Arne Van Antwerpen: Indecomposable and simple solutions of the Yang-Baxter equation

30th September, 2024

Recall that a set-theoretic solution of the Yang-Baxter equation is a tuple (X,r), where X is a non-empty set and r: X × X → X × X a bijective map such that

(r × idX) (idX × r) (r × idX) = (idX × r) (r × idX) (idX × r),

where one denotes r(x,y)=(lx(y), ry(x)). Attention is often restricted to so-called non-degenerate solutions, i.e. lx and ry are bijective. We will call these solutions for short in the remainder of this abstract. To understand more general objects, it is an important technique to study 'minimal' objects and glue them together. For solutions both indecomposable and simple solutions fit the bill for being a minimal object. In this talk, we will report on recent work with I. Colazzo, E. Jespers and L. Kubat on simple solutions. In particular, we will discuss an extension of a result of M. Castelli that allows to identify whether a solution is simple, without having to know or calculate all smaller solutions. This method employs so-called skew braces, which were constructed to provide more examples of solutions, but also govern many properties of general solutions. In the latter part of the talk, we discuss the extension of a method to construct new indecomposable or simple solutions from old ones via cabling, originally introduced by V. Lebed, S. Ramirez, and L. Vendramin to unify the known results on indecomposability of solutions.

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.

Duc-Khanh Nguyen: Application of (K-theoretic) Peterson isomorphism

17th September, 2024

The theory of symmetric polynomials plays a key role in Representation Theory, Schubert Calculus, and Algebraic Combinatorics. Fundamental rules like the Pieri, Murnaghan-Nakayama, and Littlewood-Richardson rules describe the decomposition of products of Schubert classes into Schubert classes. We focus on the decomposition of polynomial representatives of Schubert classes in homology and K-homology of the affine Grassmannian of SLn, as well as quantum Schubert classes in quantum cohomology and K-cohomology of the full flag manifold of type A. Specifically, we explore how to use the Peterson isomorphism to connect formulas between homology and quantum cohomology, and between K-homology and quantum K-cohomology, extending techniques from the work of Lam-Shimozono on Schubert classes.

Karthik Ganapathy: GL-algebras and the Noetherianity problem

16th September, 2024

Draisma recently proved that finite length polynomial representations of the infinite general linear group GL are topologically GL-noetherian, i.e., the descending chain condition holds for GL-stable closed subsets. The scheme-theoretic variant of this theorem is a major open problem in the area. I will briefly outline the rich history of this problem and provide a negative answer in characteristic 2.

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.

Alberto San Miguel Malaney: Partial Resolutions of Affine Symplectic Singularities II

9th September, 2024

We will continue to discuss partial resolutions of conical affine symplectic singularities, particularly their deformation theory and Springer theory. First we will explain the construction of the universal deformations of symplectic singularities and their partial resolutions, generalizing the Grothendieck-Springer resolution. Then we will use these universal deformations to study the Springer theory of symplectic singularities and their partial resolutions, using recent work of McGerty and Nevins. In particular, we will compute the cohomology of the fibres of the partial resolutions under suitable conditions, generalizing a result of Borho and MacPherson for the nilpotent cone. Finally, we will use partial resolutions to construct and study symplectic resolutions of symplectic leaf closures, generalizing the Springer maps from cotangent bundles of partial flag varieties to nilpotent orbit closures.

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.

Isabel Martin-Lyons: Skew Bracoids

2nd September, 2024

The skew brace was devised by Guanieri and Vendramin in 2017, building on Rump's brace. Since then, the skew brace has been central to the study of solutions to the Yang-Baxter equation, with connections to many other areas of mathematics including Hopf-Galois theory. We introduce the skew bracoid, a generalization of the skew brace which can arise as a partial quotient thereof. We explore the connection between skew bracoids and Hopf-Galois theory, as well as the more recent connection to solutions of the Yang-Baxter equation.

Alberto San Miguel Malaney: Partial Resolutions of Affine Symplectic Singularities I

26th August, 2024

Symplectic singularities are a generalization of symplectic manifolds that have a symplectic form on the smooth locus but allow for certain well-behaved singularities. They have a strong relationship to representation theory and include nilpotent cones of semisimple Lie algebras, quiver varieties, affine Grassmannian slices, and Kleinian singularities. There is a combinatorial description for partial resolutions of conical affine symplectic singularities, stemming from Namikawa's 2013 result that a symplectic resolution is also a relative Mori Dream Space. In this talk we will explore these partial resolutions in more detail, exploring their birational geometry, deformation theory, and Springer theory. In particular, we will review the definition of the Namikawa Weyl group for conical affine symplectic singularities and use birational geometry to define a generalization for their partial resolutions. We will also use this Namikawa Weyl group to classify the Poisson deformations of the partial resolutions. We will then describe how these partial resolutions fit into the framework of Springer Theory for symplectic singularities, following Kevin McGerty and Tom Nevins' recent paper, Springer Theory for Symplectic Galois Groups. Finally, we will discuss some ongoing research that stems from these ideas, inspired by parabolic induction and restriction.

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.

Pierre Catoire: The free tridendriform algebra, Schröder trees and Hopf algebras

12th August, 2024

The notions of dendriform algebras, respectively tridendriform, describe the action of some elements of the symmetric groups called shuffle, respectively quasi-shuffle over the set of words whose letters are elements of an alphabet, respectively of a monoid. A link between dendriform and tridendriform algebras will be made. Those words algebras satisfy some properties but they are not free. This means that they satisfy extra properties like commutativity. In this talk, we will describe the free tridendriform algebra. It will be described with planar trees (not necessarily binary) called Schröder trees. We will describe the tridendriform structure over those trees in a non-recursive way. Then, we will build a coproduct on this algebra that will make it a (3, 2)-dendriform bialgebra graded by the number of leaves. Once it will be build, we will study this Hopf algebra: duality, quotient spaces, dimensions, study of the primitive elements.

Łukasz Kubat: On Yang-Baxter algebras

5th August, 2024

To each solution of the Yang-Baxter equation one may associate a quadratic algebra over a field, called the YB-algebra, encoding certain information about the solution. It is known that YB-algebras of finite non-degenerate solutions are (two-sided) Noetherian, PI and of finite Gelfand-Kirillov dimension. If the solution is additionally involutive then the corresponding YB-algebra shares many other properties with polynomial algebras in commuting variables (e.g., it is a Cohen-Macaulay domain of finite global dimension). The aim of this talk is to explain the intriguing relationship between ring-theoretical and homological properties of YB-algebras and properties of the corresponding solutions of the Yang-Baxter equation. The main focus is on when such algebras are Noetherian, (semi)prime and representable.

Érica Fornaroli: Involutions of the second kind on finitary incidence algebras

29th July, 2024

Let K be a field and X a connected partially ordered set. In this talk, we show that the finitary incidence algebra FI(X,K) of X over K has an involution of the second kind if and only if X has an involution and K has an automorphism of order 2. We also present a characterization of the involutions of the second kind on FI(X,K). We conclude by giving necessary and sufficient conditions for two involutions of the second kind on FI(X,K) to be equivalent in the case where the characteristic of K is different from 2 and every multiplicative automorphism of FI(X,K) is inner.

Janina Letz: Generation time for biexact functors and Koszul objects in triangulated categories

25th July, 2024

One way to study triangulated categories is through finite building. An object X finitely builds an object Y, if Y can be obtained from X by taking cones, suspensions and retracts. The X-level measures the number of cones required in this process; this can be thought of as the generation time. I will explain the behaviour of level with respect to tensor products and other biexact functors for enhanced triangulated categories. I will further present applications to the level of Koszul objects.

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.

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.

Bertrand Toën: Geometric quantization for shifted symplectic structures

4th July, 2024

The purpose of this talk is to present an ongoing work on geometric quantization in the setting of shifted symplectic structures. I will start by recalling the various notions involved as well as the results previously obtained by James Wallbridge, who constructed the prequantized (higher) categories of a given integral shifted symplectic structure. I will then explain our main result so far: the construction of the shifted analogues of the Kostant–Souriau prequantum operators, which will be realized as a "Poisson module over a Poisson category" (a categorification of the notion of a Poisson module over a Poisson algebra). This will be obtained by means of deformation theory arguments for categories of sheaves in the setting of (derived) differential geometry. If time permits, I will discuss further aspects associated to the notion of polarizations of shifted symplectic structures.

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.