Program
Introductory courses (5 x 1h talks)
Course 1
Introduction to algebraic deformations (Abdenacer Makhlouf)
The main purpose of this course is to provide a complete introduction to algebraic deformation theory, including formal deformations introduced by Gerstenhaber based on formal power series, global deformations based on any commutative algebra, following a general theory of Schlessinger and the recent developments in quantization deformation theory connected to the celebrated work of M. Kontsevich. The course gives detailed expositions of fundamental concepts along with an introduction to recent developments in the topics using these fundamental tools. In this course, we deal with methods from Algebra, geometry and category theory.
Course 2
Composition algebras (Alberto Elduque)
After reviewing the process of the construction of the complex numbers from the real numbers, this process will be iterated to produce the algebra of quaternions. Applications of this algebra to the study of rotations in the Euclidean spaces of dimension 3 and 4 will be considered. A further iteration provides the algebra of octonions. These algebras will be generalized over arbitrary fields. Symmetric composition algebras will be introduced too, and several connections of composition algebras with exceptional simple Lie and Jordan algebras will be given.
Course 3
The aim of this series of lectures is to give an introduction to the cohomology of Leibniz algebras. The origin of the subject is the search of Jean-Louis Loday for the obstruction to the periodicity in algebraic K-theory, see [L1], [L2]. Leibniz algebras are a class of non-associative algebras where the product is written as a bracket which then satisfies the (left) self-derivation property
[x, [y, z]] = [[x, y], z] + [y, [x, z]],
see [F] for a discussion of basic properties of Leibniz algebras from the non-associative point of view. Clearly Lie algebras are special cases of Leibniz algebras (where the bracket is anticommutative). In the lectures, we will introduce the natural cohomology theory for Leibniz algebras (due to Loday and Pirashvili), discuss the significance of low degree cohomology spaces and set forth tools for the computation of this cohomology, see [P]. These tools include spectral sequences which are parallel to the Hochschild-Serre spectral sequence in Chevalley-Eilenberg cohomology for Lie algebras and which lead to vanishing theorems for semisimple, nilpotent and supersolvable Leibniz algebras (see [FW], joint work together with Jörg Feldvoss (University of South Alabama, Mobile, USA)).
[F] J. Feldvoss: Leibniz algebras as non-associative algebras, in: Nonassociative Mathematics and Its Applications, Denver, CO, 2017 (eds. P. Vojtěchovský, M. R. Bremner, J. S. Carter, A. B. Evans, J. Huerta, M. K. Kinyon, G. E. Moorhouse, J. D. H. Smith), Contemp. Math., vol. 721, Amer. Math. Soc., Providence, RI, 2019, pp. 115–149.
[FW ] J. Feldvoss, F. Wagemann: On Leibniz cohomology. arXiv:1902.06128
[L1] J.-L. Loday: Une version non commutative des algèbres de Lie: les algèbres de Leibniz. [A noncommutative version of Lie algebras: Leibniz algebras] R.C.P. 25, Vol. 44 (French) (Strasbourg, 1992), 127–151, Pr ́epubl. Inst. Rech. Math. Av., 1993/41, Univ. Louis Pasteur, Strasbourg, 1993.
[L2] J.-L. Loday: Algebraic K-theory and the conjectural Leibniz K-theory. Special issue in honor of Hyman Bass on his seventieth birthday. Part II. K-Theory 30 (2003), no. 2, 105–127.
[P] T. Pirashvili: On Leibniz homology, Ann. Inst. Fourier (Grenoble) 44 (1994), no. 2, 401–411.
Course 4
Axial algebras (Justin McInroy)
We will give an introduction to axial algebras and discuss some recent developments. Axial algebras are a new class of non-associative algebra, introduced recently by Hall, Rehren and Shpectorov, which generalise some properties found in vertex operator algebras and the Griess algebra and have a strong connection to groups. Axial algebras are generated by axes which are idempotents whose adjoint action decomposes the algebra as a direct sum of eigenspaces. The multiplication of eigenvectors is controlled by a so-called fusion law. When this is graded, it leads naturally to a subgroup of automorphisms of the algebra called the Miyamoto group. The prototypical example is the Griess algebra which has the Monster sporadic group as its Miyamoto group. Other examples include a wide number of Jordan algebras and also Matsuo algebras, which are defined from 3-transposition groups.
Course 5
Post-Lie algebras (Karel Dekimpe)
A post-Lie algebra (V, . , { , }) is a vector space over a field k equipped with two k-bilinear operators x. y and {x,y} such that
(V, { , }) is a Lie algebra.
{x,y}.z = (y.x).z - y.(x.z) - (x.y).z + x.(y.z).
x .{y,z} = {x.y,z} + {y,x.z}.
Any post-Lie algebra (V, . ,{,}) has a second associated Lie bracket [ , ] determined by
[x,y]= x.y - y.x + {x,y}.
We let (g, [ , ]) = (V, [ , ]) and (n, { , }) = (V, . , { , }) be the two Lie algebras with the same underlying vector space V and we say that "." is a post-Lie algebra structure on the pair (g, n).
In this course we will
Introduce post-Lie algebras and post-Lie algebra structures and prove some basic properties.
Explain the geometric motivation (i.e. Nil-affine actions) for studying post-Lie algebra structures.
Pay attention to the special cases of left-symmetric structures and LR-structures (or bi-commutative structures).
Investigate which algebraic conditions imply whether or not a pair of Lie algebras (g, n) can admit a post-Lie algebra structure. By ``algebraic condition'' we mean properties as ``abelian'', ``nilpotent'', ``solvable'', ``simple'', ``semisimple'', ``reductive'', .... By studying several of these cases, we can illustrate several techniques used to investigate specific types of Lie algebras.
Course 6
Evolution Algebras (Yolanda Cabrera Casado)
Introduction: a type of genetic algebras: evolution algebras. Recently a new type of genetic algebras, denominated evolution algebras, have emerged to enlighten the study of Non-Mendelian genetics, which is the basic language of the molecular Biology. In particular, evolution algebras can be applied to the inheritance of organelle genes. Basic facts about evolution algebras. We give the essential definitions about this type of non-associative algebras. We study the notions of evolution subalgebra and ideal. We show some properties about evolution algebras including and non-degeneracy. Moreover, we associate a graph to any evolution algebra which will play a fundamental role to describe the structure of the algebra. Decomposition of an evolution algebra.
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Research talks (1h talks)
Factoring Octonion Polynomials (Adam Chapman)
The decomposition of any mathematical object into irreducible building blocks is a natural research goal, and indeed this is what one often does first when one wishes to study a certain mathematical object. Such examples include prime factorization of integers, factorization of polynomials into irreducible (or linear) polynomials, and decomposition of semi-simple algebras into simple components. In this talk we shall discuss the decomposition of polynomials over the octonion algebra into linear factors. As with polynomials over fields, the roots of these polynomials play a role in this game, and therefore we shall also explain how to find them, and their relations to the left and right eigenvalues of the companion matrix.
Addition chains and numerical invariants of non-associative algebras (Alexander Guterman)
An addition chain, some times called an additive chain, for an integer $n$ is defined to be a sequence of integers 1=a_0<a_1<...<a_r=n, such that for each i>0 the equality a_i=a_j+a_k holds for some k<=j<i. For example, the sequence of Fibonacci numbers and the set of powers of two are classical examples of addition chains. These sequences are under consideration since the ancient times and used in the number of applications, for example, in algorithms of fast exponentiation. The minimal length of an addition chain for a given n is denoted by l(n). Theory of addition chains contains a lot of open problems and conjectures, for example, the famous Scholz–Brauer conjecture asserts that l(2n-1) <= n+l(n)-1. We show that addition chains are connected with numerical characteristics of non-associative algebras, in particular, their length and characteristic sequences. Several concrete applications of this theory to investigations of non--associative algebras, their properties and numerical invariants will be described in the talk. The talk is based on a joint research with Dmitry Kudryavtsev (Lomonosov Moscow State University, Moscow, Russia).
A new invariant for finite dimensional Leibniz/Lie algebras (Ana Agore)
For an n-dimensional Leibniz/Lie algebra h over a field k we introduce a new invariant A(h), called the universal algebra of h, as a quotient of the polynomial algebra k[X_ij], i, j = 1, ..., n through an ideal generated by n^3 polynomials. Furthermore, A(h) admits a unique bialgebra structure which makes it an initial object among all bialgebras coacting on h through a Leibniz/Lie algebra homomorphism. The bialgebra A(h) is the key object in approaching three classical and open problems in Lie algebra theory. First, we prove that the automorphisms group of h is isomorphic to the group of all invertible group-like elements of the finite dual of A(h). Secondly, for an abelian group G, we show that there exists a bijection between the set of all G-gradings on h and the set of all bialgebra homomorphisms from A(h) to k[G]. Finally, for a finite group G, we prove that the set of all actions as automorphisms of G on h is parameterized by the set of all bialgebra homomorphisms from A(h) to k[G]*. A(h) is also used to prove that there exists a universal commutative Hopf algebra associated to any finite dimensional Leibniz algebra h. The talk is based on a joint work with Gigel Militaru [1]
[1] A.L. Agore, G.Militaru - A new invariant for finite dimensional Leibniz/Lie algebras, Journal of Algebra 562 (2020), 390-409.
Transposed Poisson algebras (Chengming Bai)
We introduce a notion of transposed Poisson algebra which is a dual notion of the Poisson algebra by exchanging the roles of the two binary operations in the Leibniz rule defining the Poisson algebra. We interpret the close relationships between it and some structures such as Novikov-Poisson and pre-Lie Poisson algebras including the example given by a commutative associative algebra with a derivation, and 3-Lie algebras.
Algebras with slowly growing length (Dmitry Kudryavtsev)
Length is a combinatorial structural invariant of algebras, until recently studied predominantly for associative algebras. This talk will focus on the newfound results dealing with bounds on lengths of important non-associative algebras, such as Lie, Leibinz, Zinbiel and Novikov algebras. The talk is based on a joint research with Alexander Guterman (Lomonosov Moscow State University, Moscow, Russia).
Genetic algebras (Farrukh Mukhamedov)
In this talk we provide an introduction to genetic algebras and their relation to dynamical system will be discussed as well. A review on the basics of genetic algebras will be given. We will study certain algebraic properties of genetic algebras, and moreover, some open problems will be provided.
Frobenius and contact Lie algebras (Gil Salgado)
In this talk we will show the deep relationship between Frobenius Lie algebras and contact Lie algebras. We start reviewing the "double extension" process defined by Kac, which allows constructing a new Frobenius (resp. contact) Lie algebra whether we start with a Frobenius (resp. contact) Lie algebra. We answer in a positive way the following related question, given a Frobenius (resp. contact) Lie algebra: When there exists a codimension one contact (resp. Frobenius) ideal? As a final point, I will show that any bi-parabolic (or seaweed) index one Lie algebra must be a contact Lie algebra.
Local derivations and automorphisms on non associative algebras (Karimbergen Kudaybergenov)
In this talk we shall present some recent results about local derivations and automorphisms on non associative algebras (semisimple Lie algebras, nilpotent Lie algebras, Malcev algebras).
Radical rings, braces and the Yang-Baxter equation (Leandro Vendramin)
Braces were introduced by Rump to study non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation. To study non-involutive solutions one needs skew braces, a non-commutative analog of braces. In this talk, we discuss the basic properties of skew braces and how these structures are related to the Yang-Baxter equation. We also discuss intriguing connections between skew braces and structures appearing in non-commutative ring theory.
Perspectives in Specht's problem (Lucio Centrone)
We will summarize techniques and results (old and new) about the well known Specht problem. Therein, we will give an overview toward new frontiers about the problem.
Lie Automorphisms of Incidence Algebras (Mykola Khrypchenko)
It follows from the result by Đoković [1] that each Lie automorphism of the upper triangular matrix algebra T_n(K) is the sum of either an automorphism or the negative of an anti-automorphism and the Lie automorphism induced by a central-valued map annihilating all the commutators. It turns out that in the case of incidence algebras I(X,K) (which generalize T_n(K)) the structure of Lie automorphisms is more complicated. We give a full description of Lie automorphisms of the incidence algebra I(X,K) of a finite connected poset X over a field K. In particular, we show that they are in general not proper. This is a joint work with Ednei A. Santulo Jr and Érica Z. Fornaroli (Universidade Estadual de Maringá, Brazil).
[1] Đoković , D., Automorphisms of the Lie algebra of upper triangular matrices over a connected commutative ring, J. Algebra, 170, 1 (1994), 101-110.
Algebras of length one (Olga Markova)
By the length of a finite system of generators for a finite-dimensional (not necessarily associative) algebra over a field we mean the least positive integer k such that the products of length not exceeding k span this algebra (as a vector space). The maximum length for the systems of generators of an algebra is referred to as the length of the algebra. The length evaluation can be a difficult problem, for example, the length of the full matrix algebra is unknown (Paz’s Problem, 1984). The study of algebras whose length is close to the minimal value has interest in the context of computational procedures. In the talk we present a complete characterization of length one algebras over arbitrary fields in terms of a basis with a known multiplication table. In particular, we describe Jordan and flexible algebras of length one. The talk is based on a joint research with C. Martinez (University of Oviedo, Spain) and R.L. Rodrigues (University of São Paulo, Brazil).
On the lattice of restricted subalgebras of a restricted Lie algebra (Pilar Páez-Guillán)
We will discuss how a restricted Lie algebra is affected by the structure of the lattice of its restricted subalgebras. In particular, we will study some latice-theoretic properties such as being distributive, Boolean, atomistic and dually atomistic, or upper and lower semimodular. This talk is based on joint works with Nicola Maletesta, Salvatore Siciliano and David Towers.
Algebraic and geometric classification of non-associative algebras (Renato Fehlberg Júnior)
Abstract: In this talk we will present the results obtained in the classification of 5 dimensional non 2-step antiassociatives algebras and 5 dimensional Zinbiel algebras. The method for the algebraic classification is based on the calculation of central extensions of nilpotent algebras of smaller dimensions from the same variety. With this classification, in the geometric case, we compute degenerations of these algebras. This talk is based on joint works with Ivan Kaygorodov, María Alejandra Alvarez and Crislaine Kuster.
Quadratic structures on flat pseudo-Euclidean Lie algebras (Said Benayadi)
A flat pseudo-Euclidean Lie algebra (G, <,>) is a real Lie algebra G endowed with a nondegenerate symmetric bilinear form <,> such that the Levi-Civita product defined by, for any x,y,z\in G,
2<x.y,z>=<[x,y],z>-<[y,z],x>+<[z,x],y>,
is left-symmetric. That is, for any x,y,z \in G, ass(x,y,z)=ass(y,x,z), where ass(x,y,z)=(x.y).z-x.(y.z). In geometrical terms, a flat pseudo-Euclidean Lie algebra is a Lie algebra of a Lie group endowed by a left-invariant pseudo-Riemannian metric of vanishing curvature. recall that, the curvature of a pseudo-Riemannian Lie group (G, <,>) is defined by, for any x,y\in G
R(x,y)=L_{[x,y]}-[L_x,L_y], where L_x(y)=x.y. If we denote R_x(y)=y.x then L_x-R_x=ad_x and L_x is skew-symmetric.
A quadratic structure on a Lie algebra G is a nondegenerate symmetric bilinear form k such that k([x,y],z)+k(y,[x,z])=0, for any x,y,z \in G. In this talk we will give several interesting examples of Lie algebras which admit at the same time flat pseudo-Euclidean metric <,> and quadratic structure k. Such Lie algebra is called flat quadratic Lie algebra and denoted by (G,<,>,k). In this talk we study Lie algebras which admit at the same time flat pseudo-Euclidean metric <,> and quadratic structure k. Such Lie algebra is called flat quadratic Lie algebra and denoted by (G,<,>,k). Afterwards we will present some results on the structure of these Lie algebras. This is a joint work with Hicham Lebzioui.
Identities in group algebras, enveloping algebras and related structures (Salvatore Siciliano)
We discuss the main results about polynomial identities in group algebras, ordinary and restricted enveloping algebras, smash products, and symmetric Poisson algebras, with special attention to Lie identities satisfied by these algebraic structures.
A class of nonassociative algebras and their applications (Susanne Pumpluen)
Starting from a skew polynomial ring, we define a class of unital nonassociative algebras introduced by Petit in 1966 (but largely ignored so far), and discuss their structure and applications to coding theory. The algebras can be seen as canonical generalizations of the classical central simple algebras, like quaternion algebras, and some can be used surprisingly similar to their classical "cousins" in applications. For instance, these algebras can be employed to systematically build fast-decodable space-time block codes used for wireless digital data transmission, e.g. in mobile phones, laptops or portable TVs.
Zero divisor graphs of real Cayley-Dickson algebras (Svetlana Zhilina)
Due to the non-alternativity of Cayley-Dickson algebras with dimension at least 16, there appear zero divisors which are hard to study and to classify, except for some particular cases. At present most of the authors restrict their attention to the algebras of the main sequence where all parameters which determine the Cayley-Dickson construction are assumed to be equal to -1. We mention in particular the works by Moreno and Biss, Dugger, and Isaksen. Moreno's key idea was to study doubly alternative zero divisors, that is, such elements that their components are both alternative elements of the previous algebra. This notion was then extended to the Cayley-Dickson split-algebras and led to similar results. The talk is devoted to zero divisors in arbitrary real Cayley-Dickson algebras whose components satisfy some additional conditions on the norm and alternativity. We are interested in the patterns which they form in orthogonality and zero divisor graphs, and these patterns appear to be hexagonal. In the case of the algebras of the main sequence, these hexagons can be extended to the so-called double hexagons. The vertices of a double hexagon have a convenient multiplication table which has a block structure. The talk is based on a joint research with Alexander Guterman (Lomonosov Moscow State University, Moscow, Russia).
Upper triangular matrices: gradings, involutions and images of polynomials (Thiago Castilho de Mello)
In this talk we present some recent results about the algebra of n x n upper triangular matrices over an infinite field F. In the first part of the talk, we present necessary and sufficient conditions on a grading of UT_n so that it admits a homogeneous involution. We show that when this happens, all such involutions are of the same type. In particular, if UT_n admits a graded involution, all homogeneous involutions are graded. In the second part of the talk, we present a recent result about image of multilinear polynomials on UT_n and show that the analogous of the Lvov-Kaplansky conjecture for UT_n is true.
Structure of local Weyl modules for map superalgebras (Tiago Macedo)
In this talk, we will present some recent results about the structure of local Weyl modules for map superalgebras. More specificaly, we will present character and supercharacter formulas, basis for local Weyl modules for map superalgebras over sl(1,2), as well as a certain filtration by Chari-Venkatesh modules. These are results of a joint work with M. Brito and L. Calixto.