Quantum groups, combinatorics and geometry seminar

Spring 2011

Organizers: Peter Tingley and Steven Sam

Time and place: Fridays 1-3 in 2-132

mailing list: If you want to be added, let me know, preferably by email (ptingley@mit.edu)

Participant talks and abstracts

Plan: I will introduce deformations of universal enveloping (Hopf) algebras, leading to the standard Drinfeld-Jimbo 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.

  • Kashiwara's survey paper On crystal bases. This covers essentially all the crystal bases material we need, and also has a brief explanation of quantized enveloping algebras.
  • Kashiwara's earlier paper On crystal bases of the q-analogue of universal enveloping algebras. This contains Kashiwara's "grand loop argument" proving that the theory of crytsal bases actually works.
  • Chari and Pressley, Chapter 14.1 has some material, but does not contain everything we need.
  • Hong and Kang's book "An introduction to quantum groups and crystal bases" is a nice read. If nothing else, it is worth reading the introduction.

Quiver varieties: Somehow this list got to be the longest, although the seminar will be more about quantum groups then quiver varieties.

  • Alistair Savage Finite dimensional algebras and quivers. This short expository paper contains general background on the relationship between quivers and representation theory. Section 6 has a brief explanation (without proofs) of a realization of representations of g using quiver varieties which is of interest to us.
  • Alistair Savage Geometric and combinatorial realizations of crystal graphs. This contains a nice background section, and the main result of the paper (relating the geometry and combinatorics) will also be important for us.
  • Kashiwara and Saito Geometric construction of crystal bases. This is the original construction of the crystal structure on components of quiver varieties.
  • Nakajima Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras. This is Nakajima's paper introducing what are now called Nakajima quiver varieties. It is still one of the best references for a proof of the geometric construction of g-modules (see section 10, although of course it may be incomprehensible if you haven't looked at the earlier sections).
  • Lusztig Quivers, perverse sheaves, and quantized enveloping algebras. This is the paper where Lusztig introduces the nilpotent quiver variety (see section 12), which is crucial. Many of the technicalities of this paper are beyond the scope of this seminar, but the canonical bases developed here and a few earlier papers of Lusztig are one of the great successes of the geometric approach, so deserve to be mentioned.

Calendar of participant talks:



Giorgia Fortuna
Co-Poisson Hopf algebras, deformation theory of co-Poisson Hopf algebras and quantum groups
Martina Balagovic
Knot invariants and the sl(2) quantum group
David Jordan
The Peter-Weyl and Borel-Weil theorems for classical and quantum SL_N
Jethro van Ekeren
The 6-vertex model

Spring break (no meeting)
Bhairav Singh
Specialization at roots of unity
Vinoth NandaKumar
Gelfand' Tsetlin bases and crystals
Steven Sam Hall algebras and quantum groups
Taedong Yun
The Littelmann path model
Dinakar Muthiah
Chris Dodd
The geometric Satake correspondence and MV polytopes
Fun with categorifictaion!
Alejandro Morales
Cluster algebras

Feb 25, Giorgia Fortuna: Co-Poisson Hopf algebras, deformation theory of co-Poisson Hopf algebras and quantum groups.

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 R-matrix. In this talk we will discuss Poisson Lie groups, Lie bialgebras, Yang-Baxter equation and the notion of quantization of a co-Poisson 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 Jones-Conway 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 R-matrices.

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 Peter-Weyl and Borel-Weil theorems for classical and quantum SL_N.

The Peter-Weyl 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 Borel-Weil 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 Borel-Weil theorem.

March 18, Jethro van Ekeren : The 6-vertex model

After a brief description of what lattice models and partition functions are, I will describe the 6-vertex model on an N-by-M 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 6-vertex model: the presence of an R-matrix. It turns out that the R-matrix 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 "non-restricted" 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.

Gelfand-Tsetlin 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 n-fold 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 GT-bases of the irreducible components. The precise identification is given by a version of the RSK algorithm. This shows that GT-bases 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 Littlewood-Richardson rule in type A case, and later turned out to be a combinatorial way to understand the highest weight crystals for general Kac-Moody 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 Mirkovic-Vilonen basis. In the course of this discussion, I will introduce the Anderson-Kamnitzer 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 Cartan-Killing 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 Caldero-Chapoton formula.