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.