Резюме. 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.