Cyclic Frobenius algebras

TL;DR

This paper introduces cyclic Frobenius algebras, explores their structures on various algebra types, and connects them to integrable systems, providing explicit formulas for hierarchy integrals across classical, non-Abelian, and quantum contexts.

Contribution

It defines cyclic Frobenius algebras and demonstrates their existence on associative and Poisson algebras, linking them to integrable systems and hierarchies.

Findings

CF-algebras exist on associative and Poisson algebras

CF-algebra structures are found in the KdV hierarchy

Explicit formulas for hierarchy integrals are derived

Abstract

In this paper, we introduce the notion of cyclic Frobenius algebras (CF-algebras). Canonical structures of CF-algebras exist on associative and Poisson algebras. It turns out that the modern theory of integrable systems yields non-trivial examples of CF-algebras. In the theory of the KdV hierarchy there is a structure of CF-algebra which leads to explicit expressions for the first integrals of the N-th Novikov hierarchy, suitable for classical, non-Abelian and quantum cases.