Diagonal reduction algebra for 𝔬𝔰𝔭(1|2)

The problem of providing complete presentations of reduction algebras associated to a pair of Lie algebras \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$( \mathfrak{G} , \mathfrak{g} )$$\end{document} has previously been considered by Khoroshkin and Ogievetsky in the case of the diagonal reduction algebra for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak{gl}(n)$$\end{document}. In this paper, we consider the diagonal reduction algebra of the pair of Lie superalgebras \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$( \mathfrak{osp}(1|2) \times \mathfrak{osp}(1|2) , \mathfrak{osp}(1|2) )$$\end{document} as a double coset space having an associative \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathrel{\scriptstyle\lozenge} $$\end{document}-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.