• Corpus ID: 247451028

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

@inproceedings{Hartwig2022GhostCA,
  title={Ghost center and representations of the diagonal reduction algebra of \$\mathfrak\{osp\}(1|2)\$},
  author={Jonas Torbj{\"o}rn Hartwig and Dwight Anderson Williams II},
  year={2022}
}

• J. Hartwig, Dwight Anderson Williams II
• Published 15 March 2022
• Mathematics

. Reduction algebras are known by many names in the literature, including step alge- bras, 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 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 osp (1 | 2) ). As another ap- plication, we classify all finite-dimensional irreducible representations of A . Lastly, we calculate an infinite-dimensional tensor product decomposition explicitly.