School of Mathematics Colloquium

Reduced Whitehead groups of algebras.

by Dr. Nivedita Bhaskhar (UCLA, USA)

Thursday, December 20, 2018 from to (Asia/Kolkata)
at AG-69
Description
Abstract:
Any central simple algebra A over a field K is a form of a matrix algebra.
Further A/K comes equipped with a reduced norm map which is obtained by
twisting the determinant function. Every element in the  commutator
subgroup [A*, A*] has reduced norm 1 and hence lies in SL_1(A), the group
of reduced norm one elements of A. Whether the reverse inclusion holds was
formulated as a question in 1943 by Tannaka and Artin in terms of the
triviality of the reduced Whitehead group SK_1(A) := SL_1(A)/[A*,A*].

One can define Whitehead groups more generally for any isotropic group and
it turns out that the Tannaka-Artin question is a special case of the
well-known Kneser-Tits conjecture. The Whitehead group detects the
non-rationality of the underlying variety of the algebraic group and
therefore is an interesting albeit difficult invariant to study. In this
talk, we discuss these connections to rationality questions and trace the
progress towards answering the Tannaka-Artin question, which becomes
especially interesting over low cohomological dimension fields.