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Duality of operator Frobenius algebras and solution of Eisenhart-Stäckel problem in the non-diagonal case

Alexey V. Bolsinov, Andrey Yu. Konyaev, Vladimir S. Matveev

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--Stäckel 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.

Duality of operator Frobenius algebras and solution of Eisenhart-Stäckel problem in the non-diagonal case

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--Stäckel 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 -tensors commute as operator fields.

Paper Structure

This paper contains 5 sections, 14 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Frobenius algebras and their regular representations
  3. Operator Frobenius algebras and their duals
  4. Nijenhuis operators and their symmetry algebras
  5. Application of symmetry algebras to finite-dimensional integrable systems

Key Result

Theorem 1

Let $K_1,\dots, K_n: V \to V$, $\dim V=n$, be commuting operators. The subspace $\mathfrak A=\operatorname{Span}(K_1, \dots, K_n) \subset \mathrm{gl} (V)$ is the image of the regular representation of an $n$-dimensional Frobenius algebra $\mathfrak a$ (under a certain identification $\mathfrak a \si $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (36)

  • Theorem 1
  • proof
  • Lemma 2.1
  • proof
  • Remark 2.1
  • Example 2.1
  • Example 2.2
  • Theorem 2
  • proof
  • Lemma 3.1
  • ...and 26 more