Duality of operator Frobenius algebras and solution of Eisenhart-St\"ackel problem in the non-diagonal case

TL;DR

This paper introduces a duality concept for Frobenius algebras of operator fields, enabling the construction of new integrable systems and solving the Eisenhart--St"ackel problem in arbitrary dimensions.

Contribution

It defines a new duality for Frobenius algebras of operator fields and applies it to solve the Eisenhart--St"ackel problem for all Segre characteristics.

Findings

Duality preserves mutual symmetries of operator fields.

Constructs new integrable hydrodynamic systems from existing ones.

Provides a complete description of finite-dimensional integrable systems with quadratic integrals.

Abstract

We study Frobenius algebras of operator fields and introduce a novel notion of duality for them. We show that, under the assumption that the operator fields forming the Frobenius algebra are mutual symmetries, the operator fields in the dual Frobenius algebra are also mutual symmetries. This result allows one to construct new infinite-dimensional integrable systems of hydrodynamic type starting from a given one. As the main application, we solve the long-standing Eisenhart--St\"ackel problem for any Segre characteristic and in arbitrary dimension: namely, we describe all nondegenerate finite-dimensional integrable systems whose integrals are quadratic in momenta such that the corresponding $(1,1)$-tensors commute as operator fields.