Vladimir Chernousov

Motivic decomposition of isotropic projective homogeneous varieties

We give a decomposition of the Chow motive of an isotropic projective homogeneous variety of a semisimple algebraic group in terms of twisted motives of simpler projective homogeneous varieties. As an application, we prove a generalization of Rost’s nilpotence theorem.

Division algebras with the same maximal subfields

We give a survey of recent results related to the problem of characterizing finite-dimensional division algebras by the set of isomorphism classes of their maximal subfields. We also discuss various generalizations of this problem and some of its applications. In the last section, we extend the problem to the context of absolutely almost simple algebraic groups.

Conjugacy theorems for loop reductive group schemes and Lie algebras

The conjugacy of split Cartan subalgebras in the finite-dimensional simple case (Chevalley) and in the symmetrizable Kac–Moody case (Peterson–Kac) are fundamental results of the theory of Lie algebras. Among the Kac–Moody Lie algebras the affine algebras stand out. This paper deals with the problem of conjugacy for a class of algebras—extended affine Lie algebras—that are in a precise sense higher nullity analogues of the affine algebras. Unlike the methods used by Peterson–Kac, our approach is entirely cohomological and geometric. It is deeply rooted on the theory of reductive group schemes developed by Demazure and Grothendieck, and on the work of Bruhat–Tits on buildings. The main ingredient of our conjugacy proof is the classification of loop torsors over Laurent polynomial rings, a result of its own interest.

Connectedness of classes of fields and zero-cycles on projective homogeneous varieties

We study the Chow group of zero-dimensional cycles for projective homogeneous varieties of semisimple algebraic groups. We show that in many cases this group has no torsion.

The genus of a division algebra and the unramified Brauer group

Let

On the Kernel of the Rost Invariant for E 8 Modulo 3

D

Motivic decomposition of projective homogeneous varieties and the Krull-Schmidt theorem

be a finite-dimensional central division algebra over a field