Tag - Non-associative rings

Petr Vojtěchovský: Solvability and nilpotence just beyond groups

30th October, 2023

Solvability and nilpotence arise naturally from the commutator theory in congruence modular varieties. In the presence of associativity, the resulting concepts agree with the classical concepts of group theory. But the two kinds of solvability differ in loops (= not necessarily associative groups) and it is a difficult question to determine the boundary where the two theories coincide. I will review the general theory and report on recent results, particularly in Moufang loops. For instance, we will prove the Odd Order Theorem for Moufang loops for the stronger notion of solvability.

Guodong Zhou: The homotopy theory of operated algebras

16th October, 2023

The talk is a survey of our recent results on the homotopy theory of operated algebras such as Rota-Baxter associative (or Lie) algebras and differential associative (or Lie) algebras etc. We make explicit the Kozul dual homotopy cooperads and the minimal models of the operads governing these operated algebras. As a consequence the L∞ structures on the deformation complexes are described as well.

Hader Elgendy: On Jordan quadruple systems

25th September, 2023

We present the recent results on Jordan quadruple systems. We show the Peirce decomposition for a Jordan quadruple system with respect to a quadripotent. We extend the notions of the orthogonality, primitivity, and minimality of tripotents in a Jordan triple system to that of quadripotents in a Jordan quadruple system. We show the relation between minimal and primitive quadripotents in a Jordan quadruple system. We also discuss the results on complemented subsystems of Jordan quadruple systems.

Adam Chapman: Roots and Critical Points of Cayley-Dickson Algebras

18th September, 2023

We study the roots and critical points (i.e., points at which the formal derivative vanishes) of standard polynomials over Cayley-Dickson algebras. In the anisotropic real case, we prove that the critical points live inside the convex hull of the roots of the polynomial.

Bauyrzhan Sartayev: Binary perm algebras and alternative algebras

11th September, 2023

We describe the defining identities of a variety of binary perm algebras which is a subvariety of the variety of alternative algebras. Moreover, we construct a basis of the free binary perm algebra. In addition, we describe the subalgebras of binary perm algebras under commutator which has a connection with Malcev algebras.

Mykola Khrypchenko: Transposed Poisson structures

4th September, 2023

A transposed Poisson algebra is a triple (L,⋅,[⋅,⋅]) consisting of a vector space L with two bilinear operations ⋅ and [⋅,⋅], such that (L,⋅) is a commutative associative algebra; (L,[⋅,⋅]) is a Lie algebra; and the 'transposed' Leibniz law holds: 2z⋅[x,y]=[z⋅x,y]+[x,z⋅y] for all x,y,z∈L. A transposed Poisson algebra structure on a Lie algebra (L,[⋅,⋅]) is a (commutative associative) multiplication ⋅ on L such that (L,⋅,[⋅,⋅]) is a transposed Poisson algebra. I will give an overview of my recent results in collaboration with Ivan Kaygorodov (Universidade da Beira Interior) on the classification of transposed Poisson structures on several classes of Lie algebras.

Anne Moreau: Functorial constructions of double Poisson vertex algebras

30th August, 2023

To any double Poisson algebra we produce a double Poisson vertex algebra using the jet algebra construction. We show that this construction is compatible with the representation functor which associates to any double Poisson (vertex) algebra and any positive integer a Poisson (vertex) algebra. We also consider related constructions, such as Poisson reductions and Hamiltonian reductions. This allows us to provide various interesting examples of double Poisson vertex algebras, in particular from double quivers.

Csaba Schneider: Computing invariants of some nilpotent Lie algebras

21st August, 2023

I will present some interesting computations concerning polynomial and rational invariants of nilpotent Lie algebras. I will say more about standard filiform Lie algebras which appear to have the highest level of complication among the small-dimensional algebras. I will outline an implementable algorithm for the computation of generators of the field of rational invariants.

Lleonard Rubio y Degrassi: Hochschild cohomology groups under gluing idempotents

7th August, 2023

Stable equivalences occur frequently in the representation theory of finite-dimensional algebras; however, these equivalences are poorly understood. An interesting class of stable equivalences is obtained by ‘gluing’ two idempotents. More precisely, let A be a finite-dimensional algebra with a simple projective module and a simple injective module. Assume that B is a subalgebra of A having the same Jacobson radical. Then B is constructed by identifying the two idempotents belonging to the simple projective module and to the simple injective module, respectively. In this talk we will compare the first Hochschild cohomology groups of finite-dimensional monomial algebras under gluing two arbitrary idempotents (hence not necessarily inducing a stable equivalence). As a corollary, we will show that stable equivalences obtained by gluing two idempotents provide 'some functoriality' to the first Hochschild cohomology, that is, HH1(A) is isomorphic to a quotient of HH1(B).

Şehmus Fındık: Symmetric polynomials in some certain noncommutative algebras

31st July, 2023

Let F be a finitely generated free algebra in a variety of algebras over a field of characteristic zero. A polynomial in F is called symmetric, if it is preserved under any permutation of the generators. The set S(F) of symmetric polynomials is a subalgebra of F. In this talk, we examine the algebras S(F), where F is the free metabelian associative, Lie, Leibniz, Poisson algebra or the free algebra generated by generic traceless matrices or the free algebra in the variety generated by Grassmann algebras.