Contractions of Degenerate Quadratic Algebras, Abstract and Geometric

TL;DR

This paper classifies and analyzes degenerate quadratic algebras related to superintegrable systems, exploring their geometric contractions and how they can be realized within phase space, extending previous work on nondegenerate cases.

Contribution

It provides a classification of degenerate quadratic algebras, examines their geometric contractions, and identifies which can be realized as superintegrable systems, building on prior classifications of nondegenerate cases.

Findings

Classified all free degenerate quadratic algebras arising from superintegrability.

Mapped Bôcher contractions between degenerate superintegrable systems.

Identified exceptions where abstract contractions do not correspond to geometric contractions.

Abstract

Quadratic algebras are generalizations of Lie algebras which include the symmetry algebras of 2nd order superintegrable systems in 2 dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical and quantum mechanics. Distinct superintegrable systems and their quadratic algebras can be related by geometric contractions, induced by B\^ocher contractions of the conformal Lie algebra $\mathfrak{so}(4,\mathbb {C})$ to itself. In 2 dimensions there are two kinds of quadratic algebras, nondegenerate and degenerate. In the geometric case these correspond to 3 parameter and 1 parameter potentials, respectively. In a previous paper we classified all abstract parameter-free nondegenerate quadratic algebras in terms of canonical forms and determined which of these can be realized as quadratic algebras of 2D nondegenerate superintegrable systems on constant curvature spaces and Darboux spaces, and studied the relationship between B\^ocher contractions of these systems and abstract contractions of the free quadratic algebras. Here we carry out an analogous study of abstract parameter-free degenerate quadratic algebras and their possible geometric realizations. We show that the only free degenerate quadratic algebras that can be constructed in phase space are those that arise from superintegrability. We classify all B\^ocher contractions relating degenerate superintegrable systems and, separately, all abstract contractions relating free degenerate quadratic algebras. We point out the few exceptions where abstract contractions cannot be realized by the geometric B\^ocher contractions.