Seminars in Lie Theory

Test

Test

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.

Ian Tan: Tensor decompositions with applications to LU and SLOCC equivalence of multipartite pure states

30th September, 2024

We introduce a broad lemma, one consequence of which is the higher order singular value decomposition (HOSVD) of tensors defined by DeLathauwer, DeMoor and Vandewalle (2000). By an analogous application of the lemma, we find a complex orthogonal version of the HOSVD. Kraus' (2010) algorithm used the HOSVD to compute normal forms of almost all n-qubit pure states under the action of the local unitary group. Taking advantage of the double cover SL2(ℂ) × SL2(ℂ) → SO4(ℂ), we produce similar algorithms (distinguished by the parity of n) that compute normal forms for almost all n-qubit pure states under the action of the SLOCC group.

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.

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.

Artem Chernikov: Recognizing groups in Erdős geometry and model theory

15th March, 2024

Erdős-style geometry is concerned with difficult questions about simple geometric objects, such as counting incidences between finite sets of points, lines, etc. These questions can be viewed as asking for the possible number of intersections of a given algebraic variety with large finite grids of points. An influential theorem of Elekes and Szabó indicates that such intersections have maximal size only for varieties that are closely connected to algebraic groups. Techniques from model theory - variants of Hrushovski’s group configuration and of Zilber’s trichotomy principle - are very useful in recognizing these groups, and led to far reaching generalizations of Elekes-Szabó in the last decade. I will overview some of the recent developments in this area, in particular explaining how all of this is not just about polynomials and works for definable sets in o-minimal structures.