location:  Publications → journals → CJM
Abstract view

# Non-stable $K_1$-functors of Multiloop Groups

Published:2015-10-21
Printed: Feb 2016
• Anastasia Stavrova,
Department of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg, Russia
 Format: LaTeX MathJax PDF

## Abstract

Let $k$ be a field of characteristic 0. Let $G$ be a reductive group over the ring of Laurent polynomials $R=k[x_1^{\pm 1},...,x_n^{\pm 1}]$. 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, $K_1^G(R)\to K_1^G\bigl( k((x_1))...((x_n)) \bigr)$ 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.
 Keywords: loop reductive group, non-stable $K_1$-functor, Whitehead group, Laurent polynomials, Lie torus
 MSC Classifications: 20G35 - Linear algebraic groups over adeles and other rings and schemes 19B99 - None of the above, but in this section 17B67 - Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras

 top of page | contact us | privacy | site map |