Tag - Lie algebras

Oleksandr Tsymbaliuk: Lyndon words and fused currents in shuffle algebra

2nd March, 2024

Classical q-shuffle algebras provide combinatorial models for the positive half Uq(𝔫) of a finite quantum group. We define a loop version of this construction, yielding a combinatorial model for the positive half Uq(L𝔫) of a quantum loop group. In particular, we construct a PBW basis of Uq(L𝔫) indexed by standard Lyndon words, generalizing the work of Lalonde-Ram, Leclerc and Rosso in the Uq(𝔫) case. We also connect this to Enriquez's degeneration A of the elliptic algebras of Feigin-Odesskii, proving a conjecture that describes the image of the embedding Uq(L𝔫)→A in terms of pole and wheel conditions. The talk shall conclude with the shuffle interpretations of fused currents proposed by Ding-Khoroshkin.

Julia Plavnik: Semisimplification of contragredient Lie algebras

1st March, 2024

The study of symmetric categories over fields of positive characteristic has gained a lot of attention in the last couple of years. One of the key examples of this theory is the Verlinde categories Verp. Understanding the structure of these categories in-depth and how to construct algebraic structures within them are important questions. In this talk, we will start by introducing the Verlinde categories and some of their important properties. We will also give examples of some Lie algebras in symmetric categories in positive characteritic that can be obtained as semisimplification of contragredient Lie algebras in characteristic p. If time allows, we will present some interesting properties that they have. As an application, we will exhibit concrete new examples of Lie algebras in the Verlinde category.

Lucas Buzaglo: Derivations, extensions, and rigidity of subalgebras of the Witt algebra

5th February, 2024

We study Lie algebraic properties of subalgebras of the Witt algebra and the one-sided Witt algebra: we compute derivations, one-dimensional extensions, and automorphisms of these subalgebras. In particular, all these properties are inherited from the full Witt algebra (e.g. derivations of subalgebras are simply restrictions of derivations of the Witt algebra). We also prove that any isomorphism between subalgebras of finite codimension extends to an automorphism of the Witt algebra. We explain this "rigid" behavior by proving a universal property satisfied by the Witt algebra as a completely non-split extension of any of its subalgebras of finite codimension. This is a purely Lie algebraic property which I will introduce in the talk.

Yanyong Hong: Novikov bialgebras, infinite-dimensional Lie bialgebras, and Lie conformal bialgebras

29th January, 2024

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.

Friedrich Wagemann: Cohomology of semi-direct product Lie algebras

22nd January, 2024

Intrigued by computations of Richardson, our goal is to compute the adjoint cohomology spaces of Lie algebras which are the semi-direct product of a simple Lie algebra 𝔰 and an 𝔰-module. We present some theorems and conjectures in these cohomologies.

Jason Gaddis: Rigidity of quadratic Poisson algebras

18th December, 2023

The Shephard-Todd-Chevalley Theorem gives conditions for the invariant ring of a polynomial ring to again be polynomial. However, this behaviour is rarely observed for non-commutative algebras. For example, the invariant ring of the first Weyl algebra by a finite group is not isomorphic to the first Weyl algebra. In this talk, I will discuss this rigidity in the context of quadratic Poisson algebras. A key example will be those Poisson polynomial algebras with skew-symmetric structure.

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.

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.