Non-stable K₁-functors of multiloop groups

2014/04/30 by Stavrova, A.
#FOS: Mathematics #Group Theory (math.GR) #K-Theory and Homology (math.KT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1404.7587

Abstract

Let k be a field of characteristic 0. Let G be a reductive group over the ring of Laurent polynomials R=k[x₁^{± 1},...,x_n^{± 1}] containing a maximal R-torus T (equivalently, loop reductive). Assume also that every semisimple normal subgroup of G contains a two-dimensional split torus G_m². We show that the natural map of non-stable K₁-functors K₁^G(R)-> K₁^G(k((x₁))...((x_n))) is injective. This complements the surjectivity result for the same map obtained by V. Chernousov, P. Gille and A. Pianzola in arXiv:1109.5236. As a corollary, we provide a way to evaluate the difference between the full automorphism group of a Lie torus (in the sense of Yoshii-Neher) and the subgroup generated by exponential automorphisms.

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