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On the Diagonal Reduction Algebra of the Lie Superalgebra osp(1|2)

Mathematical Physics and Probability Seminar

On the Diagonal Reduction Algebra of the Lie Superalgebra osp(1|2)
Series: Mathematical Physics and Probability Seminar
Location: MATH 402
Presenter: Jonas Hartwig, Iowa State University
 
Reduction algebras (also known as step algebras, Mickelsson algebras, Zhelobenko algebras, transvector algebras, etc) are useful tools in representation theory and are associated to a reductive pair of Lie (super)algebras. In this talk we present some results about the reduction algebra $A$ associated to the diagonal embedding $\mathfrak{g}\hookrightarrow \mathfrak{g}\times\mathfrak{g}$ where $\mathfrak{g}$ is the orthosymplectic Lie superalgebra $\mathfrak{osp}(1|2)$. For $A$ we give analogs of a Harish-Chandra homomorphism, Verma modules, and Shapovalov forms. Using these, we give a complete description of the ghost center (center plus anti-center) of $A$: It is generated by two central elements and one anti-central element (analogous to the Scasimir due to Le\’{s}niewski for $\mathfrak{osp}(1|2)$). Lastly we classify all finite-dimensional irreducible representations of $A$. As an application we use an explicit lowering operator to decompose the tensor product of an infinite-dimensional irreducible representation and a finite dimensional irreducible representation into its irreducible summands.
 
This talk is based on joint work with Dwight Anderson Williams II.
 
This talk is in person, but you may also attend via Zoom:  https://arizona.zoom.us/j/85470607369