Diagonal reduction algebra for $\mathfrak{osp}(1|2)$

TL;DR

This paper studies the diagonal reduction algebra for the Lie superalgebra rak{osp}(1|2), providing a complete presentation with generators, relations, a PBW basis, Casimir-like elements, and automorphisms.

Contribution

It offers the first complete presentation of the diagonal reduction algebra for rak{osp}(1|2), including generators, relations, and structural elements, extending previous work on Lie algebras.

Findings

Complete presentation of the algebra with generators and relations

Construction of a PBW basis for the algebra

Identification of Casimir-like elements and automorphisms

Abstract

The problem of providing complete presentations of reduction algebras associated to a pair of Lie algebras $(\mathfrak{G},\mathfrak{g})$ has previously been considered by Khoroshkin and Ogievetsky in the case of the diagonal reduction algebra for $\mathfrak{gl}(n)$. In this paper we consider the diagonal reduction algebra of the pair of Lie superalgebras $\left(\mathfrak{osp}(1|2) \times \mathfrak{osp}(1|2), \mathfrak{osp}(1|2)\right)$ as a double coset space having an associative diamond product and give a complete presentation in terms of generators and relations. We also provide a PBW basis for this reduction algebra along with Casimir-like elements and a subgroup of automorphisms.