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Erschienen in:
2016 | OriginalPaper | Buchkapitel
Automorphisms of Multiloop Lie Algebras
verfasst von : Anastasia Stavrova
Erschienen in: Lie Theory and Its Applications in Physics
Verlag: Springer Singapore
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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 \(\mathrm {rank}(\varDelta )\ge 2\).