynomial evaluated on the matrix algebra or order n is always a vector subspace. A solution to this problem is known only for n=2. In this talk we will present analogous conjectures for other associative and non-associative algebras and for graded algebras. Also, we will show how we can use gradings to present a statement equivalent to the Lvov-Kaplansky conjecture.

Abstract: The celebrated problem of deformation quantization discusses deformations of Poisson algebras into associative algebras, a question that is, in the end, motivated by quantum mechanics. I shall discuss this question and some of its generalisations from the purely algebraic point of view using the theory of operads. In particular, I shall show how to prove that there are, in a strict mathematical sense, only two meaningful deformation problems for Poisson algebras, namely deforming them in the class of all Poisson algebras or all associative algebras, and there is only one meaningful deformation problem for the so called almost Poisson algebras (also sometimes known as generic Poisson algebras), namely deforming them in the class of all almost Poisson algebras. For instance, this explains the existing body of work in the mathematical physics literature asserting that some classes of non-associative star products cannot be alternative, are always flexible etc.

Abstract: The pentagon equation classically originates from the field of Mathematical Physics. Our attention is placed on the study of set-theoretical solutions of this equation, namely, maps s: X×X → X×X given by s(x, y)=(xy, θ_x(y)), where X is a semigroup and θ_x: X → X is a map satisfying two laws.  In this talk, we give some recent descriptions of some classes of solutions achieved starting from particular semigroups. Into the specific, we provide a characterization of idempotent-invariant solutions on a Clifford semigroup X, that are those for which θ_a remains invariant on the set of idempotents E(X). In addition, we will focus on the classes of involutive and idempotent solutions, which are solutions fulfilling s^2=id_{X×X} and s^2=s, respectively. 

Abstract: The notion of a double Poisson bracket on an associative algebra was introduced by M. Van den Bergh in order to induce a (usual) Poisson bracket on the representation spaces of this algebra. I will start by reviewing the basics of this theory and its relation to other interesting operations, such as Leibniz brackets and H_0-Poisson structures. I will then explain some recent results and generalisations related to double Poisson brackets.

Abstract: Set-theoretic solution of the Yang-Baxter equation is a mapping r : X×X → X×X satisfying (r×1)(1×r)(r×1) = (1×r)(r×1)(1×r). A solution r : (x, y)  → (σ_x(y), τ_y (x)) is called non-degenerate if the mappings σ_x and τ_y are permutations, for all x, y ∈ X. A solution is called involutive if r^2 = 1. If (X, r) is a non-degenerate involutive solution (X, r) then the relation ∼ defined by x ∼ y ≡ σ_x = σ_y is a congruence. A solution is of multipermutation level 2 if |(X/ ∼)/ ∼ | = 1. In our talk, we focus on these solutions and we present several constructions and properties. 

Abstract: I will talk about the development of the theory of polynomial identities initiated by important questions such as  Burnside's  asking if  every finitely generated torsion group is finite. The field was enriched by contributions of many great mathematicians. Most notably Lie rings methods were developed and used by Zelmanov in the 1990s to give a  positive solution to the restricted Burnside problem which awarded him the Fields medal. It has been of great interest to expand the theory to other varieties of algebraic structures. In particular, I will review when a group algebra or enveloping algebra satisfy a polynomial identity.

Abstract: We introduce a generalization of K-k-Schur functions and k-Schur functions via the Pieri rule. Then we obtain the Murnaghan-Nakayama rule for the generalized functions. The rule is described explicitly in the cases of K-k-Schur functions and k-Schur functions, with concrete descriptions and algorithms for coefficients. Our work recovers the result of Bandlow, Schilling, and Zabrocki for k-Schur functions, and explains it as a degeneration of the rule for K-k-Schur functions. In particular, many other special cases promise to be detailed in the future.

Abstract: Let g be a 2n-dimensional solvable Lie algebra. A complex structure on g is an endomorphism J that satisfies J^2=-Id and N_J(X,Y)=0, for every X,Y ∈ g, being N_J(X,Y):=[X,Y]+J[JX,Y]+J[X,JY]-[JX,JY]. Suppose that g simultaneously admits a complex structure J and a symplectic structure ω.  Although J and ω are initially two unrelated structures, one can ask for an additional condition involving both of them. In this sense, the pair (J,ω) is said to be a complex symplectic structure if J is symmetric with respect to ω, in the sense that ω(JX,Y)=ω(X,JY), for every X,Y ∈ g. In this talk, we will present some methods to find certain types of solvable Lie algebras (such as nilpotent or almost Abelian) admitting complex symplectic structures.

Abstract: In classification problems over the real field R first Galois cohomology sets play an important role, as they often make it possible to classify the orbits of a real Lie group. In this talk we outline an algorithm to compute the first Galois cohomology set H^1(G,R) of a complex reductive algebraic group G defined over the real field R. The algorithm is in a large part based on computations in the Lie algebra of G. This is joint work with Mikhail Borovoi.

Abstract: Etale affine representations of Lie algebras and algebraic groups arise in the context of affine geometry on Lie groups, operad theory, deformation theory and Young-Baxter equations. For reductive groups, every etale affine representation is equivalent to a linear representation and we obtain a special case of a prehomogeneous representation. Such representations have been classified by Sato and Kimura in some cases. The induced representation on the Lie algebra level gives rise to a pre-Lie algebra structure on the Lie algebra g of G. For a Lie group G, a pre-Lie algebra structure on g corresponds to a left-invariant affine structure on G. This refers to a well-known question by John Milnor from 1977 on the existence of complete left-invariant affine structures on solvable Lie groups. We present results on the existence of etale affine representations of reductive groups and Lie algebras and discuss a related conjecture of V. Popov concerning flattenable groups and linearizable subgroups of the affine Cremona group.

Abstract: In 1985, K. Saito introduced elliptic root systems as root systems belonging to a real vector space F equipped with a symmetric bilinear form I with signature (l, 2, 0). Such root systems are studied in view of simply elliptic singularities which are surface singularities with a regular elliptic curve in its resolution. K. Saito had classified elliptic root systems R with its one-dimensional subspace G of the radical of I, in the case when R/G \subset F/G is a reduced affine root system. In our joint work with A. Fialowski and Y. Saito, we have completed its classification; we classified the pair (R,G) whose quotient R/G \subset F/G is a non-reduced affine root system. In this talk, we give an overview of elliptic root systems and describe some of the new root systems we have found.

Abstract: Axial algebras are a class of non-associative algebras with a strong natural link to groups and have recently received much attention. They are generated by axes which are semisimple idempotents whose eigenvectors multiply according to a so-called fusion law. Of primary interest are the axial algebras with the Monster type (α,β) fusion law, of which the Griess algebra (with the Monster as its automorphism group) is an important motivating example. By previous work of Yabe, and Franchi and Mainardis, any symmetric 2-generated axial algebra of Monster type (α,β) is either in one of several explicitly known families, or is a quotient of the infinite-dimensional Highwater algebra H, or its characteristic 5 cover H^. We complete this classification by explicitly describing the infinitely many ideals and thus quotients of the Highwater algebra (and its cover). As a consequence, we find that there exist 2-generated algebras of Monster type (α,β) with any number of axes (rather than just 1,2,3,4,5,6,∞ as we knew before) and of arbitrarily large finite dimension. In this talk, we will begin with a reminder of axial algebras which were introduced last week. This is joint work with: Clara Franchi, Catholic University of the Sacred Heart, Milan Mario Mainardis, University of Udine.

Abstract: Extending earlier work by Ivanov on Majorana algebras, axial algebras of Monster type were introduced in 2015 by Hall, Rehren and Shpectorov in order to axiomatise some key features of certain classes of algebras related to large families of finite simple groups, such as the weight-2 components of OZ-type vertex operator algebras, Jordan algebras, and Matsuo algebras. In this talk I'll review the definition of axial algebras and the major examples. Then I'll discuss the general classification problem of the 2-generated objects and, time permitting, show its applications in some special cases related to the Monster.

Abstract: Zinbiel algebras were introduced by Loday in 1995. They are the Koszul dual of Leibniz algebras and Lemaire proposed the name of Zinbiel, which is obtained by writing Leibniz backwards. In this talk I will introduce some of their main properties, including the fact that, over any field, they are nilpotent.

Abstract: Exceptional algebraic groups are intimately related to various classes of non-associative algebras: for example, octonion algebras are related to groups of type G2 and D4, and Albert algebras to groups of type F4 and E6. This can be used, on the one hand, to give concrete descriptions of homogeneous spaces under these groups and, on the other hand, to parametrize isotopes of these algebras using said homogeneous spaces. The key tools are provided by the machinery of torsors and faithfully flat descent, working over arbitrary commutative rings (sometimes assuming 2 and 3 to be invertible). I will talk about recent work where we do this from Brown algebras and their associated Freudenthal triple systems, whose automorphism groups are of type E6 and E7, respectively. I will hopefully be able to show how algebraic and geometric properties come together in this picture.

Abstract: Exceptional algebraic groups are intimately related to various classes of non-associative algebras: for example, octonion algebras are related to groups of type G2 and D4, and Albert algebras to groups of type F4 and E6. This can be used, on the one hand, to give concrete descriptions of homogeneous spaces under these groups and, on the other hand, to parametrize isotopes of these algebras using said homogeneous spaces. The key tools are provided by the machinery of torsors and faithfully flat descent, working over arbitrary commutative rings (sometimes assuming 2 and 3 to be invertible). I will talk about recent work where we do this from Brown algebras and their associated Freudenthal triple systems, whose automorphism groups are of type E6 and E7, respectively. I will hopefully be able to show how algebraic and geometric properties come together in this picture.

Abstract: After reviewing the basic definitions of tensor categories and the notion of semisimplification of symmetric tensor categories, it will be shown how the semisimplification of the category of representations of the cyclic group of order 3 over a field of characteristic 3 is naturally equivalent to the category of vector superspaces over this field. This allows to define a superalgebra starting with any algebra endowed with an order 3 automorphism. As a noteworthy example, the exceptional composition superalgebras will be obtained, in a systematic way, from the split octonion algebra.