Wide Subalgebras of Semisimple Lie Algebras |
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Authors: | Dmitri I Panyushev |
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Institution: | 1. Institute for Information Transmission Problems of the R.A.S., Dobrushin Mathematical Laboratory, B. Karetnyi per. 19, Moscow, 127994, Russia
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Abstract: | Let G be a connected semisimple algebraic group over \({\mathbb C}\) , with Lie algebra \({\mathfrak g}\) . Let \({\mathfrak h}\) be a subalgebra of \({\mathfrak g}\) . A simple finite-dimensional \({\mathfrak g}\) -module \({\mathbb V}\) is said to be \({\mathfrak h}\) -indecomposable if it cannot be written as a direct sum of two proper \({\mathfrak h}\) -submodules. We say that \({\mathfrak h}\) is wide, if all simple finite-dimensional \({\mathfrak g}\) -modules are \({\mathfrak h}\) -indecomposable. Some very special examples of indecomposable modules and wide subalgebras appear recently in the literature. In this paper, we describe several large classes of wide subalgebras of \({\mathfrak g}\) and initiate their systematic study. Our approach is based on the study of idempotents in the associative algebra of \({\mathfrak h}\) -invariant endomorphisms of \({\mathbb V}\) . We also discuss a relationship between wide subalgebras and epimorphic subgroups. |
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