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Fixed point subalgebras of root graded Lie algebras

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  • Mathematics

Abstract

Yousofzadeh, M. Osaka J. Math. 46 (2009), 611–643 FIXED POINT SUBALGEBRAS OF ROOT GRADED LIE ALGEBRAS MALIHE YOUSOFZADEH (Received April 9, 2008) Abstract We study the subalgebra of fixed points of a root graded Lie algebra under a certain class of finite order automorphisms. As the centerless core of extended affine Lie algebras or equivalently irreducible centerless Lie tori are examples of root graded Lie algebras, our work is an extension of some recent result about the subalgebra of fixed points of a Lie torus under a certain finite order automorphism. 0. Introduction In 1955, A. Borel and G.D. Mostow [7] proved that the fixed point subalgebra (or f.p.s. for the sake of brevity) of a finite dimensional simple Lie algebra over a field F of characteristic zero, under a finite order automorphism, is a reductive Lie algebra. As a finite dimensional simple Lie algebra is an extended affine Lie algebra of nullity zero, a natural question which arises is that, what is the f.p.s. of an extended affine Lie algebra under a certain finite order automorphism? In 2005, S. Azam, S. Berman and M. Yousofzadeh [5] considered and showed that such a subalgebra has a reductive-like structure, namely it is decomposed into a sum of extended affine Lie algebras (up to existence of some isolated root spaces), a subspace of the center and a subspace con- tained in the centralizer of the core. Since the centerless core of an extended affine Lie algebra is a centerless irreducible Lie torus, a second question arises: What we can say about the fixed points of a Lie torus under automorphisms of similar nature. In 2006, S. Azam and V. Khalili [4] studied the f.p.s. of a centerless irreducible Lie torus L un- der a certain class of finite order automorphisms. They showed that the centerless core of the f.p.s. of L under an automorphism in the stated class is a direct sum of center- less irreducible Lie tori. In this article, we consider a similar question for a much more general class of Lie algebras, name

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