LECTURES ON HALL ALGEBRAS

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quantum group

projective line

hall algebra

xiao's hall

called weighted projective

categories

kac-moody algebras

hall algebras

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LECTURES ON HALL ALGEBRAS
OLIVIER SCHIFFMANN
Contents
Introduction 2
Lecture 1. 6
1.1. Finitary categories 6
1.2. Euler form and symmetric Euler form. 6
1.3. The name of the game. 7
1.4. Green’s coproduct. 9
1.5. The Hall bialgebra and Green’s theorem. 11
1.6. Green’s scalar product. 17
1.7. Xiao’s antipode and the Hall Hopf algebra. 18
1.8. Functorial properties. 20
Lecture 2. 22
2.1. The Jordan quiver. 22
2.2. Computation of some Hall numbers. 23
2.3. Steinitz’s classical Hall algebra. 25
2.4. Link with the ring of symmetric functions. 29
2.5. Other occurences of Hall algebras. 30
Lecture 3. 32
3.1. Quivers. 32
3.2. Gabriel’s and Kac’s theorems. 33
3.3. Hall algebras of quivers. 37
3.4. PBW bases ( nite type). 41
3.5. The cyclic quiver. 45
3.6. Structure theory for tame quivers. 47
3.7. The composition algebra of a tame quiver. 50
Lecture 4. 54
4.1. Generalities on coherent sheaves. 54
14.2. The category of coherent sheaves overP . 55
14.3. The Hall algebra ofP . 57
4.4. Weighted projective lines. 63
4.5. Crawley-Boevey’s theorem. 68
4.6. The Hall algebra of a weighted projective line. 71
4.7. Semistability and the Harder-Narasimhan ltration. 75
4.8. The spherical Hall algebra of a parabolic weighted projective line. 78
4.9. The Hall of a tubular weighted projective line. 79
4.10. The Hall algebra of an elliptic curve. 82
4.11. The Hall of an arbitrary curve. 85
Lecture 5. 89
5.1. Motivation. 89
5.2. The Drinfeld double. 92
5.3. Conjectures and Cramer’s theorem. 93
5.4. Example and applications. 94
5.5. The Hall Lie algebra of Peng and Xiao. 99
12 OLIVIER SCHIFFMANN
5.6. Kapranov and Toen’s derived Hall algebras. 100
Windows. 103
Appendix. 106
A.1. Simple Lie algebras 106
A.2. Kac-Moody algebras. 111
A.3. Enveloping algebras. 115
A.4. Quantum Kac-Moody algebras. 116
A.5. Loop algebras of Kac-Moody algebras. 119
A.6. Quantum loop algebras. 122
References 125
Introduction
These notes represent the written, expanded and improved version of a series
of lectures given at the winter school \Representation theory and related topics"
held at the ICTP in Trieste in January 2006, and at the summer school "Geometric
methods in representation theory" held at Grenoble in June 2008. The topic for
the lectures was \Hall algebras" and I have tried to give a survey of what I believe
are the most fundamental results and examples in this area. The material was
divided into ve sections, each of which initially formed the content of (roughly)
one lecture. These are, in order of appearance on the blackboard :
Lecture 1. De nition and rst properties of (Ringel-)Hall algebras,
2. The Jordan quiver and the classical Hall algebra,
Lecture 3. Hall algebras of quivers and quantum groups,
4. Hall of curves and quantum loop groups,
Lecture 5. The Drinfeld double and Hall algebras in the derived setting.
By lack of time, chalk, (and yes, competence !), I was not able to survey with
the proper due respect several important results (notably Peng and Xiao’s Hall Lie
algebra associated to a 2-periodic derived category [PX2], Kapranov and Toen’s
versions of Hall algebras for derived categories, see [K3], [T], or the recent theory of
Hall algebras of cluster categories, see [CC], [CK], or the recent use of Hall algebra
techniques in counting invariants such as in Donaldson-Thomas theory, see [J2],
[KS], [R4],...). These are thus largely absent from these notes. Also missing is
the whole geometric theory of Hall algebras, initiated by Lusztig [L5] : although
crucial for some important applications of Hall algebras (such as the theory of
crystal or canonical bases in quantum groups), this theory requires a rather di erent
array of techniques (from algebraic geometry and topology) and I chose not to
include it here, but in the companion survey [S5]. More generally, I apologize to all
those whose work deserves to appear in any reasonable survey on the topic, but is
unfortunately not to be found in this one. Luckily, other texts are available, such
as [R7], [R8], [H5]. There are essentially no new results in this text.
Let me now describe in a few words the subject of these notes as well as the
content of the various lectures.
Roughly speaking, the Hall, or Ringel-Hall algebra H of a (small) abelianA
categoryA encodes the structure of the space of extensions between objects inA.
In slightly more precise terms, H is de ned to be the C-vector space with a basisA
consisting of symbolsf[M]g, whereM runs through the set of isomorphism classes
of objects inA; the multiplication between two basis elements [M] and [N] is aLECTURES ON HALL ALGEBRAS 3
linear combination of elements [P ], where P runs through the set of extensions of
M by N (i.e. middle terms of short exact sequences 0! N ! P ! M ! 0),
and the coe cient of [ P ] in this product is obtained by counting the number of
ways in which P may be realized as an extension of M by N (see Lecture 1 for
details). Of course, for this counting procedure to make senseA has to satisfy
certain strong niteness conditions (which are coined under the term nitary ), but
there are still plenty of such abelian categories around. Another fruitful, slightly
di erent (although equivalent) way of thinking about the Hall algebra H is toA
consider it as the algebra of nitely supported functions on the \moduli space"
M of objects ofA (which is nothing but the set of isoclasses of objects ofA,A
equipped with the discrete topology), endowed with a natural convolution algebra
structure (this is the point of view that leads to some more geometric versions of
Hall algebras, as in [L5], [L1], [S4]).
Thus, whether one likes to think about it in more algebraic or more geometric
terms, Hall algebras provide rather subtle invariants of nitary abelian categories.
Note that it is somehow the \ rst order" homological properties of the category A
1(i.e. the structure of the groups Ext (M;N)) which directly enter the de nition
of H , butA may a priori be of arbitrary (even in nite) homological dimension.A
However, as discovered by Green [G4], whenA is hereditary , i.e. of homological
dimension one or less, it is possible to de ne a comultiplication : H ! H
HA A A
and, as was later realized by Xiao [X1], an antipode S : H ! H . These threeA A
operations are all compatible and endow (after a suitable and harmless twist which
we prefer to ignore in this introduction) H with the structure of a Hopf algebra.A
All these constructions are discussed in details in Lecture 1.
As the reader can well imagine, the above formalism was invented only after
some motivating examples were discovered. In fact, the above construction appears
in various (dis)guises in domains such as modular or p-adic representation theory
(in the form of the functors of parabolic induction/restriction), number
and automorphic forms (Eisenstein series for function elds), and in the theory
of symmetric functions. The rst occurence of the concept of a Hall algebra can
probably be traced back to the early days of the twentieth century in the work of
E. Steinitz (a few years before P. Hall was born) which, in modern language, deals
with the case of the categoryA of abelian p-groups for p a xed prime number.
This last example, the so-called classical Hall algebra is of particular interest due to
its close relation to several fundamental objects in mathematics such as symmetric
functions (see [M1]), ag varieties and nilpotent cones. After studying in some
details Steintiz’s classical Hall algebra we brie y state some of the other occurences
of (examples of) Hall algebras in Lecture 2.
The interest for Hall algebras suddenly exploded after C. Ringel’s groundbreak-
ing discovery ([R5]) in the early 1990s that the Hall algebra H of the category~RepQ
~ofF -representations of a Dynkin quiverQ (equiped with an arbitrary orientation)q
provides a realization of the positive part U(b) of the enveloping algebra U(g) of
the simple complex Lie algebra g associated to the same Dynkin diagram (to be
more precise, one gets a quantized enveloping algebra U (g), where the deformationv
parameter v is related to the order q of the nite eld F ).q
It is also at that time that the notion of a Hall algebra associated to a nitary
category was formalized (see [R6]). These results were subsequently extended to
arbitrary quivers in which case one gets (usually in nite-dimensional) Kac-Moody
algebras, and were later completed by Green. The existence of a close relationship
between the representation theory of quivers on one hand, and the structure of4 OLIVIER SCHIFFMANN
simple or Kac-Moody Lie algebras on the other hand was well-known since the
seminal work of Gabriel, Kac and others on the classi cation of indecomposable
representations of quivers (see [G1], [K1]). Hall algebras thus provide a concrete,
beautiful (and useful !) realization of this correspondence. After recalling the
forerunning results of Gabriel and Kac, we state and prove Ringel’s and Green’s
fundamental theorems in the third Lecture.
Apart from the categories ofF -representations of quivers, a large source of ni-q
tary categories of global dimension one is provided by t