Time and place: Fridays 13 in 2132
mailing list: If you want to be added, let me know, preferably by
email (ptingley@mit.edu)
Plan:
I will introduce deformations of universal enveloping
(Hopf)
algebras, leading to the standard DrinfeldJimbo quantum
groups. I will then discuss various properties of these quantum groups and
their
representation theory, with an emphasis on Kashiwara's crystal bases and
related combinatorics. Later on I'm planning to discuss geometric realizations of crystals and representations. This will mainly involve the quiver varieties of Lusztig
and Nakajima, but perhaps I'll mention other geometric constructions as
well.
I would also like to cover some side topics and applications. This will mainly be done through participant talks. Possible topics include (but are not limited to):
References: This is a brief list of books and papers that might be useful. It will likely be updated.
Quantum groups:
I have not found a concise reference for the quantum groups topics I want to
cover, although there are many good books on the subject. I am using Chari
and Pressley's book "A guide to quantum groups" as my main reference, so perhaps that
is the best choice. We will mostly cover chapters 6,8 and 9, although we
will mention topics from many other chapters as well. I have also been using
Calaque and Etingof's notes Lectures on tensor
categories as a reference for some of the results on deformations of
tensor categories which I feel help motivate the study of quantum groups.
Crystals: There are various sources covering this topic.
Quiver varieties: Somehow this list got to be the longest, although the seminar will be more about quantum groups then quiver varieties.
Calendar of participant talks:
February  
11 

None 
18 

None 
25 
Giorgia Fortuna 
CoPoisson Hopf algebras, deformation theory of coPoisson Hopf algebras and quantum groups 
March 

04 
Martina Balagovic 
Knot invariants and the sl(2) quantum group 
11 
David Jordan

The PeterWeyl and BorelWeil theorems for classical and quantum SL_N 
18 
Jethro van Ekeren 
The 6vertex model 
25 

Spring break (no meeting) 
April 

01 
Bhairav Singh 
Specialization at roots of unity 
08 
Vinoth NandaKumar 
Gelfand' Tsetlin bases and crystals 
15 
Steven Sam  Hall algebras and quantum groups 
22 
Taedong Yun 
The Littelmann path model 
29 
Dinakar Muthiah Chris Dodd 
The geometric Satake correspondence and MV polytopes
Fun with categorifictaion! 
May 

06 
Alejandro Morales 
Cluster algebras 
Consider a Hopf algebra A and consider the monoidal category of
representations of A. It is natural to ask whether or not we can define an
action of the braid group or of the symmetric group on Rep(A).In the special
case of where A is a quantized universal enveloping algebra, the existense
of such action is related to the particular structure of bialgebra that we
have chosen on the underlying Lie algebra. The study of this relationship
leads to the notion of triangular Lie bialgebra/Hopf algebra and to the
notion of Rmatrix. In this talk we will discuss Poisson Lie groups, Lie
bialgebras, YangBaxter equation and the notion of quantization of a
coPoisson Hopf algebra. In particular we will see how starting from a
triangular bialgebra we get a quantiziation of the enveloping Lie algebra
that satisfies the nice property we mentioned before.
March 4, Martina Balagovic: Knot theory and the
sl(2) quantum group.
The JonesConway polynomial is a knot invariant, assigning a Laurent
polynomial in two variables to each oriented link up to isotopy. We will
define it, and show that with two simple axioms (setting its value on the
trivial knot, and requiring it to satisfy certain skein relations, which
tell us how the polynomial changes if we try to "unknot" the knot), such
an invariant exists and is unique. The proof uses quantum enveloping
algebra U_q(sl_2), and the fact that its category of representations is
braided but not symmetric, so there are nontrivial Rmatrices.
Prerequisites: none in knot theory, very little in category theory (there
will be at least one nice picture per defined category in this talk), very
little in quantum groups (finite dimensional representations of Uq(sl2),
R matrices). I will explain all the results we will use.
March 11, David Jordan: The PeterWeyl and
BorelWeil theorems for classical
and quantum SL_N.
The PeterWeyl theorem describes the coordinate algebra O(G) of a simply
connected algebraic group G with Lie algebra as a
bimodule for the corresponding Lie algebra. We will discuss this theorem for classical groups,
and then produce the quantum analog, describing the algebra O_q(G) of
matrix coefficients for the quantized universal envelloping algebra U_q(g) as a
U_q(g)bimodule.
As an application, we present a simple proof (assuming some facts from
geometry) of the BorelWeil theorem, which constructs all the simple
representations of $G$ as spaces of global sections of certain line bundles
on the flag variety $G/B$. We will also explain the quantum analog of the
BorelWeil theorem.
March 18, Jethro van Ekeren :
The 6vertex model
After a brief description of what lattice models and partition functions
are, I will describe the 6vertex model on an NbyM grid and a simple way
to compute its partition function in the limit as M, N go to infinity. The
problem reduces to computing the largest eigenvalue of a certain matrix,
which is anything but simple. By using Bethe's ansatz and Baxter's trick we
will see that this problem can be (more or less) solved. Baxter's trick
relies on a special feature of the 6vertex model: the presence of an
Rmatrix. It turns out that the Rmatrix is exactly the one associated to quantum
affine sl_2, and other affine quantum groups provide, in principle, other
examples of exactly solvable models. Finally I will mention other things that
crop up, like frozen vs. random regions.
April 1, Bhairav Singh: Specialization at roots
of unity.
The defining relations for QUE algebras allow us to easily specialize to
any q not a root of unity. This leads to the natural question, can we make
sense of U_q(g) when q is a root of unity? It turns out there are several
answers, each obtained by constructing an integral form of U_q(g)
over C[q,q^1]. I will focus on the two most studied integral forms, the
"nonrestricted" and "restricted" forms. The first form at a root of unity
has a very large center, which is related to the dual Poisson Lie group to
the original group G, and its representation theory looks quite different
from the classical case.
The representation theory of the
second form uses constructions similar to the "classical highest weight
modules", but has some key differences. It is related to characteristic p
representations of the original Lie algebra.
April 8, Vinoth NandaKumar : Gelfand' Tsetlin bases and crystals.
GelfandTsetlin bases are bases defined for finite dimensional
irreducible representations of classical Lie algebras, and also for the
quantum group in type A. In this talk I will first define and discuss some
properties for these bases. Then I will describe a theorem which describes
that at q>0, if we decompose the nfold tensor product of the standard
representation V into irreducibles in a suitable manner, the natural bases
of the tensor product coincides with the union of the GTbases of the
irreducible components. The precise identification is given by a version of
the RSK algorithm. This shows that GTbases bear a strong relation to
crystals, and are in some sense a precursor for the crystal theory.
April 15, Steven Sam: Hall algebras and quantum groups.
Ringel defined a twisted analogue of the Hall algebra of a
quiver and showed that the upper half of the corresponding quantum group
embeds into this algebra. I'll explain his construction and how one can
get a different description of canonical bases in the finite type case.
Time permitting, I'll say something about getting a Hopf algebra out of
the Hall algebra approach.
April 22, Taedong Yun:
The Littelmann path model.
The Littelmann path model was originally developed by Littelmann in
order to generalize the character formula and the LittlewoodRichardson
rule in type A case, and later turned out to be a combinatorial way to
understand the highest weight crystals for general KacMoody algebras.
In this talk, I will briefly review the type A case and then define the
path model and Kashiwara operators. This will give a crystal structure
on a large set of piecewise linear paths in the weight space. This
crystal contains copies of the highest
weight crystals for any given dominant weight (and in fact each
appears in many different ways). Finally I will explain the connection between Littelmann path model and the Young tableau model
in the type A case.
April 29, Dinakar Muthiah:
The geometric Satake correspondence and MV polytopes.
Mirkovic and Vilonen give a proof of the geometric Satake
correspondence which provides a natural basis in each representation
of a complex reductive group. After briefly reviewing the geometric
Satake correspondence, I will discuss the geometry underlying the
MirkovicVilonen basis. In the course of this discussion, I will
introduce the AndersonKamnitzer theory of MV polytopes and explain a
surprising connection between MV polytopes and Lusztig's canonical
basis.
April 29, Chris Dodd:
Fun with categorification.
We'll discuss the notion of an sl_2 categorification, and explain how it
arises in many diverse situations in mathematics, including category O,
perverse sheaves on grassmannians, and representations of Hecke algebras.
Time permitting, we'll discuss some of the connections with canonical basis
theory, and we'll try to give a glimpse at categorification of some lie
algebras other than sl_2.
May 5, Alejandro Morales: Cluster Algebras.
Cluster algebras were introduced by Fomin and Zelevinsky in 2000 to
understand combinatorially Lusztig's theory of total positivity in algebraic
groups and canonical bases on quantum groups. These goals have still not
been reached but the study of these algebras has blossomed thanks to several
connections with a wide range of fields. One of the main features of these
algebras is that they have a distinguished set of generators called cluster
variables that are grouped into clusters that have an exchange property. In
this talk we introduce cluster algebras via examples related to quiver
representations. We then mention Fomin and Zelevinsky's classification of
cluster algebras of finite type (those with finitely many distinct cluster
variables) which is an instance of the CartanKilling classification.
Similarly, we present Gabriel's theorem which gives a similar classification
for indecomposable quiver representations. Finally, we give an explicit
connection between these two classifications via the CalderoChapoton
formula.
.