Author: Eiko

Tags: math, tannakian formalism, category theory, tensor categories, number theory

Neutral Tannakian Categories

A neutral Tannakian category over field k is a rigid k-linear abelian tensor category C

  • whose unit 1 satisfies End(1)k

  • equipped with an exact and faithful (does exact faithful = faithfully exact?) tensor functor that is called the (neutral) fiber functor

    ω:CVectk

    into the category of finite-dimensional k-vector spaces.

Example: Group Representations

The category Repk(G) of finite-dimensional representations of an affine group scheme G over k, equipped with the forgetful functor Repk(G)Vectk, is a neutral Tannakian category.

One interesting question people in representation theory have thought about is, how do we recover the group G from the category Repk(G)? This is the essence of the Tannakian formalism.

  • The first possibility one can try is to consider some symmetry group on some of the structures of the category itself, if we consider the automorphism group of the fiber functor ω, we get

    Aut(ω)=(kG)×,

    where kG is the group algebra of G. The result is nearly G but contains more terms like what we get in a polynomial.

  • If we add a ‘multiplicative’ requirement, we can get the ‘monomials’ back from (kG)×, i.e. we require the automorphisms to respect the tensor structure, asking a subset Aut(ω)Aut(ω) for which αMN=αMαN and α1=1. This time we get

    Aut(ω)=G,

    which is the group we started with owo

References

  • Tamás Szamuely, Galois groups and fundamental groups