Quantum Exceptional Group G2 and its Semisimple Conjugacy Classes

Baranov, Alexander; Mudrov, Andrey; Ostapenko, Vadim

doi:10.1007/s10468-019-09913-4

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Shapovalov Elements Of Degree1

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2023

2024

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Cited by 4 publications

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“…For all non-exceptional types of g, the entries of the matrix C participating in the route summation formula are calculated in [27], Proposition 2.2. That is also done for g 2 in [28]. This makes the above description of Shapovalov elements for such quantum groups absolutely explicit.…”

Section: Shapovalov Elements Of Degreementioning

confidence: 92%

Shapovalov elements of classical and quantum groups

Mudrov¹

2023

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Shapovalov elements θ β,m of the classical or quantized universal enveloping algebra of a simple Lie algebra g are parameterized by a positive root β and a positive integer m. They relate the highest vector of a reducible Verma module with highest vectors of its submodules.We obtain a factorization of θ β,m to a product of θ β,1 and calculate θ β,1 as a residue of a matrix element of the inverse Shapovalov form via a generalized Nigel-Moshinsky algorithm. This way we explicitly express θ β,m of a classical simple Lie algebra through the Cartan-Weyl basis in g. In the case of quantum groups, we give an analogous formulation through the entries of the R-matrix (quantum L-operator) in fundamental representations.

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Section: Shapovalov Elements Of Degreementioning

confidence: 92%