منابع مشابه
Cyclic elements in semisimple Lie algebras
where g±d 6= 0. The positive integer d is called the depth of this Z-grading, and of the nilpotent element e. This notion was previously studied e.g. in [P1]. An element of g of the form e+ F , where F is a non-zero element of g−d, is called a cyclic element, associated with e. In [K1] Kostant proved that any cyclic element, associated with a principal (= regular) nilpotent element e, is regula...
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The goal of Section 2 is to provide a proof of Theorem 2.0.1. Section 3 introduces the necessary facts about Lie algebras and representation theory, with the goal being the proof of Proposition 3.5.7 (ultimately as an application of Theorem 2.0.1), and Proposition 3.3.1. In Section 4 we prove the main theorem, using Propositions 3.3.1 and 3.5.7. In Section 5, we apply the theorem to the special...
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Let L be a finite-dimensional, semisimple Lie algebra over an algebraically closed field k of characteristic 0. Let H be a fixed Cartan subalgebra of L, and Φ be the root system. Fix a base ∆ = {α1, · · · , αl} of Φ. Let Λ denote the set of dominant, integral linear functions on H. Theorem 0.1. There is a one-to-one correspondence Λ ∼ −→ {isomorphism classes of finite-dimensional irreducible L-...
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Furthermore h was diagonalisable in every irreducible representation and H := Span(h) is obviously an abelian subalgebra. Note that h = h + 0 is the abstract Jordan decomposition of h, that H = CL(H) is the weight space of H , acting on L with the adjoint action, corresponding to the weight 0 ∈ H . Likewise, Span(e) is the weight space for the weight c · h 7→ −2c for c ∈ C, and Span( f ) is the...
متن کاملRepresentations of Semisimple Lie Algebras
This paper studies the representations of semisimple Lie algebras, with care given to the case of sln(C). We develop and utilize various tools, including the adjoint representation, the Killing form, root space decomposition, and the Weyl group to classify the irreducible representations of semisimple Lie algebras.
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ژورنال
عنوان ژورنال: Transformation Groups
سال: 2013
ISSN: 1083-4362,1531-586X
DOI: 10.1007/s00031-013-9214-0