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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
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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 loop reductive group, non-stable $K_1$-functor, Whitehead group, Laurent polynomials, Lie torus
MSC Classifications: 20G35, 19B99, 17B67 show english descriptions Linear algebraic groups over adeles and other rings and schemes
None of the above, but in this section
Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
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
 

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