Factorization of Shapovalov elements

2022/02/13 by Mudrov, Andrey · 1 citation
#FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2202.06220

Abstract

Shapovalov elements θ_β,m are special elements in a Borel subalgebra of a classical or quantum universal enveloping algebra parameterized by a positive root β and a positive integer m. They relate the canonical generator of a reducible Verma module with highest vectors of its Verma submodules. For m=1, they can be explicitly obtained as matrix elements of the inverse Shapovalov form. We extend this approach to m>1 for all β but three roots in \mathfrakg₂, \mathfrakf₄, and \mathfrake₈, presenting θ_β,m as a product of matrix elements of weight β.

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