Conference abstracts

Session S02 - Commutative Algebra and Algebraic Geometry

July 25, 15:40 ~ 16:10

## Tannaka duality for algebraic groups

### CMAT-Udelar, Uruguay   -   alvaro@cmat.edu.uy

The Chevalley's structure theorem states that any connected algebraic group over an algebraically closed field is the extension of an abelian variety by a connected affine algebraic group. In view of this result, the theory of algebraic groups has been developed along two directions: the study of linear (affine) algebraic groups and that of abelian varieties. The representation theory of affine algebraic groups plays an important role in their study: the (classical) Tannaka duality theorem guarantees that an affine algebraic group can be recovered from its category or representations.

In this talk we propose a representation theory for arbitrary algebraic groups, as follows: let $G$ be an algebraic group $G$ consider its Chevalley decomposition $1\rightarrow G_{aff}\rightarrow G\rightarrow A\rightarrow 0$. A representation of $G$ is a homogeneous vector bundle $E\rightarrow A$ together with regular action $\varphi :G\times E\to E$, linear on the fibres and such that the induced morphism $\widetilde{\varphi}: A\times A\to A$ is the product in $A$ (recall that $A\times A$ is the Albanese variety of $G\times E$). We will define the category of representations of $G$, and prove that a generalisation of Tannaka duality theorem is valid in this context, therefore allowing us to recover an algebraic group from its category of representations.

This is an ongoing joint work, partially financed by CSIC-Udelar and ANII (Uruguay).

Joint work with Pedro Luis del \'Angel (CiMat, México).