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- Title
Twisting functors and Gelfand–Tsetlin modules over semisimple Lie algebras.
- Authors
Futorny, Vyacheslav; Křižka, Libor
- Abstract
We associate to an arbitrary positive root α of a complex semisimple finite-dimensional Lie algebra a twisting endofunctor T α of the category of -modules. We apply this functor to generalized Verma modules in the category () and construct a family of α -Gelfand–Tsetlin modules with finite Γ α -multiplicities, where Γ α is a commutative ℂ -subalgebra of the universal enveloping algebra of generated by a Cartan subalgebra of and by the Casimir element of the (2) -subalgebra corresponding to the root α. This covers classical results of Andersen and Stroppel when α is a simple root and previous results of the authors in the case when is a complex simple Lie algebra and α is the maximal root of . The significance of constructed modules is that they are Gelfand–Tsetlin modules with respect to any commutative ℂ -subalgebra of the universal enveloping algebra of containing Γ α . Using the Beilinson–Bernstein correspondence we give a geometric realization of these modules together with their explicit description. We also identify a tensor subcategory of the category of α -Gelfand–Tsetlin modules which contains constructed modules as well as the category ().
- Subjects
LIE algebras; UNIVERSAL algebra
- Publication
Communications in Contemporary Mathematics, 2023, Vol 25, Issue 8, p1
- ISSN
0219-1997
- Publication type
Article
- DOI
10.1142/S0219199722500316