Seminar talk, 9 December 2020
Speaker: Alexander Mikhailov
Title: Quantisation ideals of nonabelian integrable systems
In my talk I'll discuss a new approach to the problem of quantisation of dynamical systems, introduce the concept of quantisation ideals and show meaningful examples. Traditional quantisation theories start with classical Hamiltonian systems with dynamical 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 systems defined on a free associative algebra. In this approach the quantisation problem is reduced to a description of two-sided ideals which define the commutation relations (the quantisation ideals) in the quotient algebras and which are invariant with respect to the dynamics of the system. Surprisingly this idea works rather efficiently and in a number of cases I have been able to quantise the system, i.e. to find consistent commutation relations for the system. To illustrate this approach I'll consider the quantisation problem for the non-abelian Bogoyavlensky N-chains and other examples, including quantisation of nonabelian integrable ODEs on free associative algebras.
The talk is based on: AVM, Quantisation ideals of nonabelian integrable systems, arXiv preprint arXiv:2009.01838, 2020 (Published in Russ. Math. Surv. v.75:5, pp 199-200, 2020).