Ghost center and representations of the diagonal reduction algebra of $\mathfrak{osp}(1|2)$

Abstract

Reduction algebras are known by many names in the literature, including step algebras, Mickelsson algebras, Zhelobenko algebras, and transvector algebras, to name a few. These algebras, realized by raising and lowering operators, allow for the calculation of Clebsch-Gordan coefficients, branching rules, and intertwining operators; and have connections to extremal equations and dynamical R-matrices in integrable face models. In this paper we continue the study of the diagonal reduction superalgebra $A$ of the orthosymplectic Lie superalgebra $\mathfrak{osp}(1|2)$. We construct a Harish-Chandra homomorphism, Verma modules, and study the Shapovalov form on each Verma module. Using these results, we prove that the ghost center (center plus anti-center) of $A$ is generated by two central elements and one anti-central element (analogous to the Scasimir due to Leśniewski for $\mathfrak{osp}(1|2)$). As another application, we classify all finite-dimensional irreducible representations of $A$. Lastly, we calculate an infinite-dimensional tensor product decomposition explicitly.

arXiv comment27 pages; updated introduction: references and motivation; readability; comments welcomed!

References in corpus (7)

• Mickelsson algebras and Zhelobenko operators
• On Casimir's Ghost
• Representations of centrally extended Lie superalgebra $\mathfrak{psl}(2|2)$
• Extremal projectors for contragredient Lie (super)symmetries (short review)
• Generating functions for the osp(1|2) Clebsch-Gordan coefficients
• Diagonal reduction algebra for $\mathfrak{osp}(1|2)$
• Contravariant form for reduction algebras and Pieri rule

Cited by in corpus (1)

• Symplectic Differential Reduction Algebras and Generalized Weyl Algebras