1st session
Chairman: Yunhe Sheng (Jilin University, China)
08.00 Bauyrzhan Sartayev (Narxoz University, Kazakhstan)
TITLE: On the Malcev classification for the variety of associative algebras
ABSTRACT: In this talk, we consider four types of subvarieties of the variety of associative algebras. We study these subvarieties from the point of view of operads and show their connections with well-known classes of algebras, such as dendriform algebras and noncommutative Novikov algebras. Also, we define the commutator and anti-commutator operations on these algebras and derive several identities satisfied by these operations. For the second and third types of associative algebras, we construct the bases for the corresponding free algebras. Finally, we prove that a free metabelian Lie algebra can be embedded into a free associative algebra of the second type.
09.00 Ilan Levin (Bar Ilan University, Israel)
TITLE: Measuring the associativity of loops
ABSTRACT: In group theory folklore, there is a well-known theorem: The probability that two randomly uniformly chosen elements commute in a non-abelian group G cannot exceed 5/8. The bound is attained by the Quaternion group Q_8. In this talk, we shall discuss a non-associative analog of the theorem. Namely, the probability of three randomly chosen elements associating is bounded by 43/64 in Moufang loops with nuclear commutators, with the bound attained by the Octonion loop O_16.
10.00 Pavel Kolesnikov (Sobolev Institute of Mathematics, Russia)
TITLE: Dendriform splitting and chiral algebras
ABSTRACT: Vertex algebras usually considered as a tool in mathematical physics (conformal field theory) or representation theory (infinite-dimensional Lie algebras and sporadic finite simple groups) may be thought of as of deeply generalized analogues of Poisson algebras. Namely, following B. Bakalov and V.G. Kac (2002), a vertex algebra is a breed of pre-Lie differential algebra and Lie conformal algebra structures. In this talk, we will observe several approaches to vertex algebras including the one via chiral operads. Within the latter approach, a vertex algebra is a morphism from the operad Lie to the chiral operad P^{ch}(V) constructed on a space V with a single linear operator. We apply this approach to get the class of dendriform split systems (preLie chiral, or pre-vertex algebras) and study the left adjoint functor to the forgetful functor from the category of pre-vertex algebras to preLie conformal algebras.
11.00 Coffee-Break
2nd session
Chairman: Bernhard Keller (Université Paris Cité, France)
12.00 Lucio Centrone (University of Bari, Italy)
13.00 Cindy Tsang (Ochanomizu University in Tokyo, Japan)
TITLE: Hopf--Galois structures of cyclic type on parallel extensions of prime power degree
ABSTRACT: Let L/K be a finite separable extension with Galois closure \widetilde{L}/K. We say that L'/K is parallel to L/K if L' is an immediate field of \widetilde{L}/K and [\widetilde{L}:K]=[L:K]. Note that this notion is not symmetric. We are interested in comparing the Hopf-Galois structures on L'/K with those on L/K. In this talk, I will first explain how this problem reduces to a completely group-theoretic problem that involves the study of transitive subgroups of the holomorph. After that, I will report on some new results when the type of the Hopf-Galois structures is cyclic, and the degree of the extension is a prime power. This is joint work with Andrew Darlington.
14.00 Anastasia Doikou (Heriot-Watt University, UK)
TITLE: Combinatorial Drinfel'd twists & the Yang-Baxter equation
ABSTRACT: We introduce the special set-theoretic Yang-Baxter algebra and show that it is a Hopf algebra subject to certain conditions. The associated universal R-matrix is also obtained via an admissible Drinfel'd twist. The structure of braces emerges naturally in this context by requiring the special set-theoretic Yang-Baxter algebra to be a Hopf algebra and a quasi-triangular bialgebra after twisting. The fundamental representation of the universal R-matrix yields the familiar involutive set-theoretic (combinatorial) solution of the Yang-Baxter equation. We also introduce rack Hopf-like algebras and obtain rack and quandle solutions of the YBE. We show that the same combinatorial twist cane be used to produce non-involutive set-theoretic solutions.
15.00 Coffee-Break
3rd session
Chairman: David Towers (Lancaster University, UK)
16.00 Marzia Mazzotta (University of Salento, Italy)
TITLE: Associative Pentagon Algebras
ABSTRACT: A set-theoretical solution of the Pentagon Equation can be described in terms of a pentagon algebra (S, +, *), namely, a set equipped with two binary operations where (S,+) forms a semigroup and the operations + and * satisfy additional compatibility conditions arising from the Pentagon Equation. In this talk, we introduce an algebraic framework for studying such structures and outline several strategies for constructing and classifying their solutions. Our attention will be devoted in particular to the case in which the second operation * is also associative. We present characterizations of these algebras, together with structural consequences for the interaction between the two operations. Moreover, we will present some recent results from an ongoing collaboration with Agata Pilitowska and Arne Van Antwerpen.
17.00 Maxime Fairon (Université Bourgogne Europe, France)
TITLE: Double Poisson algebra cohomology
ABSTRACT: The structure of a double Poisson algebra (as introduced by Van den Bergh) induces a structure of Poisson algebra on each of its representation algebras. For these, a cohomology theory (initiated by Pichereau and Van de Weyer) can be constructed under mild conditions, and it gets mapped to the usual Poisson cohomology of the corresponding representation algebras. My aim is to recall this original construction, and then I will explain a far-reaching generalisation leading to several other cohomology theories of a similar nature. This talk is meant to be an overview of Part 1 of arXiv:2509.21232 (joint with Daniele Valeri).
18.00 Artem Lopatin (University of Campinas, Brazil)
19.00 Coffee-Break
Next talks will be given on
European Non-Associative Algebra Seminar
[Every Monday, 15:00 (+0 UTC) ]