Spring School : “Hall algebras
and their applications”
Wuppertal, 19 March – 23 March
2007
Organizers: Andrew Hubery (Paderborn),
Markus Reineke (Wuppertal)
Hosted and funded by:
![[Group photo]](Gruppenfoto.jpg)
The Spring School:
The topic of the Spring School is
the construction of Hall algebras and their interactions with and
applications to representation theory, quantum groups and cluster
algebras.
The Spring School is intended for
Ph.D. students and (early) postdocs working on and interested in
these fields.
The Spring School will follow the
tradition of previous spring/summer schools in representation theory.
Every participant chooses a subject of the program (see below) and
gives a talk. The typical participant will not be an expert in the
subject, but working in one of the fields indicated above and
interested in jointly learning a new subject. There will be no
special prerequisities except for standard knowledge in algebra, but
participants are expected to prepare themselves and their talks in
advance. It is recommended that this is done in small groups; in
particular, the talks could/should be subdivided into smaller parts.
At the end of the week, there will
be lectures by Philippe Caldero (Lyon) on more recent developments in
the field.
Program:
Here is a detailed list of the talks, with some proposals for topics to be covered during the talks, and possible references:
Lecture 1: ???
Matrices up to conjugation. Jordan Normal Form. Finite dimensional modules for k[x].
Nilpotent matrices <-> eigenvalue zero. Finite dimensional modules for k[[x]].
Indecompoposables <-> Jordan blocks <-> positive integers.
Isomorphism classes of nilpotent matrices <-> partitions.
Definition of Hall numbers. Polynomiality. (Number of homomorphisms/automorphisms given by polys.)
First examples: Semisimple modules -> Grassmannian. Indecomposables -> single point.
Example 2.4 from [S]?
References [M], [S].
Lecture 2: ???
Definition of the Hall algebra.
Theorem of Steinitz, Hall, Macdonald: The Hall algebra is isomorphic to Q(q)[z_1,z_2,...]
where we can take as generators the semisimple modules or the indecomposable modules.
Macdonald's ring of symmetric functions. Isomorphism to Hall algebra.
Comultiplication on symmetric functions. Transfer to Hall algebra.
Identification of various functions: elementary symmetric functions, complete symmetric functions, power sum functions, `cyclic' functions.
Hopf pairing on symmetric functions/Hall algebra.
References [H1], [M], [S].
Lecture 3: Philipp Lampe, Matthias Warkentin
Quivers/path algebras. Representations/modules.
Krull-Remak-Schmidt for finite dimensional modules.
First examples: simple modules, projective modules.
Dimension vectors, representation varieties, GL action.
Relation to first lecture - Jordan quiver.
Standard resolution => heredity.
Apply hom => Ringel's four-term exact sequence => Euler form.
References [B], [CB], [H2].
Lecture 4: Philipp Lampe, Matthias Warkentin
The classification problem: classify all indecomposable representations.
Done for a single loop: first two lectures.
Solution for the linear quiver.
Gabriel's Theorem (finite representation type iff Dynkin diagram, dimension vector giving bijection between indecomposables and `positive roots')
Two loop quiver: Take Jordan block and diagonal matrix for k^n. Get n-parameter family of indecomposables.
Kac's Theorem (dimension vectors of indecomposables over alg. closed field are the positive roots)
References [B], [CB], [H2], [S].
Lecture 5: Daiva Pucinskaite
Definition of Hall numbers and twisted Hall algebra.
Associativity and unit. Grading by dimension vector.
Riedtmann's Formula.
Comultiplication (as in Lecture 1 of [S], so omitting the torus elements K_i).
Coassociativity (reduces to same formula as for associativity) and counit, again graded.
References [H2], [S].
Lecture 6: Nikolay Dichev
First examples of the multiplication and comultiplication.
Trivial quiver (single dot). Obtain quantum numbers (commutative and cocommutative bialgebra).
(Jordan quiver already done).
Rank two calculations. Type A_2. Kronecker quiver. Two cycle. Generalised Kronecker quiver.
Neither comm. nor cocomm. Also, not quite a bialgebra!
Obtain quantum Serre relations.
References [H2], [S].
Lecture 7: Torsten Hoge, Thorsten Weist
Lie Theory.
Symmetric generalised Cartan matrices <-> diagrams.
Root system \Phi, Weyl group action, bilinear form.
`free' Lie algebra \tilde{\mathfrak g} associated to GCM. Grading. Extension of bilinear form.
Definition of Lie algebra {\mathfrak g} as quotient of \tilde{\mathfrak g} via the radical of the bilinear form.
Theorem showing that the Serre relations are defining relations.
Universal enveloping algebra via generators and relations, triangular decomposition.
PBW basis and hence formula for the character of u^+:
ch u^+ := \sum_{\alpha\geq0} (\dim u^+_\alpha) e^\alpha = \prod_{\alpha\in\Phi^+} (1-e^\alpha)^{-\dim {\mathfrak g}_\alpha}
References [C], [K].
Lecture 8: Britta Aufgebauer, Johannes Engel
PDF file
Quantum Group Theory.
Definition of the `free' quantum group associated to a GCM. Hopf algebra structure. Triangular decomposition.
[L, Chapter 3], but can be done over an arbitrary field k with v\in k^\ast not a root of unity (\`a la Jantzen [J]).
Bilinear form on the positive part.
Definition of the quantum group as the quotient via the radical of this bilinear form.
Theorem
stating that the quantum Serre relations are defining relations
(essentially the same ideas used in the proof as for the Lie algebra
case).
Now compute the character for U^+_k. (not really published. soon to be in my notes [H].)
First, let A be the subalgebra of Q(v) by localising the Laurent poly ring Q[v,v^{-1}] at all roots of unity v^n-1.
Let U^+_A be the A-algebra with the same presentation as the quantum group (generators and quantum Serre relations).
The specialisation at every field k with non root of unity v gives the quantum group U^+_k constructed above.
Now,
U^+_A has a bilinear form, and any A-torsion element must be in the
radical. Since each quantum group U^+_k has non-degenerate bilinear
form, deduce that U^+_A has no A-torsion elements. Thus U^+_A is a
subalgebra of the quantum group U^+_{Q(v)}.
As in Lusztig [L, Chapter 33], we deduce that
\dim_k (U^+_k)_\alpha = \rank_A (U^+_A)_\alpha = \dim_Q u^+_\alpha
where u^+ is the universal enveloping algebra.
It follows that, for all fields k with non root of unity v, we have
ch
U^+_k = \sum_{\alpha\geq0} (\dim_k (U^+_k)_\alpha) e^\alpha = ch u^+ =
\prod_{\alpha\in\Phi^+}(1-e^\alpha)^{-\dim{\mathfrak g}_\alpha}
Addendum to Lecture 8.
Let A be the subalgebra of Q(v) given by localising the Laurent polys at v^n-1 for all n.
Let U^+_A be
the subalgebra of U^+_{Q(v)} generated by the E_i. (This contains all
divided powers, so is given by extension of scalars of Lusztig's
Q[v,v^{-1}]-form.)
Now the bilinear form on U^+_A takes values in A.
Given a field k and v\in k^\ast not a root of unity, we can form the specialisation
\bar U^+_k := U^+_A \otimes_A k.
This has generators the E_i and a bilinear form. Hence there is a natural map
\bar U^+_k -> U^+_k,
since the latter was defined by quotienting out the radical of the bilinear form.
Conversely we
have a presentation for U^+_k, and the defining relations clearly hold
in \bar U^+_k, since the generic versions hold in U^+_A. Thus we have a
map
U^+_k -> \bar U^+_k.
Hence U^+_k is isomorphic to the specialisation U^+_A \otimes_A k, so
\dim_k (U^+_k)_\alpha = \rank_A (U^+_A)_\alpha,
since U^+_A is a free A-module by definition.
Now we can use Lusztig, Chapter 33, to see that
\rank_A (U^+_A)_\alpha = \dim_Q u^+_\alpha.
Thus
ch U^+_k = ch u^+_\alpha
for all fields k.
References [J], [L] (and hopefully [H] for the last part).
Lecture 9 (2 parts): Jeanne Scott, Marcel Wiedemann
Definition of the extended (twisted) Ringel-Hall algebra.
New definition of comultiplication, inspired by the theory of quantum groups (so including the torus elements K_i).
Computation showing that the extended Ringel-Hall algebra is a bialgebra iff Green's Formula holds.
Definition of bilinear form. Adjointness of multiplication and comultiplication. Non-degeneracy.
Proof of Green's Formula.
(The
proof of Schiffmann [S] is essentially the same as Ringel's version,
which is an expanded version of Green's original proof, with a
homological interpretation. The proof in [H2] is, I think, slightly
simpler as we don't immediately construct a bijection between the two
sides (frames and crosses), but relate them to counting 3x3 diagrams
and counting fibres.)
References [H2],[S].
Lecture 10: Dagfinn Vatne
Orthonormal basis \theta_i constructed via Green's non-degenerate bilinear form.
First properties -> Borcherd's matrix.
(Extended,
twisted) Ringel-Hall algebra is now isomorphic as a Hopf algebra to the
zero-and-positive parts of the quantum group of a Borcherd's Lie
algebra \tilde{\mathfrak g} (and everything we learnt about quantum
groups for Kac-Moody Lie algebras holds for these more general Lie
algebras).
Note that the Borcherd's Lie algebra depends on the field (and possibly on the orientation).
However,
the composition algebra (the subalgebra generated by the simples) is
isomorphic to the subalgebra generated by the E_i, and this is the
quantum group associated to the Kac-Moody Lie algebra {\mathfrak g}
corresponding to the original quiver.
In particular, the character of the composition algebra is independent of both the field and the orientation of the quiver.
\theta_i is either simple or else has dimension vector lying in the fundamental region.
(This
is not contained in [SV] or [DX]. We need to show that \theta_i has
connected support. If not, then each module occurring in \theta_i
decomposes as a direct sum according to the connected components of its
support. But then \theta_i cannot be primitive, a contradiction.)
References [SV] ([H2] if I have time).
Lecture 11: Olaf Schnürer
Kac's Theorem via Ringel-Hall algebras.
The Krull-Remak-Schmidt Theorem implies that
ch H := \sum_{\alpha\geq0} (\dim H_\alpha) e^\alpha = \prod_{\alpha>0} (1-e^\alpha)^{-I(kQ,\alpha)}
where I(kQ,\alpha) equals the number of isomorphism classes of indecomposable kQ-modules of dimension vector \alpha.
On
the other hand, we know that H\cong U^+(\tilde{\mathfrak g})_k, the
positive part of the quantum group of a generalised Borcherd's matrix.
Therefore, as in the lecture on quantum groups, we have
ch H = \prod_{\beta>0} (1-e^\beta)^{-\dim \tilde{\mathfrak g}_\beta}.
We
wish to compare these, but note that they are actually with respect to
two different gradings on H. One comes from the dimension vector, the
other from the generators \theta_i.
Therefore we actually obtain that
I(kQ,\alpha) = \sum_{\bar\beta=\alpha} \dim \tilde{\mathfrak g}_\beta
where \bar\theta_i=\underline\dim\,\theta_i.
It
remains to show that this map sends \tilde\Phi^+ onto \Phi^+, the
positive roots for \tilde{\mathfrak g} and {\mathfrak g} respectively.
Have already seen that \theta_i is either simple or in fundamental
region. Follows that the map sends the fundamental region for
\tilde{\mathfrak g} to the fundamental region for {\mathfrak g}, and
that this is surjective. The Weyl groups are equal. Hence the result.
Moreover, each real root for {\mathfrak g} has a unique preimage in \tilde\Phi^+
Thus have proved Kac's Theorem for finite fields:
I(kQ,\alpha)\neq 0 iff \alpha\in\Phi^+ and \alpha real root => I(kQ,\alpha)=1.
References [DX] ([H2] if I have time).
Lectures on Kronecker modules and sheaves over P^1: Gregoire Dupont, Yann Palu
Description of the module category of the Kronecker quiver.
Calculation of some Hall polynomials.
(DMV Lectures by Crawley-Boevey, Ringel's book, Szanto's article, ...)
Ringel-Hall algebra is isomorphic to U^+(\hat{\mathfrak{sl}}_2).
Description of coherent sheaves over P^1.
Calculation of some Hall polynomials.
(Baumann-Kassel.)
Ringel-Hall
algebra is isomorphic to a subalgebra of U(\hat{\mathfrak{sl}}_2). This
subalgebra really comes from the loop construction
\mathfrak{sl}_2[t,t^{-1}].
Comparison
of categories: preinjective/regular modules correspond to sheaves
generated by global sections (i.e. O(n) for n\geq0 and the torsion
sheaves).
Comparison of algebras: isomorphism of subalgebras corresponding to the equivalence of subcategories.
Lecture 16 on the Composition monoid: Stefan Wolf
definition and associativity, specialised Serre relations.
relation to composition algebra (Dynkin, cyclic, Kronecker).
References: [Rei], [HuC], [Mah], [W]
References
[B] M. Barot, Representations of quivers, www.matem.unam.mx/barot/articles/notes_ictp.pdf .
[C] R. Carter, Lie algebras of finite and affine type, Cambridge studies in advanced mathematics 96 (2005).
[CB] W. Crawley-Boevey, Lectures on quiver representations, www.amsta.leeds.ac.uk/~pmtwc/quivlecs.pdf
[DX] A new approach to Kac's theorem on representations of valued quivers, Math. Z. 245 (2003) 183--199.
[H1]
A. Hubery, Symmetric functions and the centre of the Ringel-Hall
algebra of a cyclic quiver, Math. Z. 251, no. 3, (2005) 701--719.
[H2] A. Hubery, Lecture Notes on Ringel-Hall Algebras, wwwmath.upb.de/~hubery/RHAlg.html .
[J] J.C. Jantzen, Lectures on quantum groups, Graduate Studies in Mathematics 6 (AMS, 1996).
[K] V.G. Kac, Infinite dimensional Lie algebras, 3rd edition, Cambridge Univ. Press (1995).
[L] G. Lusztig, Introduction to quantum groups, Progress in Mathematics 110 (Birkh\"auser, 1994).
[M] I.G. Macdonald, Symmetric functions and Hall polynomials, 2nd edition, Oxford Math. Mon. (1995).
[R]
C.M. Ringel, Green's Theorem on Hall algebras, Representation theory of
algebras and related topics (Mexico City, 1994), CMS Conf. Proc. 19
(AMS, 1996) 185--245.
[S] O. Schiffmann, Lectures on Hall Algebras, math.RT/0611617 .
[SV]
B. Sevenhant and M. Van den Bergh, A relation between a conjecture of
Kac and the structure of the Hall algebra, J. Pure Appl. Algebra 160
(2001) 319--332.
Talks will start on Monday
and end on Friday afternoon. A
precise schedule of the talks will appear in due course.
References:
[McD] I.G. Macdonald , Symmetric functions and Hall polynomials, second edition, Oxford Math.Mon., (1995).
[Sch] O. Schiffmann, Lectures on Hall Algebras, math.RT/0611617
[CB] W. Crawley-Boevey, Lectures on Representations of Quivers, http://www.amsta.leeds.ac.uk/~pmtwc/quivlecs.pdf
[R] C. M. Ringel, The Hall algebra approach to quantum groups, XI Latin American School of
Mathematics (Spanish) (Mexico City, 1993), 85–114, Aportaciones Mat. Comun., 15, Soc.
Mat. Mexicana, Mxico, (1995).
[Hu] A. Hubery, Introduction to Ringel-Hall algebras, http://wwwmath.upb.de/%7Ehubery/Lecture.pdf
[Kac] V. Kac, Infinite-dimensional Lie algebras, Cambridge University Press, 1994
[Ca] R. W. Carter
[Hum] J. E. Humpheys, Introduction to Lie Algebras and Representation Theory, Springer, 1972
[KaQ] V. Kac, Infinite root systems, representations of graphs and invariant theory, Invent. Math.
56, no. 1, 57–92, (1980).
[Bor] R. E. Borcherds
[Lu] G. Lusztig, Introduction to quantum groups, Birkhäuser (1992).
[Ja] J] C. Jantzen, Lectures on quantum groups, Graduate Studies in Mathematics, 6. American
Mathematical Society, Providence, RI, (1996).
[SVB] B. Sevenhant, M. Van den Bergh, A relation between a conjecture of Kac and the structure
of the Hall algebra, J. Pure Appl. Algebra 160, 319–332, (2001).
[Gr] [Gre] J.A. Green, Hall algebras, hereditary algebras and quantum groups, Invent. Math. 120,
361-377 (1995).
[RiG] C. Ringel, Green’s theorem on Hall algebras, Representation theory of algebras and related
topics (Mexico City, 1994), 185–245, CMS Conf. Proc., 19, Amer. Math. Soc., Providence,
RI, (1996).
[DX] B. Deng, J. Xiao
[FM] E. Frenkel, E. Mukhin, The Hopf Algebra Rep U_q(gl_\infty), Selecta Math. 8, 537-635 (2002)
[Kap] M. Kapranov, Eisenstein series and quantum affine algebras, Algebraic geometry, 7. J.
Math. Sci. (New York) 84, no. 5, 1311–1360, (1997).
[BK] P. Baumann, C. Kassel, The Hall algebra of the category of coherent sheaves on the projective
line, J. reine angew. Math 533, pp. 207–233, (2001).
[BS] I. Burban, O. Schiffmann, On the Hall algebra of an elliptic curve, I, preprint
math.AG/0505148.
[Rei] M. Reineke, The monoid of families of quiver representations, Proc.
London Math. Soc. (3) 84 (2002) 663–685.
[HuC] A. Hubery, Symmetric functions and the centre of the Ringel-Hall algebra of a cyclic quiver,
Math. Z. 251, no. 3, 705–719, (2005).
[LuC] G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc.
3, no. 2, 447–498, (1990).
Schedule:
Monday, 19 March:
12:30 - 14:00 Lunch
14:00 - 15:00 Classical Hall algebras part 1
15:00 - 15:30 Coffee
15:30 - 16:30 Classical Hall algebras part 2
16:45 - 17:45 Representations of quivers part 1 (Lampe, Warkentin)
18:00 - 19:00 Dinner
Tuesday, 20 March:
08:00 - 09:00 Breakfast
09:15 - 10:15 Representations of quivers part 2 (Lampe, Warkentin)
10:15 - 10:45 Coffee
10:45 - 11:45 Hall algebras of quivers (Pucinskaite)
12:30 - 14:00 Lunch
14:00 - 15:00 Examples and q-Serre relations (Dichev)
15:00 - 15:30 Coffee
15:30 - 16:30 Lie theory (Hoge, Weist)
16:45 - 17:45 Quantum group theory (Aufgebauer, Engel)
18:00 - 19:00 Dinner
Wednesday, 21 March:
08:00 - 09:00 Breakfast
09:15 - 10:15 Comultiplication and Green's theorem part 1 (Scott, Wiedemann)
10:15 - 10:45 Coffee
10:45 - 11:45 Comultiplication and Green's theorem part 2 (Scott, Wiedemann)
12:30 - 14:00 Lunch
14:00 - 18:00 Free afternoon
18:00 - 19:00 Dinner
Thursday, 22 March:
08:00 - 09:00 Breakfast
09:15 - 10:15 Generators and relations (Vatne)
10:15 - 10:45 Coffee
10:45 - 11:45 Kac's theorem (Schnürer)
12:30 - 14:00 Lunch
14:00 - 15:00 Kronecker modules and sheaves on P1 part 1 (Dupont, Palu)
15:00 - 15:30 Coffee
15:30 - 16:30 Kronecker modules and sheaves on P1 part 2 (Dupont, Palu)
16:45 - 17:45
18:00 - 19:00 Dinner
Friday, 23 March:
08:00 - 09:00 Breakfast
09:15 - 10:15 Hall algebras and cluster algebras part 1 (Caldero)
10:15 - 10:45 Coffee
10:45 - 11:45 The composition monoid (Wolf)
12:30 - 14:00 Lunch
14:00 - 15:00 Hall algebras and cluster algebras part 1 (Caldero)
Participation:
The expected number of participants
is about 20-25.
Accomodation and full boarding will
be covered by the two funding Research Training Programs . However,
travel expenses can not be covered.
If you want to participate in the
Spring School, please send an email to A. Hubery (hubery “at”
math.uni-paderborn.de) or M. Reineke (reineke “at”
math.uni-wuppertal.de) indicating which topic you are willing to give
a talk about.
The deadline
for registration was 15 February 2007.
List of participants:
Name
|
Email address
|
|
Britta Aufgebauer
|
britta@physik.uni-wuppertal.de |
|
Philippe Caldero
|
caldero@math.univ-lyon1.fr |
|
Nikolay Dichev
|
dichev@math.upb.de |
|
Gregoire Dupont
|
dupont@math.univ-lyon1.fr |
|
Johannes Engel
|
engel@math.uni-wuppertal.de |
|
Torsten Hoge
|
Blackmuetze@gmx.de |
|
Andrew Hubery
|
hubery@math.upb.de |
|
Philipp Lampe
|
Philipp.Lampe@t-online.de |
|
Yann Palu
|
palu@math.jussieu.fr |
|
Daiva Pucinskaite
|
daiva.pucinskaite1@uni-bielefeld.de |
|
Markus Reineke
|
reineke@math.uni-wuppertal.de |
|
Olaf Schnürer
|
olaf.schnuerer@math.uni-freiburg.de |
|
Jeanne Scott
|
jscott@maths.leeds.ac.uk |
|
Dagfinn Vatne
|
dvatne@math.ntnu.no |
|
Matthias Warkentin |
matthiaswarkentin@yahoo.de |
|
Thorsten Weist
|
weist@math.uni-wuppertal.de |
|
Marcel Wiedemann
|
mat4mw@leeds.ac.uk |
|
Stefan Wolf
|
swolf@math.uni-paderborn.de |
|
Yu Ye
|
yuye@math.uni-paderborn.de |
M. Reineke, 28.03.2007