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Abstract: A W-algebra is an associative algebra constructed from a semisimple Liealgebra and its nilpotent element. This paper concentrates on the study of1-dimensional representations of these algebras. Under some conditions on anilpotent element satisfied by all rigid elements we obtain a criterium for afinite dimensional module to have dimension 1. It is stated in terms of theBrundan-Goodwin-Kleshchev highest weight theory. This criterium allows tocompute highest weights for certain completely prime primitive ideals inuniversal enveloping algebras. We make an explicit computation in a specialcase in type $E 8$. Our second principal result is a version of a parabolicinduction for W-algebras. In this case, the parabolic induction is an exactfunctor between the categories of finite dimensional modules for two differentW-algebras. The most important feature of the functor is that it preservesdimensions. In particular, it preserves one-dimensional representations. Aclosely related result was obtained previously by Premet. We also establishsome other properties of the parabolic induction functor.



Autor: Ivan Losev

Fuente: https://arxiv.org/







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