1st session:
Chairman: Shavkat Ayupov (Institute of Mathematics, Uzbekistan)
08.00 Zhanqiang Bai (Soochow University, China)
Gelfand-Kirillov dimensions of highest weight modules of simple Lie algebras [video, youtube] [pdf, slides]
Gelfand-Kirillov dimension is an important invariant, that was introduced by Gelfand and Kirillov in the 1960s. This invariant usually can measure the size of the infinite-dimensional algebraic structures. In this talk, by using Lusztig's a-function and based on our previous work, we will give an algorithm to compute the Gelfand-Kirillov dimensions of highest weight modules of exceptional type Lie algebras.
09.00 Cindy Tsang (Ochanomizu University in Tokyo, Japan)
On Grün's lemma for perfect skew braces [video, youtube] [pdf, slides]
The well-known Grün’s lemma in group theory states that the quotient of a perfect group by its center is always centerless. In this talk, we shall consider its analog in the setting of skew brace, an algebraic structure that was introduced in the study of set-theoretic solutions to the Yang-Baxter equation. Here we shall use the annihilator of a skew brace as an analog of the center of a group. Our main result is that the analog of Grün’s lemma always holds for two-sided perfect skew braces but fails in general.
10.00 Lucio Centrone (University of Bari, Italy)
On geometries arising from varieties of algebras [video, youtube]
We will construct geometric objects via varieties of algebras and we shall see how they interplay in the light of their polynomial identities.
11.00 Coffee-Break
2nd session:
Chairman: Uzi Vishne (Bar Ilan University, Israel)
12.00 Kurusch Ebrahimi-Fard (Norwegian University of Science and Technology, Norway)
A post-group theoretic perspective on the operator-valued S-transform in free probability [video, youtube]
We discuss the algebraic structure underlying Voiculescu's S-transform in operator-valued free probability. It is shown how its twisted factorisation property gives rise to post-groups, crossed morphisms, as well as pre- and post-Lie algebras. Based on joint work with T. Ringeard (arXiv:2402.16450).
13.00 Mahender Singh (IISER Mohali, India)
Idempotents of quandle rings and application to knots [video, youtube] [pdf, slides]
Quandles are non-associative algebraic structures arising from the algebraic formulation of the Reidemeister moves of planar diagrams of knots. Quandle rings were introduced recently as analogues of group rings for quandles. In this talk, we will explore the idempotents of quandle rings and their connection to quandle coverings. We show that integral quandle rings of finite-type quandles, which are non-trivial coverings of well-behaved base quandles, possess infinitely many non-trivial idempotents, and offer a complete characterization of these idempotents. Additionally, we show that integral quandle rings of free quandles contain only trivial idempotents, thereby identifying an infinite family of quandles with this property. In terms of applications to knot theory, we present explicit examples of knots where coloring with idempotents yields stronger invariants compared to the traditional quandle coloring invariant.
14.00 Antonio Peralta (University of Granada, Spain)
Maps preserving λ-Aluthge transforms on product [video, youtube]
Given λ∈[0,1], the λ-Aluthge transform of an element a in a von Neumann algebra M is defined by Δ_λ(a)=∣a∣^{λ}u∣a∣^{1−λ}, where a = u|a| is the polar decomposition of a∈M. This talk will be devoted to survey some of the main conclusions on bijective maps between von Neumann algebras commuting with the λ-Aluthge transform on products of the form ab, ab*, a∘b and a∘b*, where ∘ denotes the natural Jordan product. We shall show that all these maps are naturally linked to the Jordan structure of the von Neumann algebras. We shall also see how these problems are naturally connected with those classical studies by J. Hakeda and K. Saito on linear bijections between von Neumann algebras preserving products of the form ab and a∘b.
15.00 Coffee-Break
3rd session:
Chairman: Maxime Fairon (University of Bourgogne, France)
16.00 Valeriy Bardakov (Sobolev Institute of Mathematics, Russia)
Rota-Baxter operators on groups, ranks, and algebras [video, youtube] [pdf, slides]
Rota-Baxter operators RB-operators for commutative algebras were introduced by Baxter in 1960. Since then, the theory of Rota-Baxter operators has undergone extensive development by various authors in different fields of mathematics. In 2021 L. Guo, H. Lang, Y. Sheng defined a Rota-Baxter operator on groups and proved that if G is a Lie group and B : G→G is a Rota-Baxter operator, then the tangent map B at identity is a Rota-Baxter operator on the Lie algebra of G. In 2024 V.G. Bardakov and V.A. Bovdi introduced Rota-Baxter operators on racks and quandles. In my talk, I will give a survey of results that we have found with my colleagues during the last years and which are dedicated to RB-operators on groups, racks, and Hopf algebras.
17.00 Tatiana Gateva-Ivanova (American University in Bulgaria and IMI BAS, Sofia, Bulgaria)
Quadratic algebras and idempotent braided sets [video, youtube]
We study the Yang-Baxter algebras A(K,X,r) associated to finite set-theoretic solutions (X,r) of the braid relations. We introduce an equivalent set of quadratic relations R⊆G, where G is the reduced Grobner basis of R. We show that if (X,r) is left-nondegenerate and idempotent then R=G and the Yang-Baxter algebra is PBW. We use graphical methods to study the global dimension of n-generated PBW algebras in the general case and apply this to Yang-Baxter algebras in the left-nondegenerate idempotent case. We study the d-Veronese subalgebras for a class of quadratic algebras and use this to show that for (X,r) left-nondegenerate idempotent, the d-Veronese subalgebra A^{(d)} of A =A(K,X,r) can be identified with A(K,X,r^{(d)}), where (X,r^{(d)}) are left-nondegenerate idempotent solutions for all d≥2. We determined the Segre product in the left-nondegenerate idempotent setting. Our results apply to a previously studied class of `permutation idempotent' solutions, where we show that all their Yang-Baxter algebras for a given cardinality of X are isomorphic and are isomorphic to their d-Veronese subalgebras. In the linearised setting, we construct the Koszul dual of the Yang-Baxter algebra and the Nichols-Woronowicz algebra in the idempotent case, showing that the latter is quadratic. We also construct noncommutative differentials on some of these quadratic algebras. This talk is based on a joint work with Shahn Majid.
18.00 James Zhang (University of Washington, USA)
Poisson valuations and applications [video,youtube] [pdf, slides]
We introduce the notation of a Poisson valuation and use it to study automorphism, isomorphism, and embedding problems for several classes of Poisson algebras/fields. This is joint work with Hongdi Huang, Xin Tan, and Xingting Wang
19.00 Coffee-Break
Next talks will be given on
European Non-Associative Algebra Seminar
[Every Monday, 15:00 (+0 UTC) ]