1st session
Chairman: Shavkat Ayupov (Institute of Mathematics, Uzbekistan)
07.00 Askar Dzhumadil'daev (Institute of Mathematics and Mathematical Modeling, Kazakhstan)
Rota-Baxter algebras with non-zero weights [video, youtube] [slides, pdf]
Abstract: For an associative commutative algebra A with Rota-Baxter operator R : A → A with weight λ denote by AR an algebra with linear space A and multiplication a ◦ b = aR(b). Let AR^{−} and AR^{+} are algebra AR under Lie and Jordan commutators. If λ = 0, then the algebra AR = (A, ◦) is Zinbiel, AR^{+} is associative, and AR^{−} is Tortkara. We find polynomial identities of algebras AR, AR^{−} and AR^{+} in case λ≠0. We prove that AR^{−} is Tortkara. AR^{+} satisfies an identity of degree 5. In case λ≠ 0, the algebra AR is not associative-admissible.
08.00 Vesselin Drensky (Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Bulgaria)
The Specht problem for varieties of Z_n -graded Lie algebras in positive characteristic [video, youtube] [slides, pdf]
Abstract: Let K be a field of positive characteristic p and let UT_{p+1}(K) be the algebra of (p+1)×(p+1) upper triangular matrices. We construct three varieties of Z_{p+1}-graded Lie algebras which do not have a finite basis of their graded identities and satisfy the graded identities which in the case of infinite field define the variety generated by UT_{p+1}(K). The first variety contains the other two. The second one is locally finite. The third variety is generated by a finite dimensional algebra over an infinite field. These results are in the spirit of similar results obtained in the 1970s and 1980s for non-graded Lie algebras in positive characteristic. This is a joint project with Plamen Koshlukov and Daniela Martinez Correa.
09.00 Ievgen Makedonskyi (Beijing Institute of Mathematical Sciences and Applications, China)
Duality Theorems for current Lie algebras [video, youtube] [slides, pdf]
Abstract: We study some natural representations of current Lie algebras, called Weyl modules. They are natural analogues of irreducible representations of simple Lie algebras. There are several current analogues of classical theorems about Lie algebras where these modules «play role» of irreducible modules. In my talk, I will explain analogues of duality theorems, namely Peter-Weyl theorem, Schur-Weyl duality etc.
10.00 Coffee-Break
2nd session
Chairman: David Towers (Lancaster University, UK)
11.00 Pavel Kolesnikov (Sobolev Institute of Mathematics, Russia)
Derived nonassociative algebras: identities and embedding problem [video, youtube] [slides, pdf]
Abstract: Given a nonassociative algebra A with a derivation d, let us define its derived algebra as the same linear space A equipped with two operations of multiplication a<b = a d(b), a>b = d(a) b, for a,b in A. The purpose of this talk is to show how to derive the identities that hold on all such derived algebras provided that A ranges through a given variety of nonassociative algebras. (In particular, for the variety of associative and commutative algebras the result is very well known: the variety of Novikov algebras appears in this way.) We also study the natural embedding problem related to the functor transforming a differential algebra into its derived algebra. We state a sufficient condition that guarantees an affirmative answer to the embedding problem and show an example when the embedding problem has a negative solution.
12.00 Antonio Viruel (University of Malaga, Spain)
Permutation represention of finite groups via automorphisms of idempotent evolution algebras [video, youtube] [slides, pdf]
Abstract: In the wake of the influential work by Elduque-Labra, it is known that every finite dimensional evolution K-algebra X such that X^2=X, namely X is idempotent, has finite group of automorphisms. Building on this foundation, works of Costoya et al. show that given any finite group G, there exists an idempotent finite-dimensional evolution algebra X such that Aut(X)\cong G. Moreover, when the base field is sufficiently large in comparison to the group G, such an X can be selected to be simple. As a result, Sriwongsa-Zou propose that idempotent finite-dimensional evolution algebras can be classified based on the isomorphism type of their group of automorphisms and dimension. Within this context, we establish that the natural representation of highly transitive groups cannot be realized as the complete group of automorphisms of an idempotent finite-dimensional evolution algebra. For instance, for any sufficiently large integer n, there exists no evolution algebra X such that X^2=X, dim X=n, and Aut(X) is isomorphic to the alternating group A_n. However, we demonstrate that for any (not necessarily faithful) permutation representation p : G → S_n and any field K, there exists a finite-dimensional evolution K-algebra X such that X^2=X, Aut(X)\cong G$ and the induced representation given by the Aut(X)-action on the natural idempotents of X is p. This is a joint work with C. Costoya (U. Santiago Compostela) and Pedro Mayorga (U. Malaga).
13.00 Olivier Mathieu (University of Lyon, France)
On free Jordan Algebras [video, youtube]
Abstract: The free Jordan algebra J(m) on m generators is an elusive object. It has been determined when m=1 (folklore) and m=2 (Shirshov’s Theorem). Some partial informations are known in the case m=3, namely the space of Jordan polynomial with three variables which are linear on the last one. We will present two conjectures. Conjecture 1, which determines combinatorially the structure of the homogenous components of J(m) is elementary but mysterious. Then we present Conjecture 2 about Lie algebra cohomology of a class of free Lie algebras in a certain category. Conjecture 2 is natural, but not elementary. Our main result is that Conjecture 2 implies Conjecture 1. The proof, which is quite long, is based on the cyclicity of the Jordan operad. Conjecture 1 has been checked up to degree 15 for m=2, up to degree 7 for m=3 and up to degree 6 for m>3. In the case m=1, the conjecture is equivalent to Jacobi triple identity. For conjecture 2, the vanishing of the cohomology has been proved up to degree 3 using polynomial functors. In a recent work with J. Germoni, we found two new special identities in degree 8 and 4 variables. These identities have been checked by computer, but the interesting point is that they were predicted by our conjecture.
14.00 Coffee-Break
3rd session
Chairman: Vladimir Tkachev (Linköping University, Sweden)
15.00 Tom De Medts (Ghent University, Belgium)
Primitive axial algebras of Jordan type and 3-transposition groups [video, youtube] [slides, pdf]
Abstract: The classification of 3-transposition groups has a long history. In particular, it is a highly non-trivial fact that finitely generated 3-transposition groups are finite. We provide an alternative viewpoint on this question using the corresponding “Matsuo algebras”, a class of non-associative algebras. These are instances of primitive axial algebras of Jordan type. We prove that primitive 4-generated axial algebras of Jordan type are at most 81-dimensional (and this bound is sharp). This is joint work with Louis Rowen and Yoav Segev (to appear in Proc. AMS).
16.00 Sergey Shpectorov (University of Birmingham, UK)
Solid subalgebras in algebras of Jordan type half [video, youtube]
Abstract: Algebras of Jordan type η generalise in the axial context the class of Jordan algebras generated by primitive idempotents. In addition to these examples, arising for η=1/2, the class of algebras of Jordan type includes the Matsuo algebras, constructed in terms of 3-transposition groups for all values of η. Classification of algebras of Jordan type for η≠1/2 was completed by Hall, Rerhen and Shpectorov in 2015, with a correction by Hall, Segev and Shpectorov in 2018. The case of η=1/2 remains open. Among the known results about algebras of Jordan type half are the classification, in the above mentioned paper from 2015, of 2-generated algebras, the classification of 3-generated algebras by Gorshkov and Staroletov in 2020, and the recent (from 2023) result by De Medts, Rowen and Segev bounding the dimension of 4-generated algebras by 81. In the talk we will discuss another recent (in preparation, 2023) result on the subject, by Gorshkov, Staroletov and Shpectorov. A 2-generated subalgebra B of an algebra A of Jordan type half is called solid if every primitive idempotent from B is an axis in the entire A. Surprisingly, it turned out that, at least in characteristic zero, almost all 2-generated subalgebras are solid. More, precisely, a non-solid 2-generated subalgebra is necessarily of type 3C(1/2). Consequently, if a finite-dimensional algebras of Jordan type half has a finite automorphism group then it is either a Matsuo algebra or a factor of Matsuo algebra. The above result hints of a possibility of a geometric theory of algebras of Jordan type half.
17.00 Jason Gaddis (Miami University, USA)
Rigidity of quadratic Poisson algebras [video, youtube] [slides, pdf]
Abstract: The Shephard-Todd-Chevalley Theorem gives conditions for the invariant ring of a polynomial ring to again be polynomial. However, this behavior is rarely observed for noncommutative algebras. For example, the invariant ring of the first Weyl algebra by a finite group is not isomorphic to the first Weyl algebra. In this talk, I will discuss this rigidity in the context of quadratic Poisson algebras. A key example will be those Poisson polynomial algebras with skew-symmetric structure. This is joint work with Padmini Veerapen and Xingting Wang.
18.00 Coffee-Break
Next talks will be given on
European Non-Associative Algebra Seminar
[Every Monday, 15:00 (+0 UTC) ]