KdV hierarchies and quantum Novikov's equations

The paper begins with a review of the well known KdV hierarchy, N-th Novikov equations and its finite hierarchiey in the classical commutative case. Its finite hierarchy consists of N compatible integrable polynomial dynamical systems in C2N. Then we discuss a non-commutative version of the N-th Novikov hierarchy defined on a finitely generated free associative algebra BN with 2N generators. Using the quantisation ideals method in BN, for N = 1, 2, 3, 4, we have found two-sided homogeneous ideals QN ⊂ BN (quantisation ideals) which are invariant with respect to the N-th Novikov equation and such that the quotient algebra CN = BN /QN has a well defined Poincare–Birkhoff–Witt basis. It enables us to define the quantum N-th Novikov equation and its hierarchy on the CN. We have found N commuting quantum first integrals (Hamiltonians) and represented equations of the hierarchy in the Heisenberg form. In this paper we introduce the concept of Frobenius-Hochschild algebras and in its terms we express explicitly first integrals of the N-th Novikov hierarchy in the commutative, free and quantum cases.