On conjugacy of Cartan subalgebras in extended affine Lie algebras

Abstract

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 ′ .

Spinor groups with good reduction

“…The case where is the field of rational functions in one variable over a field and consists of the discrete valuations corresponding to all irreducible polynomials was considered by Raghunathan and Ramanathan [RR84] (see also [CGP12, Theorem 2.1]): their result implies that if is a (connected) semi-simple group over and is obtained from by the base change , then any -form that splits over (where is a separable closure of ) and has good reduction at all is obtained by base change from a certain -form of . In the same notation, a description of the -forms of that (split over and) have good reduction at all was obtained by Chernousov, Gille and Pianzola [CGP12], which played a crucial role in the proof of the conjugacy of the analogues of Cartan subalgebras in certain infinite-dimensional Lie algebras [CNPY16]. (We note that if has characteristic zero, then every semi-simple -group becomes quasi-split over , which implies that those that have good reduction at all automatically split over .…”

Non-conjugacy of maximal abelian diagonalizable subalgebras in extended affine Lie algebras

“…Assume that H is a structure MAD of E. By [CNPY,Corollary 3.2] the core E c =L⊕C of E does not depend on the choice of the EALA structure on E. Therefore, by [Ne2,Theorem 6] the image π(H ∩E c ) under the projection map π:E c →L is a k-ad-diagonalizable subalgebra of L which is the 0-component of some Lie torus structure on L. It follows from Lemma 2.5 that π(H ∩E c )=k·s. By one of the axioms of the Lie torus there exists an sl 2 -triple in L of the form (e, f, γs), for some γ ∈k.…”

Conjugacy of Cartan subalgebras in EALAs with a non-fgc centreless core

“…Our construction is a special case of [CNPY,1.4]. Thus, by [CNPY,1.5], the vector space E together with the product (1.2.3) is a Lie algebra.…”

“…Our construction is a special case of [CNPY,1.4]. Thus, by [CNPY,1.5], the vector space E together with the product (1.2.3) is a Lie algebra. Since it is obtained by interlacing the central extension 0 → C → L ⊕ C → L → 0 (obvious maps) with the abelian extension 0 → C → C ⊕ D → D → 0 (again obvious maps) we call this Lie algebra the interlaced extension given by the data (L, β, D, C, τ ) and denote it IE(L, D, C) or IE(L, β, D, C, τ ) if more precision is helpful.…”

“…We have solved Problem (B2) and thus established the Conjugacy Theorem for extended affine Lie algebras in the fgc case in [CNPY,Thm. 7.6].…”