KdV hierarchies and quantum Novikov's equations

This paper begins with a review of the well-known KdV hierarchy, the $N$-th Novikov equation, and its finite hierarchy in the classical commutative case. This finite hierarchy consists of $N$ compatible integrable polynomial dynamical systems in $\mathbb{C}^{2N}$. We discuss a non-commutative version of the $N$-th Novikov hierarchy defined on the finitely generated free associative algebra ${\mathfrak{B}}_N$ with $2N$ generators. Using the method of quantisation ideals in ${\mathfrak{B}}_N$, for $N=1,2,3,4$, we obtain two-sided homogeneous ideals ${\mathfrak{Q}}_N\subset{\mathfrak{B}}_N$ (quantisation ideals) that are invariant with respect to the $N$-th Novikov equation and such that the quotient algebra ${\mathfrak{C}}_N = {\mathfrak{B}}_N/ {\mathfrak{Q}}_N$ has a well-defined Poincare-Birkhoff-Witt basis. This allows us to define the quantum $N$-th Novikov equation and its hierarchy on ${\mathfrak{C}}_N$. We derive $N$ commuting quantum first integrals (Hamiltonians) and represent the equations of the hierarchy in the Heisenberg form. Essential for our research is the concept of cyclic Frobenius algebras, which we introduced in our recent paper. In terms of the quadratic form that defines the structure of a cyclic Frobenius algebra, we explicitly express the first integrals of the $N$-th Novikov hierarchy in the commutative, free, and quantum cases.