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Stavrova, Anastasia
Non-stable $K_1$-functors of multiloop groups
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
Categories:20G35, 19B99, 17B67

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