Constructing Casimirs for the quantum reflection equation

Abstract

We explore the quantum reflection equation in the case of a full-dimensional $n\times n$ matrix $\mathbb A$. Using the technique of $q$-determinants, we find closed expressions for the quantum Pfaffian (in the case of even $n$) and for new quantum Casimirs that are ratios of special quantum (Sklyanin) minors of $\mathbb A$. Special attention is paid to the self-adjointness property of the newly constructed Casimirs. We put forward a conjecture that, adopting a natural ansatz for a quantum deformation of a Casimir of a classical algebra, self-adjointness implies commutativity. We also explore the relationship to networks of quantum cluster variables.