Algebraic structures and Hamiltonians from the equivalence classes of 2D conformal algebras

TL;DR

This paper introduces an algebraic method to classify and construct Hamiltonians with integrals based on 2D conformal algebra subalgebras, revealing quadratic symmetry algebras without explicit differential operator realizations.

Contribution

It develops an algebraic framework for classifying centralisers of the conformal algebra's enveloping algebra and constructing associated Hamiltonians, advancing the understanding of superintegrable systems.

Findings

Classified centralisers of the conformal algebra's enveloping algebra.

Constructed Hamiltonians with algebraic integrals in a new framework.

Identified six-dimensional quadratic symmetry algebras with closed Berezin brackets.

Abstract

The construction of superintegrable systems based on Lie algebras and their universal enveloping algebras has been widely studied over the past decades. However, most constructions rely on explicit differential operator realisations and Marsden-Weinstein reductions. In this paper, we develop an algebraic approach based on the subalgebras of the 2D conformal algebra $\mathfrak{c}(2)$. This allows us to classify the centralisers of the enveloping algebra of the conformal algebra and construct the corresponding Hamiltonians with integrals in algebraic form. It is found that the symmetry algebras underlying these algebraic Hamiltonians are six-dimensional quadratic algebras. The Berezin brackets and commutation relations of the quadratic algebraic structures are closed without relying on explicit realisations or representations. We also give the Casimir invariants of the symmetry algebras. Our approach provides algebraic perspectives for the recent work by Fordy and Huang on the construction of superintegrable systems in the Darboux spaces.