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.
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.
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.
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.
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.
In 2019, Alsaody and Gille showed that, for octonion algebras over unital commutative rings, there is an equivalence between isotopes and isometric quadratic forms. This leads us to a question: can this equivalence be generalized to octonion algebras over a (not necessarily affine) scheme? We give the basic definitions of octonion algebras over schemes. We show that an isotope of an octonion algebra C over a scheme is isomorphic to a twist by an Aut(C)–torsor. We conclude by giving an affirmative answer to our question.
A pseudo-Euclidean left-symmetric algebra (A, ., {, }) is a real left-symmetric algebra (A, .) endowed with a non-degenerate symmetric bilinear form { , } such that left multiplication by any element of A is skew-symmetric with respect to { , }. We recall that a pseudo-Euclidean Lie algebra (𝔤, [ , ], { , }) is flat if and only if (𝔤, ., { , }) its underlying vector space endowed with the Levi- Civita product associated with { , } is a pseudo-Euclidean left-symmetric algebra. In this talk, We will give an inductive classification of pseudo-Euclidean left-symmetric algebras (A, ., { , }) such that commutators of all elements of A are contained in the left annihilator of (A, .), these algebras will be called pseudo-Euclidean left-symmetric L−algebras of any signature.
In this talk, I will introduce a bialgebra theory for the Novikov algebra, namely the Novikov bialgebra, which is characterized by the fact that its affinization (by a quadratic right Novikov algebra) gives an infinite-dimensional Lie bialgebra. A Novikov bialgebra is also characterized as a Manin triple of Novikov algebras. The notion of Novikov Yang-Baxter equation is introduced, whose skew-symmetric solutions can be used to produce Novikov bialgebras and hence Lie bialgebras. These solutions also give rise to skewsymmetric solutions of the classical Yang-Baxter equation in the infinite-dimensional Lie algebras from the Novikov algebras. Moreover, a similar connection between Novikov bialgebras and Lie conformal bialgebras will be introduced.
How to recover an algebra structure if the algebra does NOT satisfy any reasonable identity? How to characterize its idempotents, their spectrum, or fusion laws? In my talk, I will discuss what can be thought of as "non-associative algebra in large", imitating a well-known concept of "geometry in large". In other words, the properties of non-associative algebras which crucially depend on a complete set of idempotents. The latter is very related to the concept of generic algebras. I will explain some recent results in this direction and some unsolved problems.
We report on ongoing research about a module-theoretic construction which, when successful, yields natural extensions of infinite-dimensional modules over arbitrary algebras. Whether the construction works or not depends on the basis that one chooses to carry on such a construction. Bases that work are said to be amenable. A natural example on which one may focus is when the module is the algebra itself. For instance, a great deal of the work done so far has focused on the infinite-dimensional algebra of polynomials on a single variable. We will see that amenability and related notions serve to classify the distinct bases according to interesting complementary properties having to do with the types of relations induced on them by the properties of their change-of-basis matrices.
