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 $k[x_1^{\pm 1},\ldots ,x_n^{\pm 1}]$. 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 $\mathcal {L}$, also called a Lie torus, is naturally graded by a finite root system $\varDelta$, 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 $K_1$-functors, of simple algebraic groups over the field of iterated Laurent power series $k((x_1))\ldots ((x_n))$. 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 $\mathrm {rank}(\varDelta )\ge 2$.

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