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Fibers of characters in Gelfand-Tsetlin categories

Authors: Vyacheslav Futorny and Serge Ovsienko
Journal: Trans. Amer. Math. Soc. 366 (2014), 4173-4208
MSC (2010): Primary 16D60, 16D90, 16D70, 17B65
Published electronically: April 7, 2014
MathSciNet review: 3206456
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Abstract: For a class of noncommutative rings, called Galois orders, we study the problem of an extension of characters from a commutative subalgebra. We show that for Galois orders this problem is always solvable in the sense that all characters can be extended, moreover, in finitely many ways, up to isomorphism. These results can be viewed as a noncommutative analogue of liftings of prime ideals in the case of integral extensions of commutative rings. The proposed approach can be applied to the representation theory of many infinite dimensional algebras including universal enveloping algebras of reductive Lie algebras (in particular $ \mathrm {gl}_n$), Yangians and finite $ W$-algebras. As an example we recover the theory of Gelfand-Tsetlin modules for $ \mathrm {gl}_n$.

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Additional Information

Vyacheslav Futorny
Affiliation: Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281, São Paulo, CEP 05315-970, Brasil

Serge Ovsienko
Affiliation: Faculty of Mechanics and Mathematics, Kiev Taras Shevchenko University, Vladimirskaya 64, 00133, Kiev, Ukraine

Received by editor(s): January 19, 2010
Received by editor(s) in revised form: June 18, 2010, August 14, 2012, and August 18, 2012
Published electronically: April 7, 2014
Additional Notes: The first author was supported in part by the CNPq grant (processo 301743/2007-0) and by the Fapesp grant (processo 2010/50347-9)
The second author is grateful to Fapesp for financial support (processos 2004/02850-2 and 2006/60763-4) and to the University of São Paulo for their hospitality during his visits
Article copyright: © Copyright 2014 American Mathematical Society
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