International Conference. S. K. Lando – ＂Weight systems associated to Lie algebras＂

Описание

International Conference, dedicated to the 75th birthday of Sabir M. Gusein-Zade
16.08.25
S. K. Lando (Moscow, Russia)
Weight systems associated to Lie algebras

V. A. Vassiliev's theory of finite type knot invariants allows one to associate to such an invariant a function on chord diagrams, which are simple combinatorial objects, consisting of an oriented circle and a tuple of chords with pairwise distinct ends in it. Such functions are called weight systems. According to a Kontsevich theorem, such a correspondence is essentially one-to-one: each weight system determines certain knot invariant.

In particular, a weight system can be associated to any semi-simple Lie algebra. However, already in the simplest nontrivial case, the one for the Lie algebra sl(2), computation of the values of the corresponding weight system is a computationally complicated task. This weight system is of great importance, however, since it corresponds to a famous knot invariant known as the colored Jones polynomial.

Last few years was a period of significant progress in understanding and computing Lie algebra weight systems, both for sl(2)- and gl(N)-weight system, for arbitrary N. These methods are based on an idea, due to M. Kazarian, which suggests a recurrence for gl(N)-weight system extended to permutations. The recurrence immediately leads to a construction of a universal gl-weight system taking values in the ring of polynomials ℂ[N, C₁, C₂, C₃, …] in infinitely many variables and allowing for a specialization to gl(N)- and sl(N)-weight systems for any given value of N. A lot of new explicit formulas were obtained.

Simultaneously, Zhuoke Yang extended the construction to the Lie superalgebras gl(N|M), and, together with M. Kazarian, to other classical series of Lie algebras. It happened that certain specializations of the universal gl-weight system lead to well-known combinatorial invariants of graphs, allowing thus to extend these invariants to permutations.

Certain integrability properties of the Lie algebra weight systems will be discussed.

The talk is based on work of M. Kazarian, the speaker, and N. Kodaneva, P. Zakorko, Zhuoke Yang, and P. Zinova.

General information
The conference will be held in the Faculty of Mathematics of the HSE University (Usacheva str. 6, Moscow).
Dates: 16 - 20 June 2025
Organizers: A. Buryak, V. Medvedev, A. Skripchenko

More information:
https://math.hse.ru/modern_aspects_singularity_theory25/
https://mccme.ru/ru/modern_aspects_singularity_theory2025/

Автор

Mathematics at HSE