The Kostrikin Radical and the Invariance of the Core of Reduced Extended Affine Lie Algebras

Tocón, Maribel

doi:10.4153/cmb-2008-030-5

Cited by 1 publication

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“…For special types of EALAs, namely those for which the the root system ∆ in (2.6.1) is reduced, Corollary 3.2 is proven in [T,Th. 5.1] with a completely different method.…”

Section: Invariance Of the Corementioning

confidence: 97%

On conjugacy of Cartan subalgebras in extended affine Lie algebras

Chernousov

Neher

Pianzola

et al. 2016

Advances in Mathematics

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That finite-dimensional simple Lie algebras over the complex numbers can be classified by means of purely combinatorial and geometric objects such as Coxeter-Dynkin diagrams and indecomposable irreducible root systems, is arguably one of the most elegant results in mathematics. The definition of the root system is done by fixing a Cartan subalgebra of the given Lie algebra. The remarkable fact is that (up to isomorphism) this construction is independent of the choice of the Cartan subalgebra. The modern way of establishing this fact is by showing that all Cartan subalgebras are conjugate.For symmetrizable Kac-Moody Lie algebras, with the appropriate definition of Cartan subalgebra, conjugacy has been established by Peterson and Kac. An immediate consequence of this result is that the root systems and generalized Cartan matrices are invariants of the Kac-Moody Lie algebras. The purpose of this paper is to establish conjugacy of Cartan subalgebras for extended affine Lie algebras; a natural class of Lie algebras that generalizes the finite-dimensional simple Lie algebra and affine Kac-Moody Lie algebras.Part of the properties of an EALA (E, H) is a root space decomposition: E = α∈Ψ E α with E 0 = H. The "root system" Ψ is an example of an extended affine root system. The main question, of course, is whether Ψ is an invariant of E. In other words, if H ′ is a subalgebra of E for which the pair (E, H ′ ) is given an EALA structure, is the resulting root system Ψ ′ isomorphic (in the sense of [extended affine] root systems) to Ψ? That this is true follows immediately from the main result of our paper. 0.1. Theorem (Theorem 7.6). Let (E, H) be an extended affine Lie algebra of fgc type. Assume E admits the second structure (E, H ′ ) of an extended affine Lie algebra. Then H and H ′ are conjugate, i.e., there exists a k-linear automorphism f of the Lie algebra E such that f (H) = H ′ .

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“…For special types of EALAs, namely those for which the the root system ∆ in (2.6.1) is reduced, Corollary 3.2 is proven in [T,Th. 5.1] with a completely different method.…”

Section: Invariance Of the Corementioning

confidence: 97%