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Non-stable K1-functors of Multiloop Groups

Published online by Cambridge University Press:  20 November 2018

Anastasia Stavrova*
Affiliation:
Department of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg, Russia e-mail: anastasia.stavrova@gmail.com
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Abstract

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Let $k$ be a field of characteristic 0. Let $G$ be a reductive group over the ring of Laurent polynomials $R\,=\,k\left[ x_{1}^{\pm 1},\ldots ,x_{n}^{\pm 1} \right]$ . Assume that $G$ contains a maximal $R$ -torus, and that every semisimple normal subgroup of $G$ contains a two-dimensional split torus $\mathbf{G}_{m}^{2}$ . We show that the natural map of non-stable ${{K}_{1}}$ -functors, also called Whitehead groups, $K_{1}^{G}\left( R \right)\,\to \,K_{1}^{G}\left( k\left( \left( {{x}_{1}} \right) \right)\cdots \left( \left( {{x}_{n}} \right) \right) \right)$ is injective, and an isomorphism if $G$ is semisimple. As an application, we provide a way to compute the difference between the full automorphism group of a Lie torus (in the sense of Yoshii–Neher) and the subgroup generated by exponential automorphisms.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2016

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