## Weekly outline

• ### General

• Instructor: Wushi Goldring
• Time Period: Spring 2019, ABCD
• Registration: We will pass out a sheet of paper during the first meeting of class so that students can sign up.
• Course Room: 306 (SU, House 6)
• Course meeting times: Thursdays 10:15-12:00, starting January 24.
• Core Text: "Hodge cycles, Motives, and Shimura varieties", P. Deligne, J. Milne et. al. eds.,  Lect. Notes in Math. 900. Specifically, our starting point will be article II of the volume, "Tannakian categories" by Deligne and Milne. The volume is available for free online with an SU login: Starting from https://www.su.se/english/library/ , search for SpringerLink under "databases" and then search for "Deligne Milne" at SpringerLink
• Grading PhD students: Pass/Fail. Grades will be based on completing homework assignments. A score of at least 80% will be sufficient to pass; based on course participation and the difficulty of the problems, I may decide that a lower score is enough to pass.
• ### Motivation

#### 1. What is a Tannakian category?

##### 1.1 Abstracting the categorical properties of Rep(G)

The notion of neutral Tannakian category is modeled on the categories Rep(G) of finite-dimensional representations of an (affine) algebraic group G (or more generally an affine group scheme) over a fixed field k. Two simple examples to keep in mind with k=C=complex numbers are: (i) G a finite group, (ii) G a connected, reductive algebraic group, e.g., G=GL(n) or SO(n) or Sp(2n). Another classical setting, (iii) G a connected compact Lie group, can be reduced to (ii), so it also serves as a motivating example.

A rough analogy is given by the way abelian categories are modeled on the category of abelian groups. Some basic operations on  abelian groups and homomorphisms between are: subgroup, quotient, kernel, cokernel, direct sum etc. An abelian category is then roughly a category in which these operations continue to make sense.

Since the above operations on abelian groups apply also to representations and morphisms between them, the categories Rep(G) are abelian categories. However, they possess two additional key operations: tensor product and dual. Moreover, if we consider representations over the fixed field k, we have a forgetful functor from Rep(G) to the category Vec_k of k-vector space which associates to a representation its underlying vector space. Thus, a neutral Tannakian category C is roughly an abelian category C with a tensor product, duals and a functor to Vec_k which is supposed to generalize the forgetful functor in the case of Rep(G); such a functor (which should respect the tensor product and dual) is called a fiber functor. The more general notion of Tannakian category (not necessarily neutral) is more complicated; roughly it does not assume the existence of a fiber functor over k, but requires the weaker hypothesis that there exists a fiber functor over some field extension k' of k (possibly of infinite degree.    [It is important and nontrivial to make all of the above precise.]

1.2 Tannaka Duality: Reconstructing G from Rep(G)

Return to the analogy between the category Ab of abelian groups and general abelian categories. While a general abelian category A is an abstraction of the category Ab, there are several embedding theorems which give us a way of "going back" from A to Ab. For example, one has (cf. [Freyd, Th. 7.14]) : Every (small) abelian category is equivalent to an exact subcategory of Ab.

If we cross over from abelian to Tannakian categories, the analogy leads us to question to what extent we can go back from a general Tannakian category C to a category Rep(G) of representations. For a compact topological group G, Tannaka showed that the category Rep(G) of continuous finite-dimensional representations uniquely determines G (up to isomorphism); in fact one can reconstruct G from Rep(G). An exposition of Tannaka's result is given in [Chevalley, Chap. VI, Th. 5]. A generalization of this result to neutral Tannakian categories was given by Saavedra (based on ideas of Grothendieck): Every neutral Tannakian category is equivalent to Rep(G) for some affine group scheme G. Saavedra also claimed a generalization to general Tannakian categories (where the group G has to be replaced by a fancier object called a gerb), but his argument in the general case was found to have an important error [Deligne-Milne, 3.15]. A correct statement and proof for general Tannakian categories was finally given by Deligne [Deligne].

#### 2. Applications of Tannakian categories

##### 2.1. The Tannakian viewpoint

If G is either a compact Lie group or a connected, reductive, complex algebraic group, the categories Rep(G) have been studied for roughly a century and they are ubiquitous throughout mathematics. Even in this very classical setting, the Tannakian viewpoint has been extremely powerful. This includes: (i) Studying the representations of G from the point of view of the category-theoretic properties of Rep(G) (ii) A duality between properties of G which are defined intrinsically "inside G" and those defined by means of any representation of G and independent of the choice of representation.

An important example of (i) is the notion of a tensor generator: A representation V such that every other representation W is obtained from V by successive application of the Tannakian operations (tensor product, dual and the abelian category operations of sub, quotient, direct sums). In the two classical cases above (and more generally for any affine (linear) algebraic group), every faithful representation is a tensor generator (and for compact Lie groups, the existence of a faithful representation follows from the Peter-Weyl theorem).

Given a tensor generator V, one may hope to prove that every representation -- i.e. every element of Rep(G) -- satisfies some property P as follows: (a) Show that V has property P, (b) Show that the property P is stable under the Tannakian operations e.g. if V_1 and V_2 have property P then so does the tensor product of V_1 and V_2. If (a) and (b) hold, the property P holds for all elements of Rep(G).

Here is a classical example of (ii): For G=GL(n) (over an algebraically closed field), an element is semisimple if it is diagonalizable; at the other extreme it is called unipotent if its only eigenvalue is 1. Then a general element x admits a unique Jordan decomposition x=su, with s semisimple, u unipotent and su=us. The notions of semisimple and unipotent elements, and that of Jordan decomposition generalize to an arbitrary algebraic group [Springer, Th. 2.4.8]. These notions can be defined intrinsically, without appealing to a representation of G, but also an element of G is semismple (resp. unipotent) if and only if its image has the property under every representation of G. These two complementary ways to think of semismple and unipotent elements may be viewed as the "Tannakian viewpoint" in this special case.

##### 2.2. New groups from Tannakian categories

In the previous section, we considered groups that had been discovered long ago and applied the observation that the associated representation category Rep(G) is Tannakian. So the direction was roughly "from groups to Tannakian categories". It turns out that the other direction is at least as natural. In fact, for most pairs (G, Rep(G)), it is perhaps more likely that you will first encounter Rep(G) and then deduce the existence of G by means of the Saavedra-Deligne Theorems mentioned above. Stronger yet, there are several important groups for which there is no known direct definition (as solutions to equations, or by generators and relations etc.) and the only known definition is to first define a neutral Tannakian category C with fiber functor and then define our group as "the" group G for which Rep(G) is equivalent to C.  Some key examples are discussed in the next section.

#### 3. The ubiquity and universality of Tannakian categories

Since their introduction in the late 60'/early 70's, Tannakian categories have turned out to be ubiquitous throughout mathematics. Furthermore, they have often been used to unify different fields. Some examples:

##### 3.1. Hodge theory

The category R-HS of real Hodge structures is neutral Tannakian over R=real numbers; it is equivalent to Rep(G) where G is the multiplicative group of nonzero complex numbers C* viewed as a real group (in other words Weil restriction of G_m from C to R) and we consider representations defined over R.

The category Q-HS^{pol} of polarizable Q-Hodge structures is neutral Tannakian over Q=rational numbers. This is an example where there is no direct description of the associated group G such that Rep(G) is equivalent to Q-HS^{pol}

The categories of variation of Hodge structure (over R or Q) also give neutral Tannakian categories.

##### 3.2. Fundamental group

Several categories of local systems on a topological space (resp. scheme) are Tannakian and their associated group is closely related to (completions) of the fundamental group (resp. etale fundamental group). See [Saavedra, Chap. 6, sect. 1.1].

##### 3.3. Motives

Grothendieck defined a category of (Grothendieck) motives and conjectured that it is Tannakian. This conjecture remains open; Grothendieck observed that it would follow from the "Standard Conjectures" about algebraic cycles on algebraic varieties. Assuming that Grothendieck's category is Tannakian, the associated group would be the "motivic Galois group", a generalization of the Galois group that should control all algebraic varieties (over a field which embeds in C). Unconditional definitions of a candidate for the motivic Galois group have been given by Nori and Ayoub.

Deligne constructed a category of "motives with absolute Hodge cycles" which is Tannakian and provides a substitute for Grothendieck's category which does not require knowing the standard conjectures [Deligne-Milne, sect. 5].

##### 3.4 The Langlands Program -- automorphic representations

Langlands conjectured that the collection of all automorphic representations of all GL(n) (for all n>0) should form a Tannakian category. The associated group would be the conjectural "Langlands group".

##### 3.5 Conjectures

Several of the most outstanding open conjectures in mathematics admit formulations in terms of Tannakian categories, or follow from statements about Tannakian categories. We have already mentioned the relationship between the Standard Conjectures and Grothendieck's category of motives.

Two additional monumental conjectures in algebraic geometry, in fact more striking than the Standard Conjectures, are the Hodge and Tate conjectures. The Hodge conjecture is equivalent to the Tannakian statement that the functor "Betti realization" is fully faithful from the category of motives to the category of Q-Hodge structures. Similarly, the Tate conjecture is equivalent to the statement that the functor "l-adic realization" is fully faithful from the category of motives to the category of Q_l vector spaces with continuous Galois action.

Langlands "Principle of Functoriality" conjectures about the relationship between automorphic representations of different groups is closely related to his his conjecture (3.4) that automorphic representations form a Tannakian category.

#### References

[Chevalley] C. Chevalley "Theory of Lie groups" Vol 1

[Deligne] P. Deligne "Categories Tannakiennes" In "The Grothendieck Festschrift" Vol 2

[Deligne-Milne] P. Deligne and J. Milne "Tannakian Categories" In "Hodge cycles, motives, and Shimura varieties"  Lect. Notes Math. 900

[Freyd] P. Freyd "Abelian categories"

[Saavedra] N. Saavedra-Rivano "Categories Tannakiennes" Lect. Notes Math. 265

[Springer] T. Springer "Linear Algebraic Groups"

• ### Overview

The initial goals of the course are:

1. Understand the definition of Tannakian category

2. Study the examples Rep(G) where G is an affine algebraic group, particularly the case when G is reductive

3. Understand the classification of neutral Tannakian categories

As stated above, the core text for this will be [Deligne-Milne], specifically Sect 1-2. We will need to fill in background material from several fields and we will probably use some of the references above to do this. For example, we can use [Freyd] for background on abelian categories and [Springer] for background on algebraic groups. How much we choose to go into detail into some of the background topics depends on the participants and the teacher, what we know and what we want to learn.

If there is time left, we can look at some applications of Tannakian categories, as surveyed above. Again, which applications we choose to cover depends on the interest of the participants and the teacher. Several of the applications are discussed in [Deligne-Milne] and in the other articles of the same volume LNM900. I think it would be natural to try to say something about Deligne's application to "motives with absolute Hodge cycles".

• ### Lecture 1: Introduction

1. Abelian categories: (a) The "special" abelian categories Ab of abelian groups and Mod_R of (left) modules over a ring R. (b) The Mitchell embedding theorem relating a general abelian category to the special ones: Any small abelian category admits an exact faithful embedding into Ab and an exact, fully faithful embedding into Mod_R for some ring R.

Tannakian categories are special abelian tensor categories, so we would like to now pass from abelian categories to abelian tensor categories. This may be viewed as the category version "categorification" of passing from abelian groups to rings: passing from one operation "+" to two: "+" and "tensor product"

2. Neutral Tannakian categories: Overview of the definition and main classification theorem [In later lectures I will go back and rigorously define the objects]: First, given a field k, one has the notion of an abelian, rigid, k-linear tensor category; rigid means that duals exist and k-linear means that the Hom groups are k-vector spaces. A fiber functor on such a category is a faithful, exact, k-linear tensor functor to the category Vec_k of k-vector spaces.

A neutral Tannakian category is an abelian, rigid, k-linear tensor category C such that End(1)=k and which admits a fiber functor. [De-Mi 2.19]

The analogue of Mod_R being an abelian category is: Let G be an affine group scheme over k. Then the category Rep_k(G) of finite-dimensional representations of G (homomorphisms of group schemes to GL(V) for V finite dimensional over k) is neutral Tannakian with fiber functor the forgetful functor (assigning the underlying vector space to a representation).

Theorem (Saavedra, [De-Mi 2.11]): Let C be a neutral Tannakian category with fiber functor \omega. Then

(i) The functor Aut^{\otimes}(\omega) [which we will need to define] is represented by an affine group scheme G

(ii) There is a natural equivalence between Rep_k(G) and C.

3. How are the group-theoretic properties of G reflected in the category theoretic properties of C?

4. Some first geometric examples: Examples from Hodge theory: The category R-HS of real Hodge structures is semisimple and Tannakian, the group G is S=Deligne torus=Res_{C/R} G_m. The category Q-HS of Q-Hodge structures is not semisimple. It is Tannakian over Q and there is no simple way to describe the group G, other than via the theorem. In particular the group is not algebraic i.e. not of finite type over Q. Next time: The category Q-HS^pol of polarizable Q-Hodge structures is semisimple Tannakian; the group G is again very large  and not of finite type over Q.

• ### Lecture 2: Further examples and motivation from Hodge theory

• 