An introduction to HALL ALGEBRAS a categorification of quantum groups

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An introduction to

HALL ALGEBRAS

a categorification of quantum groups

Sjoerd Beentjes

(s/n: 5922143)

Supervisor: Prof. Dr. E.M Opdam Second supervisor: Dr. R.R.J. Bocklandt

Master’s Thesis

MSc Mathematical Physics

University of Amsterdam Faculty of Science

Korteweg-de Vries Institute for Mathematics Science Park 904

1090Gl, Amsterdam The Netherlands

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Abstract

This thesis aims to give an introduction to the theory of Hall algebras as described by Ringel [48], Schiffmann [49], and as later generalised by Joyce [31, 32]. Hall algebras are certain self-dual topological Hopf algebras that one may associated to abelian categories over a finite field

k that satisfy quite restrictive finiteness properties. The examples that are treated in this thesis

are the Hall algebra of the category of nilpotent k-linear representations of a quiver, and the Hall algebra of the category of coherent sheaves on the projective line. The product structure of the Hall algebraH_A associated to such an abelian category_{A encodes how two objects of A} may build up a third one. On the other hand, the coproduct structure describes how an object of A breaks up into two smaller objects. This building up and breaking down pertains to the extenstion structure of the category, and methods of homological algebra naturally play a role in the theory.

We present a simple example of Joyce’s theory of motivic Hall algebras. These may be associ-ated to C-linear abelian categories satisfying some finiteness conditions. We treat the example of finite-dimensional modules over a finite-dimensional complex algebra. This approach uses techniques from algebraic geometry such a representation varieties and constructible functions. Besides treating some examples, we present Ringel’s Theorem (and shortly touch upon Bridgeland’s generalisation) as an application of the finite field theory. Both theorems describe a categorification of quantised universal enveloping algebras of certain Kac-Moody algebras. The theory of motivic Hall algebras is illustrated similarly, and we obtain the analog in characteristic zero of Ringel’s Theorem which is originally due to Schofield.

We end with a short discussion on what is ‘motivic’ about the motivic Hall algebra.

Information

Title: An introduction to Hall algebras: a categorification of quantum groups Author: Sjoerd Beentjes, sjoerd.beentjes@gmail.com, s/n 5922143

Supervisor: Prof. Dr. E.M. Opdam Second corrector: Dr. R.R.J. Bocklandt Master coordinator: Prof. Dr. S. Shadrin End date: August 18, 2014

Korteweg-de Vries Institute for Mathematics Facult of Science, University of Amsterdam

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Acknowledgements

First and foremost I would like to thank Eric Opdam for doing a great job as a supervisor. Not only for clarifing and resolving any technical difficulties I have had, but especially for painting a broad and vast mathematical landscape, and in doing so showing that some mountains are just more interesting to climb than others.

Secondly, I would like to thank Raf Bocklandt for being the second supervisor, and for very helpful discussions about the theory of quivers and some aspects of algebraic geometry.

Lastly, I would like to express my sincerest gratitude towards all regular and irregular inhabitants of the master rooms of both mathematics and theoretical physics. Thank you for all the interesting and sporadic coffee breaks, and for the realisation that in the end, struggling together is far more rewarding than struggling alone. In particular, I would like to thank Bart L., Bart S., Jeroen Z., Jason van Z., Didier C. and Gerben O., mostly for laughs and discussions not pertaining to theses. And finally, one more thanks to Bart L. for putting off his own breakthrough by proof-reading this thesis.

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Introduction

This thesis aims to give an introduction to the theory of Hall algebras as described by Ringel [48], Schiffmann [49], and as later generalised by Joyce [31, 32]. Historically, Steinitz (and later Hall) associated an associative unital algebra to the set of all abelian p-groups, where p is a prime number. The product structure of the algebra encodes the extensions of one abelian

p-group by another, and the resulting algebra is isomorphic to MacDonald’s ring of symmetric

functions.[43]

Ringel generalised this construction by associating an algebra to certain abelian categories.[48] We refer to this as the classical Hall algebra, with is the topic of the first part of this thesis. As a vector space, these algebras have a natural basis consisting of the isomorphisms classes of objects of the relevant category. Given two such classes, their product is defined as the formal sum of isomorphism classes of objects that are an extension of the former by the latter. The relevant coefficient ‘counts’ in how many non-equivalent ways such an extension occurs.

For this counting procedure to make sense, the abelian category must satisfy quite restrictive finiteness properties. This naturally leads one to consider categories that are linear over a finite field k. The first example of such an abelian category is the category Rep_k( ~Q) of nilpotent k-linear representations of a quiver ~Q. A second one is the abelian category Coh(V ) of coherent sheaves on a smooth projective variety V over k.

Since the extension structure of the relevant category directly enters the Hall algebras defin-ition, the global dimension of the category plays an important role in the algebraic structure. It turns out that for categories with global dimension less than or equal to one, the associated Hall algebra may be equipped with the structure of a coassociative counital coalgebra. Green has shown in [23] that the product and coproduct are compatible (after a small twist), making the Hall algebra into a self-dual (topological) bialgeba. Later, Xiao proved in [56] that this bialgebra is in fact a Hopf algebra.

Although the assumptions on the abelian category are quite restrictive, there are still two important classes of examples to which the theory applies.

The first example is the above mentioned category of quiver representation. Ringel has proven that the quantised universal enveloping algebra of the standard positive Borel subalgebra of the derived Kac-Moody Lie algebra associated to the quiver ~Q embeds as a Hopf algebra into

a certain extended Hall algebra of _Rep_k( ~Q).[48] This is an example of a categorification of an

algebraic object. Indeed, a quantum group is realised as an invariant of a suitable category. The interest of such a categorification is that categorical relations amongst quiver representations can be translated via this embedding into algebraic relations in the quantum group. This for

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CONTENTS

example yields a quantum PBW basis of the quantum group by pulling back the natural basis of isomorphism classes of objects of_Rep_k( ~Q).

The second class of examples is the category of coherent sheaves on a smooth projective curve

X. Their associated Hall algebras were studied by Kapranov who found links with quantum

affine algebras. Moreover, he interpreted the Hall algebra in the context of automorphic forms over the function field of the curve X.[35]

The above situation is in fact that of the classical Hall algebra. There is a more geometric way of thinking about the product structure. One can interpret it as a convolution product of finitely supported functions on an appropriate ‘moduli space of objects’ of the abelian category A.[11] This approach has allowed Lusztig to construct his celebrated canonical basis [41], which has manifold applications in representation theory and quantum group theory.

Recently, Joyce has replaced the naive approach of counting extensions classes in _{A by} the more sophisticated methods of motivic integration in a series of papers [31, 32]. This

motivic Hall algebra can be associated to certain C-linear categories, thus paving the way to

applications of the Hall algebra in characteristic zero. Furthermore, the global dimension of the abelian category no longer forms an obstruction to the Hall algebra being a bialgebra. For certain nice categories such as that of coherent sheaves on a complex CalabiYau threefold -there exist so-called integration maps to well-known rings. Bridgeland has used such maps in [9] to compute Donaldson-Thomas invariants of Calabi-Yau threefolds.

Let us briefly summarize the contents of this thesis. The first chapter starts out with some preliminaries on abelian categories, extensions in abelian categories and derived func-tors. Moreover, it recalls the definition of the associated Grothendieck group. Afterwards, the classical Hall algebra is defined and proven to be associative and unital. Green’s theorem is presented, proving the Hall algebra to be a self-dual twisted bialgebra, and the compatibility is discussed. Finally, the Xiao’s antipode is introduced and some functorial properties of the Hall algebra are discussed.

In the second chapter the theory of quivers and their representations is recalled. We explain why we consider nilpotent representations and consider the associated Hall algebras. After a short intermezzo on quantum groups, Ringel’s Theorem is stated and proven. Finally, we shortly describe Bridgeland’s Theorem which is, in some sense, a complete categorification of the relevant quantum group.

The third chapter treats the example of the abelian categoryCoh(X) of coherent sheaves on a smooth projective curve X over a finite field. We prove with some care that this category has a well-defined Hall algebra, and we treat the example of the projective line in detail. There turns out to be a strong analogy between the Hall algebra of the projective line and the Hall algebra of the Kronecker quiver. Following the papers [4, 15], we describe and explain the analogy. This requires a short introduction to the theory of derived categories. As a case in point, the derived category of abelian categories of global dimension zero and one are treated in detail; this contains the example of_Coh(P1). Finally, this machinery is applied to recover the Drin’feld-Beck isomorphism U_ν(slb₂) → U_ν(L sl₂).

The fourth chapter departs from the classical situation and introduces (motivic) Hall al-gebras of C-linear categories. First, we show that one may define Hall alal-gebras equivalently by considering a convolution product on the set of finitely supported function on the set of

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CONTENTS

isomorphisms classes of the abelian category. Then we leave the classical situation for good and focuss on the category of finite-dimensional modules over a finite-dimensional complex algeba. Some theory on algebraic groups and algebraic group actions is introduced to constructed a suitable moduli space of representations that is an affine variety quotiented out by an algeb-raic group action. Instead of finitely supported functions on this moduli space, we consider constructible functions on the variety that are invariant under the algebraic action. Following [11], we introduce a convolution product by means of (a baby example of) motivic integration. We calculate some examples and reprove a theorem by Schofield that is the characteristic zero equivalent of Ringel’s Theorem.

We finish this thesis by a short discussion on what is motivic about the motivic Hall algebra, and we point the reader to further literature on the matter.

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Chapter 1 Hall Algebras

In the first part of this section, we will describe the categorical setting in which Hall algebras in the sense of Schiffmann [49] are defined. They are assigned to abelian categories satisfying certain finiteness conditions. We will also provide some background on necessary concepts. The basic algebraic structure of a Hall algebra encodes the various ways in which one object of the associated abelian category can be extended by another. It is introduced in the second section. Alternatively, given an object, one can ask how such an object can be build up as an extension of smaller objects. This is achieved by Green’s coproduct, defined in the third section. It equips the Hall algebra with a topological coalgebra structure. It turns out that one numerical equality shows that the (co)multiplication is (co)associative. Interestingly enough, these structures are only compatible in the sense of bialgebras for hereditary abelian categories after a certain twist. To conclude the third section, we consider an antipode and a certain bialgebra pairing for these bialgebras, upgrading them to self-dual Hopf algebras. In the fourth and last section, some questions on the functorial behaviour of Hall algebras are considered.

1.1 Preliminaries

For the reader’s convenience, we gather here some necessary material on abelian categories and extensions. Most is taken from [54], which is our standard reference for matters of homological algebra. The part on derived functors is taken from [26, III].

1.1.1. Hall algebras in the sense of Schiffmann are associated to abelian categories. Let k

be a commutative ring with unit. Recall that a category _{A is called k-linear when all} hom-sets are k-modules, and when composition is k-bilinear. A k-linear category is called additive when it contains a zero object, and when finite direct sums exist.1 An additive category is called abelian when every morphism f ∈ Hom_A(X, Y ) has a kernel and a cokernel, and when the canonical morphism ¯f : Coim(f ) → Im(f ) is an isomorphism. Recall that by definition

Im(f ) = Ker (Y → Coker(f )), and Coim(f ) = Coker (Ker(f ) → X).

1_{Note that this is enough structure to guarantee that finite direct products exist too, and that they are}

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1.1 Preliminaries

1.1.2. The concept of an abelian category is modeled on the categoryAb of abelian groups, in which all of the above properties are present. A more general example of an abelian category is the category R-_{Mod of left-modules over a commutative, unital ring R.}2 On the other hand, a non-example is the category Vect (X) of locally free sheaves on a smooth variety X over an algebraically closed field. To wit, take X = C, let E = Cn×C be the trivial rank n vector bundle on X, and consider the vector bundle endomorphism f : E → E given by f (x, t) = (tx, t). It is well known that vector bundles over a connected base space are of constant rank. However, this fails for Ker(f ), since its rank is everywhere zero but at the origin where it is n > 0. However, the category _{Vect (X) ís exact in the sense of Quillen [45]. As we will only encounter exact}

subcategories of abelian categories, we simply define a subcategory E of an abelian category A to be exact if it is additive and if the inclusion functor is fully faithful and exact (it is additive and preserves short exact sequences). In the present situation,_{Vect (X) is an exact subcategory} of the abelian category of coherent sheaves on X.

Convention. We will always assume our abelian categories to be small, that is a category of

which the collection Ob(A) of objects is a proper set. This will always be the case in the examples we will discuss. By the Freyd-Mitchell Embedding Theorem, there then exists an associative unital ring R and an exact, fully faithful functor I : A → R-Mod that embeds A as a full subcategory in the sense that Hom_A(M, N ) ∼= Hom_R-_Mod(I (M), I (N)) for all M, N in_{A. This implies that all results from homological algebra proved in the category of modules} over such a ring remain valid in any small abelian category. See [54, 1.6] for more on this.

Let R be a commutative unital ring, let _{A denote an R-linear abelian category. One can} introduce the concept of an extension of an object of A by another in two seemingly different ways.

1.1.3. Let M, N be two objects ofA. An extension of M by N is an equivalence class of pairs (ξ, E) consisting of an object E in _{A and a short exact sequence ξ : 0 → N → E → M → 0.} Two such extensions (ξ, E), (ξ0, E0) are called equivalent if there is a commutative diagram

ξ : 0 N E M 0

ξ0: 0 N E0 M 0

∼ =

Note that the 5-lemma implies that the middle map is an isomorphism. We denote by Ex(M, N ) the set of equivalence classes of extensions of M by N . It can be given an R-module structure by means of the Baer sum. (Recall that _{A is an R-linear category.) An extension in Ex(M, N )} is called split if it is equivalent to 0 → N → M ⊕ N → M → 0, where the second map is (0,1_N); the equivalence class of the split extension is the zero element in Ex(M, N ) with respect to the Baer sum. An abelian category in which all extensions (or all short exact sequences) are split is called semi-simple.

2

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1.1 Preliminaries

1.1.4. Let M, N be objects of A. The association hM := HomA(M, −) : A → R-Mod is well

known to define a left-exact covariant functor, that is in general not right-exact. Using the concept of derived functors, originally introduced by Grothendieck in his famous paper [24], one can systematically measure the extent to which hM fails to be right-exact. For a general

left-exact covariant functor F : A → B between abelian categories, this is done as follows. Recall that an object I of_{A is called injective if h}I := Hom_A(−, I) is exact. We say that_{A has enough}

injectives if every object can be embedded in an injective object. In that case, for every object N there exists a complex (I•, d•) of injective objects and an embedding : N → I0, together called an injective resolution of N , such that the sequence

0 → N −→ I 0−→ Id0 1 −→ Id1 2 −→ . . .d2

is exact. The quotient hi(I•) := Ker(di)/ Im(di−1) is called the ith cohomology object associated to this sequence. The ith right derived functor RiF : A → B of F is then defined as RiF (N) :=

hi(F (I•)). Theorem 1.1A of [26, p. 204] summarizes the properties of these derived functors. They are additive, covariant functors, independent of the injective resolutions chosen, zero on injective objects (for i > 0), there is a natural isomorphism of functors F ∼= R0F , and to every short exact sequence in _{A is associated a long exact sequence in B, and this association} is moreover natural in the short exact sequence.3

In the present case, we write Exti_A(M, −) := RihM and call Exti_A(M, N ) the ith Ext-group of

M by N . The previous theorem yields in particular the following: given a short exact sequence

0 → N0→ N → N00→ 0 in A, we obtain a natural long exact sequence

0 → h_M(N0) → h_M(N ) → h_M(N00)−→ Extδ0 _A1(M, N0) → Ext1_A(M, N ) → Ext1_A(M, N00)−→ . . .δ1 The natural morphisms {δi}_i>0 are referred to as connecting homomorphisms.

1.1.5. One can show that there is a 1-1 correspondence Ex(M, N ) ↔ Ext1_A(M, N ) for all objects

M, N ofA by constructing a map Φ : Ex(M, N ) → Ext1A(M, N ), and verifying explicitly that it

is bijective. We recall the definition of this map here, referring the reader to [54, p. 76-78] for the proof. Let (ξ, E) be a representative of an element α of Ex(M, N ). As explained above, we obtain a long exact sequence associated to the short exact sequence ξ : 0 → N → E → M → 0 by deriving the functor h_M = Hom_A(M, −). Part of this long exact sequence is

. . . → Hom_A(M, E) → Hom_A(M, M )_{−→ Ext}δ0 1

A(M, N ) → . . . .

The map is then defined as Φ(α) := δ0(1_M).

This result implies that we can use techniques and ideas related to both definitions of extensions. Both definitions have their advantages and disadvantages. Although the latter set-up is very general, it depends on the category having enough injectives. Then again, the relevant maps and module structure can be made quite explicit. The former set-up is intuitively

3_{There is a dual theory for left-exact contravariant functors. Here one resolves projectively, using objects P}

called projective such that hP := HomA(P, −) is exact, to similarly arrive at right derived functors. Analogous

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1.1 Preliminaries

how we want to think of an extension of an object by another one, but the definition of Baer sum is quite involved. We will make use of both definitions as we go along.4

We arrive at a first definition important for the algebraic properties of Hall algebras.

Definition 1.1.6. Let A be a k-linear abelian category. The global or homological dimension of _{A is the positive integer or +∞ defined by}

gl. dim(_{A) := sup{d ∈ N : Ext}d_A(X, Y ) 6= 0 for some X, Y ∈ Ob(_A)}. We say that _{A is of finite global dimension when gl. dim(A) < ∞.}

1.1.7. If A has global dimension d ∈ N, all Ext-groups in degree bigger than d vanish. In particular, if gl. dim(_{A) = 0 all degree one Ext-groups vanish. By the correspondence mentioned} in 1.1.5 this means that all groups Ex(M, N ) are trivial. Thus all extensions are of the form 0 → N → M ⊕ N → M → 0: they are split. So any abelian category of global dimension zero is a semi-simple category.5 We will see that their associated Hall algebras are particularly simple, which is to be expected as they encode the complexicity of the categories’ extensions.

More often than not we will restrict our attention to categories with small global dimension.

Definition 1.1.8. An abelian category A is called hereditary if gl. dim(A) 6 1.

1.1.9. The nomenclature is derived from the following fact. Recall that an object P of A is called projective if h_P = Hom_A(P, −) is an exact functor or, equivalently, if Ext1_A(P, −) = 0. Let Q be a projective object, and let 0 → P → Q → Q/P → 0 be the short exact sequence associated to the inclusion of a subobject in Q. Let A be some object of the hereditary category A. Note that the long exact sequence associated to HomA(−, A) abuts after six terms. So

0 → hA(Q/P ) → hA(Q) → hA(P ) → Ext1_A(Q/P, A) → Ext1_A(Q, A)

| {z }

=0

→ Ext1_A(P, A) → 0.

Hence P inherits projectivity from Q. So in a hereditary category, subobjects of projective objects are again projective. A dual statement holds for quotients of injective objects, these are again injective. It turns out that this feature characterizes hereditary categories. (CITE)

Remark. Let A be an object of an abelian category A. Recall that the projective dimension pd(A) is the minimum integer n (or +∞) such that there exists a projective resolution of

A by projectives: 0 → Pn → . . . → P1 → P0 → A → 0. Dually, A’s injective dimension

id(A) is the minimum integer n (or +∞) such that there exists an injective resolution of A: 0 → A → I0→ I1 → . . . → In→ 0. We record the following well-known theorem for future use:

Theorem 1.1.10 (Global Dimension Theorem). In any essentially small abelian category A

the following numbers are identical:

4

Yoneda has given a definition of higher Ext-groups in the spirit of the first one, without assuming the category has enough injectives. We won’t be needing them, however, since in the present thesis we will focus our attention on hereditary categories: those where all Ext-groups with i > 1 vanish.

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1.1 Preliminaries

1. projective dimension sup{id(A) | A ∈ Ob(A)};

2. injective dimension sup{pd(A) | A ∈ Ob(_A)}; 3. global dimension gl. dim(A).

Proof. See [54, p. 91-95].

Definition 1.1.11. Following [55, II.6.1.1], the definition of the Grothendieck group K (A) associated to A is the free abelian group presented as having one generator M for every object

M of A, with the relation M = M0+ M00 if there exists an extension of M0 by M00 (or of M00 by M0, since the group is abelian).

Consequently, 0 is the neutral element of the group _{K (A). Furthermore, if M ' N in A} thenM = N inK (A). In particular, one may also present this group by one generator for each

isomorphism class of objects of_{A. We write Iso(A) for the set}6 of such classes, and [M ] for the iso-class of an object M of_{A. Finally, M ⊕ N = M ⊕ N for all M, N objects of A.}

1.1.12. Fix a ground field k. Let A be a k-linear abelian category of finite global dimension. Suppose that all Hom-sets and Ext-sets are finite-dimensional over k, so for all object M, N of A and for all i > 0 we have

dim_kExti_A(M, N ) < ∞ .

Note that by the discussion in 1.1.4, we know that Hom_A(M, −) ∼= R0Hom_A(M, −) ≡ Ext0_A(M, −) naturally as functors. This implies in particular that dim_kHom_A(M, N ) = dim_kExt0_A(M, N ).

In this setting, we can define a k-bilinear form on Ob(_{A) as follows. Let M, N ∈ Ob(A), set} hM, N ia:=

∞

X

i=0

(−1)idim_kExti_A(M, N ), (1.1)

which is well-defined sinceA is of finite global dimension. Note that k-bilinearity follows from the fact that_{A is a k-linear category. We claim that this form descends to the Grothendieck group.} To see this, suppose 0 → N0 → N → N00_{→ 0 is a short exact sequence, so [N ] = [N}0_{] + [N}00_{] in}

K (A). We obtain a long exact sequence of derived functors 0 → h_M(N0) → h_M(N ) → h_M(N00)−→ Extδ0 1

A(M, N0) → Ext1A(M, N ) → Ext1A(M, N00)

δ1

−→ . . . which implies that hM, N0i_a+ hM, N00i_a= hM, N i_aby comparing dimensions. So the k-bilinear form descends to the Grothendieck group in the second variable, and we may write hM,N ia.

To show that the same holds for the first variable, we refer to Theorem 2.7.6 of [54] which states that the Hom-functor is right balanced in the sense that for all i_{> 0,}

Exti_A(M, N ) ≡RiHom_A(M, −)(N ) ∼=RiHom_A(−, N )(M ) for all M, N ∈ Ob(_A). This implies that we also obtain a long exact sequence for any short exact sequence in the first variable of Hom_A(−, −), whence h−, −ia also descends to the Grothendieck group in the first

6_{Recall that}

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1.2 The Hall algebra

variable. In total, we have a k-bilinear form h−, −ia : K (A) × K (A) → Z called the additive

Euler form. Its symmetrised version is denoted by (M , N )a:= hM , N ia+ hN , M ia.

Sometimes we will instead use the multiplicative Euler form, the bilinear form

hM , N im:= ∞ Y i=0 # Exti_A(M, N )(−1) i!1/2 , (1.2)

which similarly descends to the Grothendieck group of _{A. We will denote by (M , N )}_m = hM , N imhN , M im its symmetrised variant. The relation between these two forms is given by

hM , N im= q1/2hM ,N ia where #k = q note that we have chosen a square root of q.

Definition 1.1.13. An abelian categoryA is called finitary if | ExtiA(M, N )| < ∞ for all objects

M, N of A and for all i > 0. If A is linear over a finite field Fq, this is equivalent to requiring

dim_F_qExti_A(M, N ) to be finite for all M, N and for i_{> 0.}

1.1.14. Our main examples of finitary abelian categories are the category of nilpotent

repres-entations of a finite quiver over a finite field (chapter 2), and the category of coherent sheaves on a projective variety over a finite field (chapter 3). One easily sees that the first category constitutes an example. We will see later that the second does so too.

1.2 The Hall algebra

In this section, we will define the Hall algebra associated to any finitary abelian category_{A. The} multiplicative structure will be shown to be associative by reinterpreting isomorphism classes of objects as certain functions on the naive moduli space of objects of _{A. Moreover, we will} show that this multiplication pertains to filtrations of a given object with fixed quotients. As an example, we consider the Hall algebra of a semi-simple category, and ofVectk.

Convention. Throughout the rest of this section we fix an essentially small finitary abelian

category A. Moreover, we fix a finite field k of order q and a square root v = q1/2_{. Lastly,}

Iso(_{A) will denote the set of isomorphism classes of objects of A, and we will simply write Hom} and Exti for the sets of morphisms and extensions in _A.

1.2.1. Some relevant numbers By the assumption thatA be finitary, there are some important finite numbers related to extensions. Let P_M,NE denote the set of short exact sequences in_{A of} the form 0 → N → E → M → 0, and let P_M,NE denote its cardinality; note that this number is finite since it is bounded above by | Hom(N, E)|| Hom(E, M )|. For any object M of _{A, its} automorphism group Aut(M ) ⊂ End(M ) is finite, and we will write aM = | Aut(M )| for this

number. Furthermore, define the following set

FE M,N = L 6 E | E/L ' M, L ' N , and denote by FE

M,N its cardinality. These numbers are most often referred to as Hall numbers.

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Proof. There is a natural mapP_M,NE →FE

M,N given by sending (f, g) to the subobject Im(f ) =

Ker(g) of E. Its fibre is the orbit of (f, g) under the natural action of Aut(M ) × Aut(N ) on the set P_M,NE via (α, β, (f, g)) = (f ◦ α−1, β ◦ g). Since this action is free, the result follows.

Secondly, we can relate Hall numbers to counting certain extension classes. To be precise, let Ext1(M, N )_E ⊆ Ext1_{(M, N ) be the subset of equivalence classes of extensions of M by N}

with middle term isomorphic to E. The relation of the cardinality of this set to Hall numbers is given by Riedtmann’s formula [29, p. 26].

Lemma 1.2.3 (Riedtmann). Let M , N , E be objects of A. We have the formula

F_M,NE = | Ext 1_{(M, N )} E| | Hom(M, N )| aE aMaN . (1.3)

As an example of a typical argument involving techniques of homological algebra, we will give the proof of this formula following the exposition by Hubery [29].

Proof. Recall definition 1.1.3 of equivalent extensions of M by N . There is a natural map

PE M,N 3 0 −→ N −→ Ef −→ M −→ 0g 7→ [(f, g)] ∈ Ext1(M, N )_E,

sending a short exact sequence to its equivalence class of extensions. The fibre above a class [(f, g)] consists of those short exact sequences given by pairs of maps (f0, g0) = (θf, gθ−1) for some θ ∈ Aut(E). This defines an action of Aut(E) onPE

M,N with quotient Ext1(M, N )E. We

claim that the stabilizer of a pair (f, g) ∈P_M,NE has cardinality | Hom(M, N )|.

Let θ ∈ Stab(f, g), so that θf = f and gθ = g. We will apply the left-exact functors Hom(−, E) and Hom(M, −) to the short exact sequence defined by (f, g). This yields

(i) firstly the following part of a long exact sequence

0 −→ Hom(M, E) g

∗

−→ Hom(E, E) f

∗

−→ Hom(N, E)−→ Extδ 1(M, E) −→ . . .

where f∗(θ − 1) = (θ − 1)f = 0 implies θ − 1 ∈ Ker(f∗) = Im(g∗). Hence, there exists a unique morphism φ ∈ Hom(M, E) such that g∗φ = φg = θ − 1. Furthermore, 0 = g(θ − 1) = gφg so that gφ = 0 since g is an epimorphism.

(ii) secondly the following part of a long exact sequence

0 −→ Hom(M, N )−→ Hom(M, E)f∗ −→ Hom(M, M )g∗ −→ Extδ 1(M, N ) −→ . . . where the fact that gφ = g∗φ = 0 implies φ ∈ Ker(g∗) = Im(f∗). Hence, we find a unique

morphismθ ∈ Hom(M, N ) such that f∗θ = f θ = φ.

In total, we have obtained an injective map Stab(f, g) → Hom(M, N ), sending θ 7→ θ such that

θ = 1 + f θg. This map is also surjective. To wit, let ψ ∈ Hom(M, N ) and define the map θψ = 1 + f ψg. Since gf = 0, we find that θψf = f and gθψ = g so that θψ stabilizes (f, g)

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indeed. Furthermore, it implies that (f ψg)2 = 0 so that 1 − f ψg ∈ Hom(E, E) is an inverse to

θψ. Hence, θψ is an automorphism of E as required. The orbit-stabilizer theorem then yields

P_M,NE = | Ext

1_{(M, N )}

E|

| Hom(M, N )| · aE which completes the proof by an application of the previous lemma.

1.2.4. Given the above formulae, we are now in a position to define an algebraic structure

on Iso(_{A) that encodes the extensions E of an object M by an object N or, equivalently, the} number of subjects of E isomorphic to N with quotient isomorphic to M . The surprising fact is that this multiplication is associative.

Definition 1.2.5 (Hall algebra). As a vector space over C, the Hall algebra HA ofA is defined

as

HA :=

M

[M ]∈Iso(A)

C · [M ].

One can define a (naive) multiplication on H_A by setting [M ] ? [N ] := X

[E]∈Iso(A)

F_M,NE · [E] (1.4)

where M , N are two objects of _{A. Following Schiffmann [49] and Green [23], one can} alternat-ively define a multiplicative structure on H_A by introducing a slight twist via

[M ] · [N ] := hM, N im

X

[E]∈Iso(A)

F_M,NE · [E]. (1.5)

As was noted by Ringel in [47], the latter product produces a direct relation with certain quantum groups when _{A is the category of representations of a certain quiver in that it allows} one to recover the quantum Serre relations. This will be made clear later on.

Remark. There are a number of things that can be said about these definitions.

1. Although slighly different, the former multiplication is associative if and only if the second is. This essentially boils down to multiplicativity in the second variable of the Euler form.

2. By a direct application of Riedtmann’s formula (1.3), the product can be rewritten in terms of alternative generators_{JM K = a}M[M ] as

JM K ? JN K = X

[E]∈Iso(A)

| Ext1(M, N )E|

| Hom(M, N )| ·JE K.

From this description, it is clear that the Hall algebra essentially encodes the first order

extension structure of the underlying categoryA. Thus heuristically, hereditary categories are in some sense the ideal candidates to associate Hall algebras to. Indeed, only hereditary categories allow for a comultiplication that is compatible with multiplication in the sense of bialgebras. This is precisely the reason why the twist in the second product is inserted7, as will be shown in the next section.

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3. One can interpret these products as a convolution product, by identifying H_A with the space Fun(Iso(_{A)) of finitely supported functions on the set of isomorphism classes of A} via [M ] 7→1_M. Here, 1_M denotes the characteristic function of [M ] which equals one on the isomorphism class of M and zero elsewhere. The product (1.4) becomes

(f · g)(E) = X

N 6E

f (E/N )g(N ) where f, g ∈ Fun(Iso(_{A)), E ∈ Ob(A)}

as follows directly from lemma 1.2.2. This more geometric approach is the one taken by Lusztig in [42]. Schiffmann uses it in [49, p. 7] to prove the product’s associativity.

Proposition 1.2.6. The Hall algebra HA equipped with the product (1.4) and unit i : C → HA

given by i(c) = c[0] is an associative complex algebra. Furthermore, the formula

X

F_L,MX F_X,NE =X

X

F_L,XE F_M,NX (1.6)

holds for all objects L, M, N and X of _{A. The summation runs over the set Iso(A).}

Remark. We follow the proof as in [29, Lemma 2.2], originally due to Ringel, which shows

that the assumption that _{A be abelian can be relaxed to the assumption that A be an exact}

category in the sense of Quillen. This will be important later, when one considers the exact

subcategory _{Vect (X) of locally free sheaves on X of the abelian category Coh(X) of coherent} sheaves on X, where X is some smooth projective k-scheme. Furthermore, this approach is based on a categorical push-pull construction that is for example used in Joyce’s approach to Motivic Hall algebras [32, 9].

Proof. Let M, N, L be objects of _{A. Comparing both orders of multiplying yields}

[L] ? ([M ] ? [N ]) =X X F_M,NX [L] ? [X] = X X,E F_L,XE F_M,NX [E] ([L] ? [M ]) ? [N ] =X X F_L,MX [X] ? [N ] = X X,E F_L,MX F_X,NE [E]

whence ? is associative if and only if we have the identity (1.6) for all objects L, M, N, E of _A. But by lemma 1.2.2, this is equivalent to the identity

X X P_L,MX P_X,NE aX =X X P_L,XE P_M,NX aX for all L, M, N, E of_A. (1.7)

To prove this identity, we will construct a bijection between the following two sets

a X PX L,M ×PX,NE Aut(X) ←→ a Y PE L,Y ×PM,NY

Aut(Y ) for all L, M, N, E ofA, (1.8)

where the action of Aut(X) onP_L,MX ×PE

X,N is given by ξ · ((a, b), (f, g)) := ((ξa, bξ−1), (f, ξg)).

By assumption, g is an epimorphism so that ξg = g implies ξ =1X. Hence, the action of Aut(X)

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There is a natural map P_L,MX ×PE X,N → ` Y PE L,Y ×PM,NY

/ Aut(Y ) given by the

pull-back construction. This can be depicted as follows. For ((a, b), (f, g)) ∈P_L,MX ×PE

X,N, consider

the commutative diagram

N N Y E L M X L f0 _f a0 g0 b0 g a b

where Y is the pull-back of (g : E → X, a : M → X). Note that the assignment ((a, b), (f, g)) 7→ ((a0, b0), (f0, g0)) yields a well-defined map because the pull-back is unique up to unique isomorph-ism. Moreover, the image only depends on ((a, b), (f, g)) up to the action of Aut(X). Thus, we obtain a map from left to right in (1.8).

The map in the other direction is induced by the dual push-out construction. Given an element ((a0, b0), (f0, g0)) ∈P_L,YE ×PY

M,N, we obtain the same commutative diagram as above

by push-out, save the fact that X is now the push-out of (g0 : Y → M, a0 : Y → E). Using the fact that both pull-back and push-out are unique up to unique isomorphism allows one to show that these two maps are inverses to each other. In other words, the commutative square

Y E

M X

a0

g0 g a

is both a pull-back and a push-out square, so it is Cartesian.

Finally, it is clear that [0] ? [M ] = [M ] = [M ] ? [0], so i is a unit indeed.

1.2.7. Natural grading By construction, the product [M ] ? [N ] yields a sum running over

equi-valence classes of short exact sequences of the form 0 → N → E → M → 0. It therefore preserves the class of objects in the Grothendieck group, since E = M + N in K (A). This implies that the Hall algebra is graded by the Grothendieck group of _{A in such a way that the} multiplication respects this grading. ThusH_A decomposes as aK (A)-graded algebra as

HA = M α∈K (A) HA[α] where HA[α] := M [M ]∈Iso(A):M =α C · [M ].

In fact, the class of any object in the Grothendieck group is tautologically the class of an object. For example, if M is an object in _{A, then there does not exists an object N in A such that}

N = −5M inK (A). This implies that the Hall algebra is actually graded by the non-negative

coneK>0(A) in the Grothendieck group, which consists solely of classes of objects, not of formal

differences or scalar multiples of such. In particular, each class in the Grothendieck group has an object as its representative.

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1.2.8. Filtrations The multiplicative structure of the Hall algebra is related to flags or filtrations

of objects. This can be seen as follows. First, expand the definition of Hall numbers by setting FE

M1,...,Mr := {0 = Ur+1 ( Ur ( . . . ( U1 = E | Ui/Ui+1

∼

= Mi}, (1.9)

where E, M₁, . . . , MR are objects of A. Denote the cardinality of this set by FME1,...,Mr. From

associativity of the multiplication, or more precisely from formula (1.6), we deduce that

[M₁] ? . . . ? [Mr] =

X

[E]∈Iso(A)

F_ME₁_,...,M_r · [E].

In this sense, the Hall algebra encodes the combinatorially challenging problem of how a finite number of subobjects build up a fixed object in an abelian category.

1.2.9. Example LetA be a semi-simple category, and let S = {Si}i∈I denote its set of simple

ob-jects. Recall that semi-simplicity means that any object is a direct sum of its simple subobjects or, equivalently that all extension groups vanish. Moreover, the Schur lemma implies that any morphism between simple objects is either zero or an isomorphism. Hence, the endomorphism ring of a simple object S is a division algebra over k. Since k is a finite field and all Hom-sets are finite, Wedderburn’s Little Theorem implies furthermore that End(S) is a finite field extension of k. Write Di= End(Si). Thus, the Euler form of two simple objects equals

hS_i, Sjia= dimkHom(Si, Sj) =

(

[Di : k] if i = j

0 if i 6= j. It is clear that FSi⊕Sj

Si,Sj = 1 when i 6= j. Hence [Si] ? [Sj] = [Si⊕ Sj] = [Sj] ? [Si] when i 6=j, and

similarly for the twisted multiplication ·, since hS_i, Sjim = 1. For powers of a simple object, we

use Riedtmann’s formula (1.3). We find

FSi⊕Si Si,Si = 1 |Di| | Aut(Si⊕ Si)| (|Di| − 1)2 ,

where we have used the fact that D_i is a field to arrive at a_S_i = |D_i| − 1. One can show that Aut(Si⊕ . . . ⊕ Si) ∼= GLn(Di) as groups,

where n ∈ N. Write di= |Di|. Recall that an invertible n×n matrix with entries in the finite field

Diis precisely determined by choosing n linearly independent vectors in Dni. A straightforward

counting argument then shows that | GL_n(D_i)| = (dn_i − 1)(dn

i − di) . . . (dni − d n−1

i ). In particular,

| Aut(Si⊕ Si)| = (d2i − 1)(d2i − di) = di(di+ 1)(di− 1)2 which results in

[S_i] ? [S_i] = (d_i+ 1)[S_i⊕ S_i] and [S_i] · [S_i] = d1/2_i (d_i+ 1)[S_i⊕ S_i].

In conclusion, H_A is a free commutative polynomial algebra in the variables {Si}i∈I. As to

its grading, the class of an object M ∼=L

i∈ISini in the Grothendieck group K (A) is given by

M =P

i∈IniSi. It encodes the degree of the corresponding polynomial if we put deg(Si) = 1.

Remark. Note that it is rather exceptional for a Hall algebra to be commutative, since normally

Ext1(M, N ) 6∼= Ext1(N, M ). In the above example, it is due to the fact thatA is semi-simple. Hall algebras of categories of global dimension one or greater are never commutative.

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1.3 The Hall Hopf algebra

In [23], Green equipped the Hall algebra associated to a finitary, hereditary algebra over a finite field with a comultiplication, encoding the dual operation of breaking down a given object as extension of two smaller objects. This construction turns the Hall algebra into a coassociative, counital coalgebra, essentially by virtue of the same numerical equality (1.6) used to prove associativity. A natural question to ask is wether or not these operations are compatible in the sense of bialgebras. A necessary condition for compatibility is that the category A be of global

dimension at most one. The reason for this will become clear in the next subsection. With this

condition in mind, we use the following

Convention. In this section, A will denote a hereditary category, so gl. dim(A) 6 1.

However, this condition alone is not sufficient. It turns out that some form of twisting or braiding of the (co)multiplication is necessary. We will discuss several approaches to solve this discrepancy. Afterwards, we mention Xiao’s antipode, showing that H_A is in fact a Hopf algebra8. Finally, the Hopf algebras obtained as Hall algebra of some abelian category turn out to be self-dual, as can be seen by a natural Hopf pairing they carry. This final piece of structure is discussed in the last part of this section.

1.3.1 Green’s coproduct

In this section, we will equipH_A with the structure of a bialgebra. For the reader’s convenience, we recall here the definition of a bialgebra. For further definitions and related results, we refer the reader to the excellent [37].

Definition 1.3.1. Let (A, µ, η) be an associative k-algebra with multiplication µ : A ⊗kA → A

and unit η : k → A. It is a bialgebra if it comes equipped with two algebra morphisms ∆ : A → A ⊗_kA and : A → k, called comultiplication and counit respectively, satisfying

(i) (Coassociativity) (∆ ⊗1A)∆ = (1A⊗ ∆)∆.

(ii) (Counitality) ( ⊗1A)∆ =1A= (1A⊗ )∆.

We say that the bialgebra A is cocommutative if ∆ = τ ∆ where τ (x ⊗ y) = y ⊗ x. The triple (A, ∆, ) is called a (coassociative counital) coalgebra. A linear subspace C ⊆ A is a subcoalgebra with respect to (∆|C, |C) if ∆(C) ⊆ C ⊗ C. A morphism of bialgebras is an algebra morphism

f : A → B such that Bf = A and (f ⊗ f )∆A= ∆Bf .

1.3.2. Example Let (g, [−, −]) be a semi-simple Lie algebra, and let U(g) denote its universal

enveloping algebra. Recall that as a k-vector space, U(g) is the tensor algebra of g divided out by the idealI generated by the set {x⊗y−y⊗x−[x, y] : x, y ∈ g}. This forces the embedding9

ι : g → U(g) to be a morphism of Lie algebras, when we equip U(g) with the commutator bracket.

Multiplication is defined on pure tensors by concatenating them and extended k-linearly to the

8

Note that antipodes are unique when they exist. As a slogan: having an antipode is a property of a bialgebra, not an extra structure on it.

9

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entire algebra. It is clearly associative. The unit map η : k → U(g) is given by η(c) = c. As for the coalgebra structure, define

∆ : U(g) −→ U(g) ⊗kU(g) by ∆(x) = x ⊗ 1 + 1 ⊗ x, and (x) = 0

for all x ∈ g. It follows from [37, Theorem V.2.4] that this extends uniquely to a bialgebra structure on U(g) by defining the actions of ∆ and on general elements of U(g) by

∆(x1⊗ . . . ⊗ xn) = ∆(x1· . . . · xn) := ∆(x1) · . . . · ∆(xn)

and similarly for the counit, where x₁, . . . , xn ∈ g. Coassociativity follows straightforwardly,

but one can wonder if counitality holds. For x ∈ g we find

( ⊗1_A)∆(x) = (x) ·1_A(1) + (1) ·1_A(x) =1_A(x) and similarly for the other equality. Thus, (U(g), µ, η, ∆, ) is a bialgebra.

Note that the coproduct is cocommutative. This will not be the case for the quantised universal envelopping algebra of g which is intimately related to Hall algebras as we will see in the next chapter. On the other hand, the product is not commutative. Generically, k-vector spaces may allow for many different structures of algebra or coalgebra, but the requirement that (∆, ) be algebra morphisms with respect to (µ, η) is quite restrictive.

1.3.3. LetA be a finitary category of arbitrary (but finite) global dimension. The dual operation of multiplication in H_A is breaking an object down in all possible ways. Although there are only finitely many extensions between two objects in _{A, there might be infinitely many ways} to break up an object in two pieces. Since this produces sums with infinitely many non-zero terms, we account for this by introducing completions ofH_A and H_A ⊗H_A.

Definition 1.3.4. A finitary abelian categoryA is said to satisfy the finite subobjects condition if every object of_{A has only finitely many subobjects.}

We will see that for such categories, there is no need pass to the completed Hall algebras. For the general case however, we introduce the following

Definition 1.3.5. The completed Hall algebra of a finitary categoryA as above is defined as Hc

A :=

Y

[M ]∈Iso(A)

C · [M ].

Furthermore, for classes α, β in the non-negative coneK>0(A) of the Grothendieck group of A,

we define HA[α] ˆ⊗HA[β] := Y M =α,N =β C · [M ] ⊗ C · [N ] and HA⊗ˆHA := Y α,β∈K>0(A) HA[α] ˆ⊗HA[β].

Effectively, we are allowing formal infinite expressions of the formP

M,NcM,N[M ] ⊗ [N ]. Higher

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Before addressing these matters of convergence somewhat further, consider the following

Definition 1.3.6 (Coproduct). Consider HA as a complex vector space. Given [E] ∈ Iso(A),

define its topological coproduct by

∆_?([E]) := X

M,N

F_M,NE aMaN aE

[M ] ⊗ [N ] ∈H_A⊗ˆH_A (1.10)

The word “topological” refers to the fact that this map takes on values in the completion ofH_A⊗ HA. Following Schiffmann [49] and Green [23], one can alternatively define a comultiplicative

structure on H_A by introducing a slight twist via ∆([E]) := X M,N hM, N imFM,NE aMaN aE [M ] ⊗ [N ] ∈H_A⊗ˆH_A. (1.11)

Note that in terms of the alternative generators_{JM K = a}M[M ], the coefficient of the termJM K ⊗ JN K in the coproduct of JE K is again simply (hM , N im)F

E

M,N. In this sense, comultiplication

is really dual to the multiplication. A different motivation for this definition will be given in section 1.3.2.

Proposition 1.3.7 (Hall coalgebra). The data (H_A, ∆, ) equip H_A with the structure of a

topological coassociative counital coalgebra. The counit :H_A _{→ C is given by ([M]) = δ}_{[M ],[0]}. Proof. First of all, for coassociativity to make sense we must verify that the two maps

(∆ ⊗1_A)∆, (1_A⊗ ∆)∆ :H_A 7→H_A⊗ˆH_A⊗ˆH_A

are well-defined. The reason for this is that the image of an element under ∆ may already consist of infinitely many terms, so it is not obvious the above formulae produce an expression with finite coefficients in the end. This is arranged for by the finitary assumption on_{A. Indeed,} the only terms in the image of ∆ that may contribute to the coefficient of [L] ⊗ [M ] ⊗ [N ] in (∆ ⊗1A)∆ (resp. (1A⊗ ∆)∆) are of the form [A] ⊗ [N ] for some extension A of L by M (resp.

of the form [L] ⊗ [B] for some extension B of M by N ), and there are only finitely many such extensions.

Secondly, note that ∆? is coassociative if and only ∆ is, because of the multiplicativity of

the Euler form in the second variable. Expressing the coassociativity condition of ∆ in terms of the alternative generators_{JN K yields the expression}

(∆ ⊗1_A)∆(_{JE K) =} X L,M,X,N F_L,MX F_X,NE _{JLK ⊗ JM K ⊗ JN K} ? = X M,N,L,X F_M,NX F_L,XE _{JLK ⊗ JM K ⊗ JN K = (1}A⊗ ∆)∆(JE K).

Thus the coproducts are coassociative if and only ifP

XFL,MX FX,NE =

P

XFM,NX FL,XE . But this

is equation (1.6), proven to hold in proposition 1.2.6. Counitality is immediate, essentially because there is only one extension class in Ext1(M, 0) for any object M of_{A. This concludes} the proof.

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1.3.8. Natural grading The coproduct respects the grading ofHA by the (positive cone in the)

Grothendieck group_{K (A), that is it sends} ∆(H_A[γ]) ⊆ Y

α+β=γ

HA[α] ˆ⊗HA[β],

where α + β = γ in K (A). Thus (H_A, ∆, ) is a K (A)-graded coalgebra.

1.3.9. Filtrations Recall the definition of the generalized Hall numbers in 1.2.8. Since HA is

a coassociative coproduct, we can unambiguously speak of the r-th iterated coproduct of an element [E] ∈ Iso(_{A). A repeated application of formula (1.6) shows that}

∆r(_{JE K) =} X

M1,...,Mr

F_ME₁_,...,M_r_JM1K ⊗ . . . ⊗ JMrK, (1.12) where the summation runs over Iso(_{A). Note that in contrast to the expression for repeated} multiplication in paragraph 1.2.8, the repeated comultiplication is expressed most neatly in terms of the generators_{JE K = a}E[E]. Reverting to the generators [E] would introduce an extra

factor of (aM1. . . aMr)/aME in front of each summand.

1.3.10. Finite subobjects condition When does the coproduct of a given object [E] ∈ Iso(A) take on values in the non-completed algebraH_A ⊗H_A? This only occurs when the set

{FE

M,N ∈ Z>0 : [M ], [N ] ∈ Iso(A)}

is finite, i.e. when it contains only finitely many non-zero numbers. The set F_M,NE contains subobjects of E satisfying certain conditions. Thus the above set is finite if and only if E has only finitely many subobjects. A finitary abelian category satisfying this requirement gives rise to a proper coalgebra structure on H_A as defined above, not only a topological one. To run ahead of matters, the category of finite-dimensional representations over a finite field of finite quivers satisfies this condition, but the category of coherent sheaves on a smooth projective scheme over a finite field does not.

1.3.11. Let us concretely consider the simplest of semi-simple categories, namely the abelian

category _{A = Vect}_k of finite-dimensional k-vector spaces.10 This category is clearly finitary. Any short exact sequence

0 → U → V → W → 0

of k-vector spaces splits, by choosing a complement of U in V . So gl. dim(A) = 0 indeed. The zero-dimensional vector space is the zero object of this category, and the unique (up to isomorphism) one-dimensional vector space k is the only simple object. In fact, two vector spaces are isomorphic if and only if they have the same dimension. Hence _{K (A) ∼}_{= Z under} taking dimensions, and each class in the Grothendieck contains only one isomorphism class of objects (namely the one of that dimension). Write u_m= [km] for this element in H_A. Then

um? un= Fm,nn+mun+m

10

This category is equivalent to that of finite-dimensional k-representations of the quiver of type A1, consisting

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where F_m,nn+m denotes the number of n-dimensional linear subspaces of kn+m. Since k is a finite field, this number is given by the well-known number of points of the Grassmannian Gr(n, n+m) over k. Thus F_m,nn+m= | Gr(n, n + m)| = n + m n ! + = [n + m]+! [n]+![m]+! (1.13)

where we have used the q-quantum numbers [n]₊ = 1 + q + q2_{+ . . . + q}n−1 _{= (q}n_{− 1)/(q − 1)}

and [n]+! = [n]+[n − 1]+. . . [1]+. As we saw in example 1.2.9, HA is commutative and it is

isomorphic to the polynomial algebra k[X].

What about its coalgebra structure? Note that | Aut(km)| = (qm−1)(qm_{−q) . . . (q}m_−qm−1₎

as follows from the observation that an invertible m × m matrix with entries in a finite field k is precisely determined by choosing m linearly independent vectors in k. This can be rewritten as am= q(

m

2)(q − 1)m[m]₊!, and a quick calculation then shows that

∆?(ur) =

X

m+n=r

q−nmum⊗ un.

So H_A is also cocommutative11. But how does the comultiplication interact with the algebraic structure of H_A? The question is wether or not ∆(XY ) = ∆(X)∆(Y ) for X, Y ∈ H_A (we do not fix a specific (co)multiplication for now). This requires a choice of multiplication on HA⊗ˆHA, and we declare (a ⊗ b)(c ⊗ d) = (ac ⊗ bd). In more sophisticated terms, the underlying

monoidal category in which the Hall algebra lives has a trivial braiding (see [38] for these notions). Writing out both ∆_?(u_r? us) and ∆?(ur) ? ∆?(us) and applying a quantum analogue

of a certain well-known binomial formula, we arrive at

∆_?(u_r? us) = X a,b,c,d : m+n=r+s q−ab−cd−ad m a ! + n b ! + um⊗ un ∆_?(u_r) ? ∆_?(u_s) = X a,b,c,d : m+n=r+s q−ab−cd m a ! + n b ! + um⊗ un,

where a + b = r, c + d = s, a + c = m, b + d = r. So even in this simplest of examples, the comultiplication is not compatible with the multiplication, but the discrepancy is only a factor. Note that the Euler form onVectk is given by hum, unia= dimkHom(km, kn) = mn, so passing

to the twisted (co)multiplication (·, ∆) would only introduce an extra factor of qmn/2 in each term of the sum. This is not enough to make the coproduct into an algebra morphism.

1.3.12. The problem of compatibility of the algebra and coalgebra structure of HA can be

re-solved in several ways. Which of these is more natural depends on one’s agenda. Historically, the problem was resolved in two steps. In the famous paper [47], Ringel twisted the multiplication onH_A using the Euler form as in equation (1.5). His motivation was to strengthen the analogy with quantum groups by realizing the socalled quantum Serre relations (or Drin’feld-Jimbo rela-tions) in the Hall algebra in the case thatA is the module category of a quiver.12 _{Moreover, this}

11_{This holds more generally for the Hall algebra of any semi-simple category}

A, as one easily checks.

12

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twist makes the algebraic structure ofH_A independent of the quiver’s orientation. Inspired by this work, Green proved in [23] the existence and compatibility of the coproduct ∆ onH_A; note that this is the naive coproduct (1.11) twisted by the Euler form. However, this compatibility requires one of the following three extra adjustments:

1. In the original article, Green introduces a twist in the multiplication of H_A⊗ˆH_A using the Euler form onK (A). To wit, he defines (A ⊗ B) · (C ⊗ D) = (B, C)m(AC) ⊗ (BD) to

account for the fact that “C jumps over B”. This ad hoc solution solves the problem, but makesH_A into a twisted bialgebra instead of an actual bialgebra object. It also suggests a non-trivial braiding requirement as alluded to earlier.13

2. Baez and Walker argue in [2] that indeed, this is ad hoc solution neglects the deeper fact that both H_A and the corresponding quantum group might not be bialgebras in the underlying symmetric monoidal category_Vect_k, but they are in a certain braided monoidal category_VectK where K is the Grothendieck group_{K (A). The latter is the category of} K-graded vector spaces, where the braiding is precisely given by BV,W(v ⊗ w) = (v, w)mw ⊗ v

for v ∈ V , w ∈ W . Pictorially, this again boils down to “C jumping over B”, but now with an appropriate braid factor. In this category,H_A is an actual bialgebra object. This is the more natural approach when one aims for a categorification14 of quantum groups by means of Hall algebras. For more on this, see Walker’s thesis [53].

3. Let g = n−⊕ h ⊕ n+_{denote the triangular decomposition of a semi-simple Lie algebra. To}

strengthen the connection between the Hall algebra associated to the underlying Dynkin diagram and the associated quantum group U+_q(g), one can consider an extended Hall algebra. It is defined by H_Ae =H_A ⊗_C_{C[K (A)] where the (co)multiplication is extended} in such a way that ∆e is a morphism of algebras without twisting the multiplication on the tensor product He

A⊗ˆHAe. This is the nicest approach for our intents and purposes,

since we both get a proper bialgebra object and a strong analogy with quantum groups.

Let us first show that the comultiplication is a morphism of algebras following Green’s original proof. Then we will define the extended Hall algebra and prove that it is a bialgebra object without twisting the multiplication ofH_A⊗ˆH_A. Henceforth, gl. dim(_{A) 6 1.}

1.3.13. Following Green The multiplication onHA⊗ˆHA is defined for elements x, y, z, w ofHA

as (x ⊗ y) · (z ⊗ w) := (y, z)_mxz ⊗ yw. Since comultiplication takes values in the completed Hall

algebra H_A⊗ˆH_A, it is not clear wether or not products of the form ∆([M ]) · ∆([N ]) produce well-defined elements of this Hall algebra, in that their coefficients in C might be infinite.

Definition 1.3.14. A product (P

iai ⊗ bi) · (Pjcj ⊗ dj) of elements of HA⊗ˆHA is called

convergent if for every pair of elements [M ], [N ] ∈ Iso(_{A) the coefficient of [M ] ⊗ [N ] in the}

product is non-zero for only finitely many values of (i, j).

13

Note that a few years earlier, Ringel already introduced such a twist in the multiplication so as to have it precisely match with the multiplication of U+

q(g); see [47].

14

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Note that this means that such a product may very well be a sum of infinitely many linearly independent terms - this lives in the completion after all. We are merely excluding infinite coefficients, as such an element does not exist in C. Fortunately, no serious problems occur, as shown in the following

Lemma 1.3.15 (p. 12,[49]). For M1, M2 ∈ Ob(A), ∆([M1]) · ∆([M2]) is convergent.

Proof. We will determine an upper bound on the number of objects L_i, Ni that can contribute

to a term [R] ⊗ [S] in the product ∆([M1]) · ∆([M2]). First of all, the coefficient of [Ni] ⊗ [Li]

can only be non-zero in ∆([M_i]) if there exists a short exact sequence 0 → L_i → M_i→ N_i → 0 where i = 1, 2. Secondly, the coefficient of [R] in [L₁] · [L₂] can only be non-zero if there exists a short exact sequence 0 → L2 → R → L1→ 0, and that of [S] can only be non-zero in [N1] · [N2]

if there exists a short exact sequence 0 → N₂ → S → N₁ → 0. When all these conditions are satisfied, we see that

L1 ' Im(R L1,→ M1), N2 ' Im(M2  N2 ,→ S)

L2 ' Ker(M2  N2,→ S), N1' Ker(R L1 ,→ M1).

Hence, the L_i, Ni for i = 1, 2 are isomorphic to images or kernels of morphisms R → M1 and

M2 → S. But since A is finitary, there are only fintely many choices for such objects.

Theorem 1.3.16. The map ∆ :HA −→HA⊗ˆHA is a morphism of algebras with respect to the

twisted multiplication on H_A⊗ˆH_A.

Let us spell out what this means for objects [M ], [N ] ∈ Iso(_{A). We find that} ∆([M ] · [N ]) = hM, N i_mX E P_M,NE aMaN ∆([E]) = hM, N i_mX X,Y hX, Y i_mX E P_M,NE P_X,YE aMaNaE [X] ⊗ [Y ] ∆([M ]) · ∆([N ]) = X A,B,C,D hA, BimhC, Dim P_A,BM P_C,DN aMaN ([A] ⊗ [B]) · ([C] ⊗ [D]) = X A,B,C,D hA, BimhC, Dim(B, C)m P_A,BM P_C,DN aMaN [A] · [C] ⊗ [B] · [D] = X X,Y X A,B,C,D hA, B + Ci_mhB + C, Di_m(B, C)_m | {z } K(A,B,C,D) P_A,BM P_C,DN P_A,CX P_B,DY aMaNaAaCaBaD [X] ⊗ [Y ]

Note that for a term in the latter sum to be non-zero, we need the following equalities in_{K (A):} [A] + [B] = [M ], [C] + [D] = [N ], [A] + [C] = [X], [B] + [D] = [Y ]

as one can read of directly from the expression above. Substituting these in hM, N i_mhX, Y i_m yields

hM, N i_mhX, Y i_m = hA, Di2_mK(A, B, C, D) = q−hA,Dia_{K(A, B, C, D)}

by the relation between the additive and multiplicative Euler form. This reveals the core of Green’s work in [23], and shows that ∆ is an algebra morphism if and only if we have the following

An introduction to HALL ALGEBRAS a categorification of quantum groups