Generalized quadratic commutator algebras of PBW-type

Abstract

In recent years, various nonlinear algebraic structures have been obtained in the context of quantum systems as symmetry algebras, Painlevé transcendent models, and missing label problems. In this paper, we treat all these algebras as instances of the class of quadratic (and higher degree) commutator bracket algebras of Poincaré–Birkhoff–Witt type. We provide a general approach for simplifying the constraints arising from the diamond lemma and apply this in particular to give a comprehensive analysis of the quadratic case. We present new examples of quadratic algebras, which admit a cubic Casimir invariant. The connection with other approaches, such as Gröbner bases, is developed, and we suggest how our explicit and computational techniques can be relevant in other contexts.