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Automorphisms of Multiloop Lie Algebras

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Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 191)

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 \(\mathrm {rank}(\varDelta )\ge 2\).

Keywords

  • Galois Ring
  • Simple Algebraic Group
  • Twisted Form
  • Reductive Group Scheme
  • Nondegenerate Symmetric Bilinear Form

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Acknowledgements

The author was supported at different stages of her work by the postdoctoral grant 6.50.22.2014 “Structure theory, representation theory and geometry of algebraic groups” and the travel grant 6.41.667.2015 at St. Petersburg State University, by the J.E. Marsden postdoctoral fellowship of the Fields Institute, and by the RFBR grants 14-01-31515-mol_a, 13-01-00429-a, 16-01-00750-a.

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Correspondence to Anastasia Stavrova .

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Stavrova, A. (2016). Automorphisms of Multiloop Lie Algebras. In: Dobrev, V. (eds) Lie Theory and Its Applications in Physics. LT 2015. Springer Proceedings in Mathematics & Statistics, vol 191. Springer, Singapore. https://doi.org/10.1007/978-981-10-2636-2_40

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