Cyclic Frobenius algebras
V.M. Buchstaber, and A.V. Mikhailov

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.
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