Tag - Non-associative rings

Louis Rowen: Weakly primitive axial algebras

6th May, 2024

In earlier work, we studied the structure of primitive axial algebras of Jordan type (PAJs), not necessarily commutative, in terms of their primitive axes. In this paper we weaken primitivity and permit several pairs of (left and right) eigenvalues satisfying a more general fusion rule, bringing in interesting new examples such as the band semigroup algebras and various non-commutative examples. Also, we broaden our investigation to the case of 2-generated algebras for which only one axis satisfies the fusion rules. As an example we describe precisely the 2-dimensional axial algebras and the 3-dimensional and 4-dimensional weakly primitive axial algebras of Jordan type (weak PAJs), and we see, in contrast to the case for PAJs, that there are higher-dimensional weak PAJs generated by two axes. We also prove a theorem that enables us to reduce weak PAJs to uniform components.

Erik Darpö: Non-associative algebras in an associative context

29th April, 2024

For any associative algebra A, the left regular representation is an embedding of A into its linear endomorphism algebra End(A). In this talk, I shall explain how this elementary observation can be generalised to a (less elementary) structure result for general non-associative algebras. The describes the category of unital, not necessarily associative, algebras in terms of associative algebras with certain distinguished subspaces.

Stéphane Launois: Derivations of quantum algebras

22nd April, 2024

I will report on joint work in progress with Samuel Lopes and Isaac Oppong where we aim to compute the derivations of quantum nilpotent algebras, a class of non-commutative algebras which includes in particular the positive part of quantized enveloping algebras and quantum Schubert cells.

Paola Stefanelli: Płonka sums of set-theoretical solutions of the Yang-Baxter equation

15th April, 2024

The Płonka sum is one of the most significant composition methods in Universal Algebra introduced by Jerzy Płonka in 1967. In particular, Clifford semigroups have turned out to be the first instances of Płonka sums of groups. In this talk, we illustrate a method for constructing set-theoretical solutions of the Yang-Baxter equation that is inspired by the notion of the Płonka sums. Moreover, we will show how to obtain solutions of this type by considering dual weak braces, algebraic structures recently studied and described in a joint work with Francesco Catino and Marzia Mazzotta.

Bernard Rybołowicz: On affine nature of trusses

8th April, 2024

In this presentation, I will introduce the audience to ternary algebras called heaps and trusses. Specifically, I will familiarize the audience with modules over trusses, highlighting differences with modules over rings. The main point will be to show the close relationship between modules over trusses and affine spaces over rings. I will illustrate that modules over trusses occupy a position between modules over rings and affine spaces over rings.

Samuel Lopes: Torsion-free representations of Smith algebras

25th March, 2024

We will discuss representations of the Smith algebra which are free of finite rank over a subalgebra which plays a role analogous to that of the (enveloping algebra of the) Cartan subalgebra of the simple Lie algebra 𝔰𝔩2. In the case of rank 1 we obtain a full description of the isomorphism classes, a simplicity criterion, and a combinatorial algorithm to produce all composition series and the multiplicities of the simple factors.

Pedro Fagundes: The L’vov-Kaplansky conjecture and some of its variations

18th March, 2024

The L'vov-Kaplansky conjecture claims that the image of a multilinear polynomial on the full matrix algebra is a vector space. Positive results concerning the conjecture are known only for small cases (polynomials of small degree or matrices of small size). Besides presenting the main results on the L'vov-Kaplasnky conjecture, in this talk we also will discuss some of its variations such as images of multilinear polynomials on some subalgebras of the full matrix algebra with additional structure (gradings, involutions, graded involutions).

María Alejandra Alvarez: On S-expansions and other transformations of Lie algebras

11th March, 2024

The aim of this work is to study the relation between S-expansions and other transformations of Lie algebras. In particular, we prove that contractions, deformations and central extensions of Lie algebras are preserved by S-expansions. We also provide several examples and give conditions so transformations of reduced subalgebras of S-expanded algebras are preserved by the S-expansion procedure.

Michael Turner: Skew Axial Algebras of Monster Type

4th March, 2024

Given a 2-generated primitive axial algebra of Monster Type, it has been shown that it has an axet which is regular or skew. With all the known examples being regular, it was proposed if any axial algebra were skew and if so, can they be classified. We will begin by defining axial algebras and axets, before producing examples of axial algebras with skew axets. We will finish by stating the complete classification of these skew axial algebras and mention how it was proven.

Kang Lu: A Drinfeld presentation of twisted Yangians via degeneration

2nd March, 2024

We formulate a new family of algebras, twisted Yangians (of split type) in current generators and relations, via degeneration of Drinfeld presentations of affine iquantum groups (associated with split Satake diagrams). These new algebras admit PBW type bases and are shown to be a deformation of twisted current algebras. For type AI, it matches with the Drinfeld presentation of twisted Yangian obtained via Gauss decomposition. We conjecture that our twisted Yangians are isomorphic to twisted Yangians constructed in RTT presentation.