Tag - Non-associative rings

Sergey Shpectorov: Solid subalgebras in algebras of Jordan type half

18th December, 2023

Algebras of Jordan type η generalize 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. The 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 turns 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 algebra 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.

Tom De Medts: Primitive axial algebras of Jordan type and 3-transposition groups

18th December, 2023

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).

Olivier Mathieu: On free Jordan Algebras

18th December, 2023

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 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.

Antonio Viruel: Permutation representations of finite groups via evolution algebras

18th December, 2023

In the wake of the influential work by Elduque-Labra, it is known that every finite-dimensional evolution K-algebra X such that X2 = X, namely X is idempotent, has a 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) ≅ 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 X2 = X, dim X = n, and Aut(X) is isomorphic to the alternating group An. However, we demonstrate that for any (not necessarily faithful) permutation representation ρ : G → Sn and any field K, there exists a finite-dimensional evolution K-algebra X such that X2 = X, Aut(X) ≅ G and the induced representation given by the Aut(X)-action on the natural idempotents of X is ρ.

Ievgen Makedonskyi: Duality Theorems for current Lie algebras

18th December, 2023

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 the role of irreducible modules. In my talk, I will explain analogues of duality theorems, namely Peter-Weyl theorem, Schur-Weyl duality etc.

Vesselin Drensky: The Specht problem for varieties of ℤn-graded Lie algebras in positive characteristic

18th December, 2023

Let K be a field of positive characteristic p and let UTp+1(K) be the algebra of (p+1)×(p+1) upper triangular matrices. We construct three varieties of ℤp+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 UTp+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.

Askar Dzhumadil’daev: Rota-Baxter algebras with non-zero weights

18th December, 2023

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+ be the 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 the case λ ≠ 0. We prove that AR− is Tortkara. AR+ satisfies an identity of degree 5. In the case λ ≠ 0, the algebra AR is not associative-admissible.

Susanne Pumpluen: A way to generalize classical results from central simple algebras to the nonassociative setting

11th December, 2023

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 behavior in the Brauer monoid. Time permitting, we also discuss some - hopefully interesting - particularities of this newly defined Brauer monoid.

Amir Fernández Ouaridi: On the simple transposed Poisson algebras and Jordan superalgebras

4th December, 2023

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.

Artem Lopatin: Polynomial invariants for 2-dimensional algebras

20th November, 2023

For every 2-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.