Tag - Non-associative rings

Jorge Garcés: Maps preserving the truncation of triple products on Cartan factors

8th July, 2024

We generalize the concept of truncation of operators to JB*-triples and study some general properties of bijections preserving the truncation of triple products in both directions between general JB*-triples. In our main result, we show that a (not necessarily linear nor continuous) bijection between atomic JBW*-triples preserving the truncation of triple products in both directions (and such that the restriction to each rank-one Cartan factor is a continuous mapping) is an isometric real linear triple isomorphism.

Nurlan Ismailov: On the variety of right-symmetric algebras

1st July, 2024

The problem of the existence of a finite basis of identities for a variety of associative algebras over a field of characteristic zero was formulated by Specht in 1950. We say that a variety of algebras has the Specht property if any of its subvariety has a finite basis of identities. In 1988, A. Kemer proved that the variety of associative algebras over a field of characteristic zero has the Specht property. Specht’s problem has been studied for many well-known varieties of algebras, such as Lie algebras, alternative algebras, right-alternative algebras, and Novikov algebras. An algebra is called right-symmetric if it satisfies the identity (a,b,c) = (a,c,b) where (a,b,c) = (ab)c − a(bc) is the associator of a, b, c. The talk is devoted to the Specht problem for the variety of right-symmetric algebras. It is proved that the variety of right-symmetric algebras over an arbitrary field does not satisfy the Specht property.

Rita Fioresi: Quantum principal bundles on quantum projective varieties

24th June, 2024

In non-commutative geometry, a quantum principal bundle over an affine base is recovered through a deformation of the algebra of its global sections: the property of being a principal bundle is encoded by the notion of Hopf-Galois extension, while the local triviality is expressed by the cleft property. We examine the case of a projective base X in the special case X = G/P, where G is a complex semisimple group and P is a parabolic subgroup. The quantization of G will then be interpreted as the quantum principal bundle on the quantum base space X, obtained via a quantum section.

Paul Laubie: Combinatorics of free pre-Lie algebras and algebras with several pre-Lie products sharing the Lie bracket

17th June, 2024

Using the theory of algebraic operads, we give a combinatorial description of free pre-Lie algebras (also known as left-symmetric algebras) with rooted trees. A numerical coincidence hints a similar description for algebras with several pre-Lie products sharing the Lie bracket using rooted Greg trees which are rooted trees with black and white vertices such that black vertices have at least two children. We then show that those Greg trees can be used to give a description of the free Lie algebras.

Erhard Neher: Corestriction

10th June, 2024

Corestriction is an important technique in the theory of central-simple associative algebras over a field. Given a finite étale extension K/F, e.g. a Galois extension, corestriction associates a central-simple associative F-algebra with every central-simple associative K-algebra. In this talk, I will give an introduction to corestriction over fields, applicable to non-associative algebras. Towards the end of my talk, I will indicate why it is of interest to generalize corestruction to schemes and sketch how this can be done.

Claudemir Fideles: Graded identities in Lie algebras with Cartan gradings: an algorithm

3rd June, 2024

The classification of finite-dimensional semisimple Lie algebras in characteristic 0 represents one of the significant achievements in algebra during the first half of the 20th century. This classification was developed by Killing and Cartan. According to the Killing-Cartan classification, the isomorphism classes of simple Lie algebras over an algebraically closed field of characteristic zero correspond one-to-one with irreducible root systems. In the infinite-dimensional case, the situation is more complicated, and the so-called algebras of Cartan type appear. It is somewhat surprising that graded identities for Lie algebras have been relatively few results to that extent. In this presentation, we will discuss some of the results obtained thus far and introduce an algorithm capable of generating a basis for all graded identities in Lie algebras with Cartan gradings. Specifically, over any infinite field, we will apply this algorithm to establish a basis for all graded identities of U1, the Lie algebra of derivations of the algebra of Laurent polynomials K[t,t-1]], and demonstrate that they do not admit any finite basis.

Christopher Drupieski: The Lie superalgebra of transpositions

28th May, 2024

In this talk I will report on work, joint with Jonathan Kujawa, to answer a series of questions originally posed by MathOverflow user WunderNatur in August 2022: Considering the group algebra ℂSn of the symmetric group as a superalgebra (by considering the even permutations in Sn to be of even superdegree and the odd permutations in Sn to be of odd superdegree), and then in turn considering ℂSn as a Lie superalgebra via the super commutator, what is the structure of ℂSn as a Lie superalgebra, and what is the structure of the Lie sub-superalgebra of ℂSn generated by the transpositions? The non-super versions of these questions were previously answered by Ivan Marin, with very different results. Time permitting, some thoughts on analogues of these questions for Weyl groups of types B/C and D may also be discussed.

Yvain Bruned: Novikov algebras and multi-indices in regularity structures

27th May, 2024

In this talk, we will present multi-Novikov algebras, a generalization of Novikov algebras with several binary operations indexed by a given set, and show that the multi-indices recently introduced in the context of singular stochastic partial differential equations can be interpreted as free multi-Novikov algebras. This is parallel to the fact that decorated rooted trees arising in the context of regularity structures are related to free multi-pre-Lie algebras.

Anastasia Doikou: Parametric set-theoretic Yang-Baxter equation: p-racks, solutions and quantum algebras

20th May, 2024

The theory of the parametric set-theoretic Yang-Baxter equation is established from a purely algebraic point of view. We introduce generalizations of the familiar shelves and racks named parametric (p)-shelves and racks. These objects satisfy a 'parametric self-distributivity' condition and lead to solutions of the Yang-Baxter equation. Novel, non-reversible solutions are obtained from p-shelf/rack solutions by a suitable parametric twist, whereas all reversible set-theoretic solutions are reduced to the identity map via a parametric twist. The universal algebras associated to both p-rack and generic parametric set-theoretic solutions are next presented and the corresponding universal R-matrices are derived. By introducing the concept of a parametric coproduct we prove the existence of a parametric co-associativity. We show that the parametric coproduct is an algebra homomorphism and the universal R-matrices intertwine with the algebra coproducts.

Tomasz Brzezinski: Lie brackets on affine spaces

13th May, 2024

We first explore the definition of an affine space which makes no reference to the underlying vector space and then formulate the notion of a Lie bracket and hence a Lie algebra on an affine space in this framework. Since an affine space has neither distinguished elements nor additive structure, the concepts of antisymmetry and Jacobi identity need to be modified. We provide suitable modifications and illustrate them by a number of examples.