Tag - Lie algebras

Andrey Lazarev: Cohomology of Lie coalgebras

15th July, 2024

Associated to a Lie algebra 𝔤 and a 𝔤-module M is a standard complex C*(𝔤,M) computing the cohomology of 𝔤 with coefficients in M; this classical construction goes back to Chevalley and Eilenberg of the late 1940s. Shortly afterwards, it was realized that this cohomology is an example of a derived functor in the category of 𝔤-modules. The Lie algebra 𝔤 can be replaced by a differential graded Lie algebra and M – with a dg 𝔤-module with the same conclusion. Later, a deep connection with Koszul duality was uncovered in the works of Quillen (late 1960s) and then Hinich (late 1990s). In this talk I will discuss the cohomology of (dg) Lie coalgebras with coefficients in dg comodules. The treatment is a lot more delicate, underscoring how different Lie algebras and Lie coalgebras are (and similarly their modules and comodules). A definitive answer can be obtained for so-called conilpotent Lie coalgebras (though not necessarily conilpotent comodules). If time permits, I will also discuss some topological applications.

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.

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.

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.

Jie Du: The quantum queer supergroup via their q-Schur superalgebras

28th May, 2024

Using a geometric setting of q-Schur algebras, Beilinson-Lusztig-MacPherson discovered a new basis for quantum 𝔤𝔩n (i.e., the quantum enveloping algebra Uq(𝔤𝔩n) of the Lie algebra 𝔤𝔩n) and its associated matrix representation of the regular module of Uq(𝔤𝔩n). This beautiful work has been generalized (either geometrically or algebraically) to quantum affine 𝔤𝔩n, quantum super 𝔤𝔩m|n, and recently, to some i-quantum groups of type AIII.

In this talk, I will report on a completion of the work for a new construction of the quantum queer supergroup using their q-Schur superalgebras. This work was initiated 10 years ago, and almost failed immediately after a few months’ effort, due to the complication in computing the multiplication formulas by odd generators. Then, we moved on testing special cases or other methods for some years and regained confidence to continue. Thus, it resulted in a preliminary version which was posted on arXiv in August 2022.

The main unsatisfaction in the preliminary version was the order relation used in a triangular relation and the absence of a normalized standard basis. It took almost two more years for us to tune the preliminary version up to a satisfactory version, where the so-called SDP condition, involving further combinatorics related to symmetric groups and Clifford generators, and an extra exponent involving the odd part of a labelling matrix play decisive roles to fix the problems.

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.

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.

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.

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.