Tag - Lie algebras

Pablo Zadunaisky: Clebsch-Gordan revisited

19th November, 2024

By an ultra classical result, the tensor product of a simple representation of 𝔤𝔩n(ℂ) and its defining representation decomposes as a direct sum of simple representations without multiplicities. This means that for each highest weight, the space of highest weight vectors is 1-dimensional. We will give an explicit construction of these highest weight vectors, and show that they arise from the action of certain elements in the enveloping algebra of 𝔤𝔩n(ℂ) + 𝔤𝔩n(ℂ) on the tensor product. These elements are independent of the simple representation we started with, and in fact produce highest weight vectors in several other contexts.

Carla Rizzo: Differential identities, almost polynomial growth and matrix algebras

11th November, 2024

Let F be a field of characteristic zero, L a Lie algebra over F, and A an L-algebra - that is, an associative algebra over F with an action of L induced by derivations. This action of L on A can be extended to an action of its universal enveloping algebra U(L), leading to the concept of L-identities or differential identities of A: polynomials in variables xu:= u(x), where u ∈ U(L), that vanish under all substitutions of elements from A. Differential identities were first introduced by Kharchenko in 1978, and, in later years, subsequent work by Gordienko and Kochetov has spurred a renewed interest in both their structure and quantitative properties. In this talk, I will present recent results on the differential identities of matrix L-algebras, with a particular focus on their classification and growth behaviour.

Raschid Abedin: Classification of D-bialgebras via algebraic geometry

28th October, 2024

In a now classic paper, Belavin and Drinfeld categorized solutions to the classical Yang-Baxter equation (CYBE), an equation crucial to the theory of integrable systems, into three classes: elliptic, trigonometric and rational. It is possible to reproduce this result by geometrizing solutions of the CYBE and then applying algebro-geometric methods. In this talk, we will explain how this approach can be used to categorize Lie bialgebra structures on power series Lie algebras, as well as non-associative generalizations of these structures: D-bialgebra structures on more general power series algebras.

Ilya Chevyrev: Pre-Lie algebras in stochastic PDEs

21st October, 2024

In this talk, I will discuss a general method to renormalize singular stochastic partial differential equations (SPDEs) using the theory of regularity structures. It turns out that, to derive the renormalized equation, one can employ a convenient multi-pre-Lie algebra. The pre-Lie products in this algebra are reminiscent of the pre-Lie product on the Grossman-Larson algebra of trees, but come with several important twists. For the renormalization of SPDEs, the important feature of this multi-pre-Lie algebra is that it is free in a certain sense.

Ignacio Bajo: Quadratic Lie algebras admitting 2-plectic structures

23rd September, 2024

A 2-plectic form ω on a Lie algebra is a 3-form on the algebra such that it is closed and non-degenerate in the sense that, for every non-zero x, the bilinear form ω(x, ·, ·) is not identically zero. We will study the existence of 2-plectic structures on the so-called quadratic Lie algebras, which are Lie algebras admitting an ad-invariant pseudo-Euclidean product. It is well-known that every centreless quadratic Lie algebra admits a 2-plectic form but not many quadratic examples with non-trivial centre are known. We give several constructions to obtain large families of 2-plectic quadratic Lie algebras with non-trivial centre, many of them among the class of nilpotent Lie algebras. We give some sufficient conditions to assure that certain extensions of 2-plectic quadratic Lie algebras result to be 2-plectic as well. For instance, we show that oscillator algebras can be naturally endowed with 2-plectic structures. We prove that every quadratic and symplectic Lie algebra with dimension greater than 4 also admits a 2-plectic form. Further, conditions to assure that one may find a 2-plectic which is exact on certain quadratic Lie algebras are obtained.

Jörg Feldvoss: Semi-simple Leibniz algebras

16th September, 2024

Leibniz algebras were introduced by Blokh in the 1960s and rediscovered by Loday in the 1990s as non-anticommutative analogues of Lie algebras. Many results for Lie algebras have been proved to hold for Leibniz algebras, but there are also several results that are not true in this more general context. In my talk, I will investigate the structure of semi-simple Leibniz algebras. In particular, I will prove a simplicity criterion for (left) hemi-semidirect products of a Lie algebra 𝔤 and a (left) 𝔤-module. For example, in characteristic zero every finite-dimensional simple Leibniz algebra is such a hemi-semidirect product. But this also holds for some infinite-dimensional Leibniz algebras or sometimes in non-zero characteristics. More generally, the structure of finite-dimensional semi-simple Leibniz algebras in characteristic zero can be reduced to the well-known structure of finite-dimensional semi-simple Lie algebras and their finite-dimensional irreducible modules. If time permits, I will apply these structure results to derive some properties of finite-dimensional semi-simple Leibniz algebras in characteristic zero and other Leibniz algebras that are hemi-semidirect products.

Dominique Manchon: Post-Lie algebras, post-groups and Gavrilov’s K-map

9th September, 2024

Post-Lie algebras appeared in 2007 in algebraic combinatorics, and independently in 2008 in the study of numerical schemes on homogeneous spaces. Gavrilov's K-map is a particular Hopf algebra isomorphism, which can be naturally described in the context of free post-Lie algebras. Post-groups, which are to post-Lie algebras what groups are to Lie algebras, were defined in 2023 by C. Bai, L. Guo, Y. Sheng and R. Tang. Although skew-braces and braided groups are older equivalent notions, their reformulation as post-groups brings crucial new information on their structure. After giving an account of the above-mentioned structures, I shall introduce free post-groups, and describe a group isomorphism which can be seen as an analogon of Gavrilov's K-map for post-groups.

Jonas Deré: Simply transitive NIL-affine actions of soluble Lie groups

26th August, 2024

Although not every 1-connected soluble Lie group G admits a simply transitive action via affine maps on ℝn, it is known that such an action exists if one replaces ℝn by a suitable nilpotent Lie group H, depending on G. However, not much is known about which pairs of Lie groups (G,H) admit such an action, where ideally you only need information about the Lie algebras corresponding to G and H. In recent work with Marcos Origlia, we show that every simply transitive action induces a post-Lie algebra structure on the corresponding Lie algebras. Moreover, if H has nilpotency class 2 we characterize the post-Lie algebra structures coming from such an action by giving a new definition of completeness, extending the known cases where G is nilpotent or H is abelian.

Yuly Billig: Quasi-Poisson superalgebras

19th August, 2024

In 1985, Novikov and Balinskii introduced what became known as Novikov algebras in an attempt to construct generalizations of Witt Lie algebra. To their disappointment, Zelmanov showed that the only simple finite-dimensional Novikov algebra is 1-dimensional (and corresponds to Witt algebra). The picture is much more interesting in the super case, where there are many more generalizations of Witt algebra, called superconformal Lie algebras. In 1988 Kac and Van de Leur gave a conjectural list of simple superconformal Lie algebras. Their list was amended with a Cheng-Kac superalgebra, which was constructed several years later. However, Novikov superalgebras are not flexible enough to describe all simple superconformal Lie algebras. In this talk, we shall present the class of quasi-Poisson algebras. Quasi-Poisson algebras have two products: it is a commutative associative (super)algebra, a Lie (super)algebra, and has an additional unary operation, subject to certain axioms. All known simple superconformal Lie algebras arise from finite-dimensional simple quasi-Poisson superalgebras. In this talk, we shall present basic constructions, describe the examples of quasi-Poisson superalgebras, and mention some results about their representations.

Ioannis Dokas: On Quillen-Barr-Beck cohomology for restricted Lie algebras

22nd July, 2024

In this talk we define and study Quillen-Barr-Beck cohomology for the category of restricted Lie algebras. We prove that the first Quillen-Barr-Beck’s cohomology classifies general abelian extensions of restricted Lie algebras. Moreover, using Duskin-Glenn’s torsors cohomology theory, we prove a classification theorem for the second Quillen-Barr-Beck cohomology group in terms of 2-fold extensions of restricted Lie algebras. Finally, we give an interpretation of Cegarra-Aznar’s exact sequence for torsor cohomology.