Automorphisms of multiloop lie algebras

Abstract

Multiloop Lie algebras are twisted forms of classical (Chevalley) simple Lie algebras over a ring of Laurent polynomials in several variables k[x±11 , … , x±1n].These algebras occur as centreless cores of extended affine Lie algebras (EALA’s) which are higher nullity generalizations of affine Kac-Moody Lie algebras. Such a multiloop Lie algebra L, also called a Lie torus, is naturally graded by a finite root system Δ, and thus possess a significant supply of nilpotent elements. We compute the difference between the full automorphism group of L and its subgroup generated by exponents of nilpotent elements. The answer is given in terms of Whitehead groups, also called non-stable K1-functors, of simple algebraic groups over the field of iterated Laurent power series k((x1)) … ((xn)). As a corollary, we simplify one step in the proof of conjugacy of Cartan subalgebras in EALA’s due to Chernousov, Neher, Pianzola and Yahorau, under the assumption rank(Δ) ≥ 2.