An online session of the Algebra and Logic Seminar will be held on April 15, 2022 (Friday) at 1:00 pm (UTC+3).

A talk on:

Quantisation of free associative dynamical systems.
Bi-quantum structure of the stationary KdV hierarchy.
Non-deformation quantisation of the Volterra hierarchy

will be delivered by Alexander V. Mikhailov (University of Leeds, UK).

Abstract. Traditional quantisation theories start with classical Hamiltonian systems with variables taking values in commutative algebras and then study their non-commutative deformations, such that the commutators of observables tend to the corresponding Poisson brackets as the (Planck) constant of deformation goes to zero. I am proposing to depart from dynamical systems defined on a free associative algebra A. In this approach the quantisation problem is reduced to the problem of finding of a two-sided ideal J ⸦ A satisfying two conditions: the ideal J has to be invariant with respect to the dynamics of the system and to define a complete set of commutation relations in the quotient algebras A_J = A / J.

To illustrate this approach I’ll consider the quantisation problem for N-th Novikov equations and the corresponding finite KdV hierarchy. I will show that stationary KdV equations and Novikov’s equations admit two compatible quantisations, i.e. two distinct commutation relations between the variables, such that a linear combination of the corresponding commutators is also a valid quantisation rule leading to the Heisenberg form of quantum equations. The picture is very similar to the bi-Hamiltonian structure in the case of classical integrable equations.

Also, I am going to discuss quantisation of the Bogoyavlensky family of integrable systems. In particular, I will show that odd degree symmetries of the Volterra chain admit two quantisations, one of them is a well-known quantisation of the Volterra chain, and another one is new and not a deformation quantisation.

The talk is partially based on:

AVM, Quantisation ideals of nonabelian integrable systems, arXiv:2009.01838, 2020 (Published in Russ. Math. Surv. v.75:5, pp 199-200, 2020)

V. M. Buchstaber and AVM, KdV hierarchies and quantum Novikov’s equations, arXiv:2109.06357v2, 2021.

The seminar will be held via Zoom and anyone can join at:

https://us02web.zoom.us/j/85137375021?pwd=RE5QczdFTE1xL1R6MnI2b1lkcGczQT09

Topic: Онлайн семинар на секция “Алгебра и логика”
Time: Apr 15, 2022 01:00 PM Sofia
Meeting ID: 851 3737 5021
Passcode: 035647

Algebra and Logic Department, IMI-BAS
http://www.math.bas.bg/algebra/seminarAiL/
============================== =====================