Reduced anisotropic unitary Whitehead groups of henselian division algebras with special residue fields of their centers

The reduced anisotropic unitary Whitehead groups of henselian division algebras with unitary involutions are computed in the cases where the centers of residue algebras are of special types.

On reduced anisotropic unitary Whitehead groups of henselian division algebras which are either unramified or tamely ramified with commutative residue algebras

Reduced anisotropic unitary Whitehead groups of henselian division algebras, which are either unramified or tamely ramified with commutative residue algebra, are computed.

Non-stable K1-functors of Multiloop Groups

Let k be a field of characteristic 0. Let G be a reductive group over the ring of Laurent polynomials . Assume that G contains amaximal R-torus, and that every semisimple normal subgroup of G contains a two-dimensional split torus G2m. We show that the natural map of non-stable K1-functors, also called Whitehead groups, KG1(R) → KG1 ( k((x1)) … ((xn))) 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.