### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 099, 32 pages      arXiv:1708.02280      https://doi.org/10.3842/SIGMA.2017.099

### Contractions of Degenerate Quadratic Algebras, Abstract and Geometric

Mauricio A. Escobar Ruiz a, Willard Miller Jr. b and Eyal Subag c
a) Centre de Recherches Mathématiques, Université de Montreal, C.P. 6128, succ. Centre-Ville, Montréal, QC H3C 3J7, Canada
b) School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455, USA
c) Department of Mathematics, Pennsylvania State University, State College, Pennsylvania, 16802 USA

Received August 09, 2017, in final form December 26, 2017; Published online December 31, 2017

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 Bôcher 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 Bôcher 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 Bôcher 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 Bôcher contractions.

Key words: Bôcher contractions; quadratic algebras; superintegrable systems; conformal superintegrability; Poisson structures.

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