Algebra

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.

There is an equivalence between the category of simplicial abelian groups and the category of differential graded abelian groups called the Dold-Kan equivalence. There is also a class of curious objects called Crossed-Simplicial Groups defined by a distributive law between a collection of groups indexed by the natural numbers and the simplicial category $\Delta$. There have been attempts to extend the Dold-Kan to crossed-simplicial setting by explicitly constructing extensions of the differential graded side. But these are few and far between. In this talk, I'll start by a new but a weakened analogue of the Dold-Kan by using certain induction and restriction functors, and by passing to the homotopy categories on both sides. Then show that this homotopical version of the equivalence extends to the crossed-simplicial setting.