# Mikhailov A. Quantisation of free associative dynamical systems (abstract)

Speaker: Alexander Mikhailov

Title: Quantisation of free associative dynamical systems

Bi-quantisation of stationary KdV hierarchy and Novikov's equations and non-deformation quantisation of the Volterra sub-hierarchy.

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 ${\displaystyle {\mathfrak {A}}}$. In this approach the quantisation problem is reduced to description of two-sided ideals ${\displaystyle {\mathfrak {J}}\subset {\mathfrak {A}}}$ satisfying two conditions: the ideals have to be invariant with respect to the dynamics of the system and to define a complete set of commutation relations in the quotient algebras ${\displaystyle {\mathfrak {A}}_{\mathfrak {J}}={\mathfrak {A}}\diagup {\mathfrak {J}}}$.

To illustrate this approach I'll consider the quantisation problem for ${\displaystyle N}$-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 i s 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.

I'll discuss quantisation of the Bogoyavlensky family of integrable ${\displaystyle N}$--chains:

${\displaystyle {\frac {du_{n}}{dt}}=\sum _{k=1}^{N}(u_{n+k}u_{n}-u_{n}u_{n-k}),\qquad n\in \mathbb {Z} ,\qquad \qquad \qquad \qquad (1)}$

quantisation of their symmetries and modifications. In particular, I will show t hat odd degree symmetries of the Volterra chain (${\displaystyle N=1}$ in (1)) admit two quantisations, one of them corresponds to known quantisation of the Volterra chain, and another one is new and not deformational.

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:2109.06357, 2021.

Event: Diffieties, Cohomological Physics, and Other Animals, 13-17 December 2021, Moscow.