Generalized quadratic commutator algebras of PBW-type

TL;DR

This paper introduces a unified approach to quadratic and higher-degree commutator algebras of PBW type, providing new examples and computational techniques relevant for quantum symmetry structures.

Contribution

It offers a general method for simplifying constraints in quadratic algebras and presents new algebraic examples with cubic Casimir invariants, connecting to computational algebra.

Findings

New quadratic algebras with cubic Casimir invariants

Unified framework for nonlinear algebraic structures in quantum systems

Connection established with Gröbner bases techniques

Abstract

In recent years, various nonlinear algebraic structures have been obtained in the context of quantum systems as symmetry algebras, Painlev\'{e} transcendent models and missing label problems. In this paper we treat all of these algebras as instances of the class of quadratic (and higher degree) commutator bracket algebras of PBW 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 Gr\"{o}bner bases is developed, and we suggest how our explicit and computational techniques can be relevant in other contexts.