Equivariant noncommutative motives

Kontsevich’s Noncommutative Numerical Motives

In this note we prove that Kontsevich’s category NCnum(k)F of noncommutative numerical motives is equivalent to the one constructed by the authors in [14]. As a consequence, we conclude that NCnum(k)F is abelian semi-simple as conjectured by Kontsevich.

Jacobians of Noncommutative Motives

In this article one extends the classical theory of (intermediate) Jacobians to the “noncommutative world”. Concretely, one constructs a Q-linear additive Jacobian functor N 7→ J(N) from the category of noncommutative Chow motives to the category of abelian varieties up to isogeny, with the following properties: (i) the first de Rham cohomology group of J(N) agrees with the subspace of the odd ...

Noncommutative Artin Motives

In this article we introduce the category of noncommutative Artin motives as well as the category of noncommutative mixed Artin motives. In the pure world, we start by proving that the classical category AM(k)Q of Artin motives (over a base field k) can be characterized as the largest category inside Chow motives which fully-embeds into noncommutative Chow motives. Making use of a refined bridg...

Equivariant KK-theory and noncommutative index theory

Equivariant noncommutative index on braided sphere

To some Hecke symmetries (i.e. Yang-Baxter braidings of Hecke type) we associate ”noncommutative varieties” called braided spheres. An example of such a variety is the Podles’ nonstandard quantum sphere. On any braided sphere we introduce and compute an ”equivariant” analogue of Connes’ noncommutative index. In contrast with the Connes’ construction our version of equivariant NC index is based ...