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Durch graduierte Lie-Algebren definierte Paar-Algebren

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Abstract

Pair algebras which have a non degenerate (left- and right-) invariant bilinear form and for which the inner derivation algebra is completely reducible are characterised by pairs (C,μ), where C is a n×n matrix satisfying certain conditions and μ is a sequence of n integers equal to 0 or 1. They occur as pair algebras of type (S(C,μ)−1,S(C,μ)1), xuy=[[x,u],y], where (S(C,μ)r)rεℤ is the gradation induced by μ. in the Kac-Moody algebraS(C). If C is an affin Cartan matrix (as in the case of Lie triple systems), there exists a finite dimensional simple Lie algebrag and a θ ε Aut (g), ord θ=m<∞ such that the pair algebra is isomorphic to the pair algebra (g −1,g 1), xuy=[[x,u],y] (product ing), whereg i. is the eigenspace of θ of eigenvalue ωi, ω a primitive m-th root of unity.

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  1. Fachbereich Mathematik der Universität, Hölderlinstraße 3, D-5900, Siegen, Bundesrepublik Deutschland

    Wolfgang Hein

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  1. Wolfgang Hein
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Hein, W. Durch graduierte Lie-Algebren definierte Paar-Algebren. Manuscripta Math 52, 97–130 (1985). https://doi.org/10.1007/BF01171488

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