Algebra

Given a (Dynkin) quiver $Q$ one can associate both a simple Lie algebra 𝔤 and the category of representations $Rep(Q)$ of $Q$. Early on it was realised that both associated objects are related, as for example beautifully illustrated by Gabriel's theorem. In this talk we will consider two associated categories of representations: (i) (some quotient of) the derived category $D^b(Rep(Q))$ and (ii) the finite dimensional representations of the quantum loop algebra $U_q(L𝔤)$. Although both look quit differently, we will delve into wished to be understood ties. The presented story will be one of categorifications of a common algebra with a rich combinatorial structure: a cluster algebra. The category (i) yields a so-called additive categorification and (ii) a monoidal one.  In the first half of the talk we will give a gentle and minimalistic introduction to the various objects and concepts mentioned. In the second half we will give an intuitive overview of recent conjectural connections and then finish by (very briefly) mentioning some ongoing contributions.

In this talk, I will show that a ring morphism $p:A ⟶ B$ satisfying certain mild assumptions induces a derived endomorphism of $A$ and a derived endomorphism of $B$, which are closely related. In fact, the derived endomorphism of $A$ is the twist around the restriction of scalars functor, and the derived endomorphism of $B$ is the corresponding cotwist. These endomorphisms are autoequivalences in certain settings, one of which is that of Frobenius exact categories. More precisely, assume that $A$ is the endomorphism algebra of an object in a Frobenius exact category satisfying mild assumptions and B is the corresponding stable endomorphism algebra. Then, if $B$ is "$n$-relatively spherical", I will show that both the twist and cotwist are equivalences. In fact, when $B$ is finite dimensional, "3-relatively spherical" is equivalent to self-injective, and the cotwist turns out to be a shift of the Nakayama autoequivalence of $B$. This technology can be used to construct new derived autoequivalences for very singular varieties.

We discuss the algebraic structure of KLR algebras by way of the diagrammatic Hecke categories of maximal parabolics of finite symmetric groups. Combinatorics (in the shape of Dyck tableaux) plays a huge role in understanding the structure of these algebras. Instead of looking only at the sets of Dyck tableaux (which enumerate the q-decomposition numbers) we look at the relationships for passing between these Dyck tableaux. In fact, this “meta-Kazhdan-Lusztig combinatorics” is sufficiently rich as to completely determine the Ext-quiver and relations presentation of these algebras.

Recall that a combinatorial solution of the Yang-Baxter equation is a tuple $(X,r)$, where $X$ is a non-empty set and $r: X \times X \rightarrow X \times X$ a (bijective) map such that on $X^3$ it holds that $$ (r \times \operatorname{id}_X) (\operatorname{id}_X \times r) (r \times \operatorname{id}_X) = (\operatorname{id}_X \times r)(r \times \operatorname{id}_X) (\operatorname{id}_X \times r).$$ One can then define the structure group $G(X,r)$/monoid $M(X,r)$ of a solution as the group/monoid generated by X with defining relations $xy = uv$ if $r(x,y)=(u,v)$. In this talk we talk we zoom in on the relation between $G(X,r)$ and a second group structure on this set, stemming from the behaviour of $r^2$. This led to the introduction of skew braces by Rump, and Guarnieri and Vendramin. Recall that a skew brace is a set $B$ with two group structures $(B,+)$ and $(B,\circ)$ that interact via a skew left distributivity condition, i.e. for any $a,b,c \in B$ one has that $a\circ (b+c) = (a \circ b) – a + (a\circ c)$. It turns out that these structures both generate and govern solutions. We will report in some recent advancements relating properties of skew braces to properties of its associated solution.

In the second part of the talk we will focus on the subclass of finite non-degenerate solutions. In the first part we discuss recent work on the structure of the monoid $M(X,r)$ and its monoid algebra $KM(X,r)$, where $K$ is an arbitrary field. In particular, we highlight the importance of the divisibility structure of $M(X,r)$ on the prime ideals of its algebra. Furthermore, we discuss how the homological properties of $KM(X,r)$ are akin to those of the polynomial algebra in several commuting variables. Concretely, a bound on the Gelfand-Kirillov and classical Krull dimension will be discussed. Moreover, some further homological properties can be shown to be equivalent to $r$ being an involution.

By a well-known procedure, usually referred to as "taking the classical limit", quantum systems become classical systems, equipped with a Hamiltonian stucture (symplectic or Poisson). From the deformation quantisation theory we know that a formal deformation of a commutative algebra $\mathcal{A}$ leads to a Poisson bracket on $\mathcal{A}$ and that the classical limit of a derivation on the deformation leads to a Hamiltonian derivation on $\mathcal{A}$ defined by the Poisson bracket. In this talk I present a generalisation of it for formal deformations of an arbitrary noncommutative associative algebra $\mathcal{A}$ [1]. I will show that a deformation leads to a commutative Poisson algebra structure on $\Pi(\mathcal{A}) := Z(\mathcal{A}) × (\mathcal{A}/Z(\mathcal{A}))$ and to the structure of a $\Pi(\mathcal{A})$-Poisson module on $\mathcal{A}$, where $Z(\mathcal{A})$ denotes the centre of $\mathcal{A}$. The limiting derivations are then still derivations of $\mathcal{A}$, but with the Hamiltonians belong to $\Pi(A)$, rather than to $A$. We illustrate our construction with several cases of formal deformations, coming from known quantum algebras, such as the ones associated with the Kontsevich integrable map, the quantum plane, the quantised Grassmann algebra and quantisations of the Volterra hierarchy [2, 3, 4].

This talk is based on a joint work with Pol Vanhaecke [1].
References
[1] Alexander V. Mikhailov and Pol Vanhaecke. Commutative Poisson algebras from deformations of noncommutative algebras. Lett. Math. Phys., 114(5), 1-51, 2024, arXiv:2402.16191v2.
[2] Alexander V. Mikhailov Quantisation ideals of nonabelian integrable systems. Russ. Math. Surv., 75(5):199, 2020, (arXiv:2009.01838), 2020).
[3] Sylvain Carpentier, Alexander V. Mikhailov and Jing Ping Wang. Quantisation of the Volterra hierarchy. Lett. Math. Phys., 112:94, 2022, (arXiv:2204.03095).
[4] Sylvain Carpentier, Alexander V. Mikhailov and Jing Ping Wang. Hamiltonians for the quantised Volterra hierarchy. Nonlinearity, 37(9), 095033 2024, arXiv:2312.12077

NOTES: extra seminar this week on an unusual day (Friday); no seminar next week.
Unrefinable partitions, arising quite unexpectedly in a combinatorial problem in group theory, represent a special subset of integer partitions into distinct parts, constrained by an additional additive relationship between the parts. Despite being a natural combinatorial object, they remain relatively unexplored in the literature, with only a few known properties and results.
In this talk, we explore the foundational aspects of unrefinable partitions, showing some of their initial properties. We will present an algorithm designed to efficiently test for unrefinability in a given partition. By establishing a bound on the largest part in such partitions, we introduce the concept of maximal unrefinable partitions, a subclass with its own distinctive structure. We will show how to count such maximal unrefinable partitions using explicit bijections, providing a clearer understanding of their combinatorial structure.

Brauer graph algeras are finite dimensional algebras constructed from the combinatorial data of a graph called a Brauer graph. Kauer proved that derived equivalences of Brauer graph algebras can be obtained from the move of one edge in the corresponding Brauer graph. Moreover, this derived equivalence is entirely described thanks to a tilting object which can be interpreted in terms of silting mutation. In this talk, I will be interested in skew Brauer graph algebras which generalize the class of Brauer graph algebras. These algebras are constructed from the combinatorial data of a Brauer graph where some edges might be "degenerate". I will explain how Kauer's results can be generalized for the move of multiple edges and to the case of skew Brauer graph algebras.

The positive integers that come up in the solutions of the Diophantine equation $x^2+y^2+z^2=3xyz$ are called Markov numbers. They play an important role in the theory of rational expansions and come with a 100+ year old open problem called the Uniqueness conjecture. Recently discovered connections to cluster algebras have revitalized the theory, leading to generalizations and deformations motivated by the cluster model. In this talk, we will use oriented posets to construct a combinatorial model for $q$-deformed Markov numbers. We will also discuss future directions and further avenues of research.

This is based on the equally named paper which is on arxiv and to appear in Journal of Algebra. I will discuss the connection with the representations of reductive groups and Lie algebras: the hyper algebra, the restricted enveloping algebra and their centres. Then I will move to divided power algebras, truncated polynomial algebras and their invariants. I will discuss some results ($GL_n$ only) from my paper, a conjecture concerning the so-called “restriction property” and the connection with “special symmetrization map” from Okounkov-Olshanskii. If time allows, I will mention extensions of the above results to several matrices and vectors and covectors.

Nichols algebras appear in several areas of mathematics, from Hopf algebras and quantum groups to Schubert calculus and conformal field theories. In this talk, I will review the main problems related to Nichols algebras and discuss some recent classification theorems.