Quantum Novikov equations

In the first part of the talk, we will discuss the well-known Korteweg-de Vries hierarchy in the free associative algebra of an infinite number of variables. For each natural number N, we give an explicit description of the noncommutative version of the N-th Novikov equation and its first integrals in a free associative algebra of 2N variables. In the second part of the talk, we will introduce the N-th quantum Novikov equations and describe their first integrals. Using the examples, we will show how work the general method of quantization ideals, recently introduced by A.V. Mikhailov. In our case, we are talking about a two-sided ideal in the free associative algebra of 2N variables, which is invariant under the noncommutative N-th Novikov equation in this algebra. A factor by such ideal defines a dynamical system in an associative algebra AN of 2N variables with the additive Poincare–Birkhoff–Witt basis. In the third part of the talk, we will introduce a polynomial invertible transformation of the algebra AN, which transforms the N-th quantum Novikov equation and the corresponding quantum hierarchy to the standard Heisenberg form. As result we will obtain the operator representation of explicitly given quantum Hamiltonians. The talk is based on the results obtained jointly with A.V. Mikhailov.