1. CJM 2015 (vol 68 pp. 150)
 Stavrova, Anastasia

Nonstable $K_1$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 twodimensional
split torus $\mathbf{G}_m^2$.
We show that the natural map of nonstable $K_1$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 YoshiiNeher)
and the subgroup generated by
exponential automorphisms.
Keywords:loop reductive group, nonstable $K_1$functor, Whitehead group, Laurent polynomials, Lie torus Categories:20G35, 19B99, 17B67 
