Abstract
The conjugacy of split Cartan subalgebras in the finite dimensional simple case (Chevalley) and in the symmetrizable Kac-Moody case (Peterson-Kac) are fundamental results of the theory of Lie algebras. Among the Kac-Moody Lie algebras the affine algebras stand out. This paper deals with the problem of conjugacy for a class of algebras --extended affine Lie algebras-- that are in a precise sense higher nullity analogues of the affine algebras. Unlike the methods used by Peterson-Kac, our approach is entirely cohomological and geometric. It is deeply rooted on the theory of reductive group schemes developed by Demazure and Grothendieck, and on the work of J. Tits on buildings
Highlights
Let g be a split simple finite-dimensional Lie algebra over a field k of characteristic 0
One of the most elegant ways of establishing that this does not happen, that the type of the root system is an invariant of g, is the conjugacy theorem of split Cartan subalgebras due to Chevalley: all split Cartan subalgebras of g are conjugate under the adjoint action of G(k) where G is the split connected group corresponding to g
Variations of this theme are to be found on the seminal work of Peterson and Kac on conjugacy of “Cartan subalgebras” for symmetrizable Kac–Moody Lie algebras [35]
Summary
Let g be a split simple finite-dimensional Lie algebra over a field k of characteristic 0. One of the most elegant ways of establishing that this does not happen, that the type of the root system is an invariant of g, is the conjugacy theorem of split Cartan subalgebras due to Chevalley: all split Cartan subalgebras of g are conjugate under the adjoint action of G(k) where G is the split connected group corresponding to g Variations of this theme are to be found on the seminal work of Peterson and Kac on conjugacy of “Cartan subalgebras” for symmetrizable Kac–Moody Lie algebras [35]. (2) Since the affine and extended affine algebras are closely related to finitedimensional simple Lie algebras, a proof of conjugacy ought to exist that is faithful to the spirit of finite-dimensional Lie theory That this much is true for toroidal Lie algebras (which correspond to the “untwisted case” in this paper) has been shown in [33]. The main ingredient of the proof of conjugacy is the classification of loop reductive torsors over Laurent polynomial rings given by Theorem 14.1
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