Speaker
Description
Shapovalov elements $\theta_{\beta,m}$ in the classical or quantum universal enveloping algebra of the negative Borel subalgebra of a simple Lie algebra are parameterized by a positive root $\beta$ and a positive integer $m$. They relate the canonical generator of a reducible Verma module with highest vectors of its Verma submodules. We obtain a factorization of $\theta_{\beta,m}$ to a product of $\theta_{\beta,1}$, where the latter is calculated as a re-scaled matrix element of the inverse Shapovalov form
via a generalized Nigel-Moshinsky algorithm. This way we explicitly express all Shapovalov elements for a classical simple Lie algebra through the Cartan-Weyl basis. In the case of quantum groups, an analogous presentation is available through matrix elements of the universal R-matrix in a representation that goes to adjoint in the classical limit.
