April 16, 2013  
12 p.m.  Surge 284 
Julie Bergner
Triangulated categories

April 18, 2013  
12 p.m.  Surge 284 
Julie Bergner
Triangulated categories

April 25, 2013  
12 p.m.  Surge 284 
Milen Yakimov (Louisiana State University)
Quantum cluster algebra structures on quantum nilpotent algebras Abstract.
Cluster Algebras and their quantum counterparts play an important role in
representation theory, combinatorics and topology. In relation to
noncommutative algebra there are several open problems on the existence of
cluster algebra structures on certain families of quantized coordinate
rings. We will describe a result that proves the existence of quantum
cluster algebra structures on a very general, axiomatically defined class
of quantum nilpotent algebras. This has a broad range of applications,
among which are a proof of the BerensteinZelevinsky conjecture for
quantum double Bruhat cells, construction of quantum cluster algebra
structures on
quantum unipotent groups in full generality, and others.

May 14, 2013  
12 p.m.  Surge 284 
Yuri Bazlov (University of Manchester, United Kingdom)
The Kostant Clifford algebra conjecture Abstract.
Let $\mathfrak g$ be a complex simple Lie algebra and $\mathfrak h$
its Cartan subalgebra. The Clifford algebra $C(\mathfrak g)$ of
$\mathfrak g$ admits a HarishChandra map, which turns out to map
primitive $\mathfrak g$invariants in $C(\mathfrak g)$ to $\mathfrak h$.
I will discuss a conjecture of Kostant which says that the image of a
certain alternating invariant of degree $2m+1$ under this map is the zero
weight vector of the simple $(2m+1)$dimensional module of the principal
$\mathfrak{sl}_2$triple in the Langlands dual of $\mathfrak g$.
My original proof of this conjecture was found to be incomplete if
$\mathfrak g$ is not of type $A$.
A complete proof was subsequently given by Joseph and AlekseevMoreau.

May 16, 2013  
12 p.m.  Surge 284 
Mathew Lunde
Oral exam

May 21, 2013  
12 p.m.  Surge 284 
Jonas Hartwig
Category $\mathscr O$

January 29, 2013  
12 p.m.  Surge 284 
Jacob West
Triangulated categories

January 31, 2013  
12 p.m.  Surge 284 
Jacob West
Triangulated categories

February 5, 2013  
12 p.m.  Surge 284 
Philip Hackney
Triangulated categories

February 7, 2013  
12 p.m.  Surge 284 
Wee Liang Gan
Triangulated categories

February 14, 2013  
12 p.m.  Surge 284 
Wee Liang Gan
Triangulated categories

February 19, 2013  
12 p.m.  Surge 284 
Liping Li
Triangulated categories

February 21, 2013  
12 p.m.  Surge 284 
Liping Li
Triangulated categories

February 26, 2013  
12 p.m.  Surge 284 
Liping Li
Triangulated categories

February 28, 2013  
12 p.m.  Surge 284 
Liping Li
Triangulated categories

March 5, 2013  
12 p.m.  Surge 284 
Catharina Stroppel (Universität Bonn, Germany, and University of Chicago)
(Walled) Brauer algebras and parabolic KazdhanLusztig polynomials Abstract.
I this talk I wil briefly recall occurrences of parabolic
KazhdanLusztig polynomials of hermitian symmetric cases and related
them to the representation theory of Brauer and walled Brauer algebras.
In particular we will show that these algebras are Koszul.

March 7, 2013  
12 p.m.  Surge 284 
Catharina Stroppel (Universität Bonn, Germany, and University of Chicago)
Quiver Hecke algebras and $q$Schur algebras Abstract.
In this talk I will introduce the geometry of quiver Hecke algebras and
their generalizations to $q$Schur algebras. As a result we show that
decomposition numbers of cyclotomic Schur algebras are governed by Fock
space.

March 12, 2013  
12 p.m.  Surge 284 
Jonas Hartwig
Category $\mathscr O$

March 14, 2013  
12 p.m.  Surge 284 
Éric Vasserot (Université Paris 7, France)
Cherednik algebras and affine category $\mathscr O$ Abstract.
We will prove that the category $\mathscr O$ of Cherednik algebras of cyclotomic
type is equivalent, as a highest weight category, to the parabolic affine
category $\mathscr O$ of type $A$ at a negative level. This implies several conjectures
concerning the category $\mathscr O$ of Cherednik algebras.

October 4, 2012  
12 p.m.  Surge 284 
Liping Li
Representations of algebras with group actions I: A generalized Koszul theory Abstract.
Group algebras and path algebras of quivers are two important types of finitedimensional algebras. Many homological and combinatorial tools have been developed to study their representations. In this talk I will describe representations of some algebras with local group structures or local group actions. Examples includes category algebras of finite EI categories and skew group algebras. In particular, I will introduce a generalized Koszul theory for locally finite graded algebras with nonsemisimple degree 0 parts. This generalized theory preserves many classical results such as the Koszul duality, and has a close relation to the classical theory.

October 9, 2012  
12 p.m.  Surge 284 
Liping Li
Representations of algebras with group actions II: Representations of finite EI categories Abstract.
A finite EI category is a small category with finitely many morphisms such that every endomorphism is an isomorphism. They includes finite groups, posets, quivers as examples, and are studies in Topology (transformation groups), group representations (transporter categories, orbit categories, fusion systems), homological algebra (support varieties). In this talk I will describe representations of finite EI categories, characterize finite EI categories with hereditary category algebras, and discuss their representation types (if time allows).

12 p.m.  Surge 284 
Jonas Hartwig
Twisted generalized Weyl algebras Abstract.
This is an introductory talk about a class of associative algebras constructed from a base ring
$R$, a set of ring automorphisms of $R$, and a set of distinguished elements of $R$. This class
contains many interesting algebras (skew group algebras with free abelian group,
(quantized) Weyl algebras, (quantized) enveloping algebra of $\mathfrak{sl}_2$,
certain MickelssonZhelobenko step algebras, etc.). I will review some historical
background which led to their definition, some structural results (consistency relations,
Cartan matrices) and their representation theory (simple weight modules).

October 23, 2012  
12 p.m.  Surge 284 
Jonas Hartwig
Weyl subalgebras and Dynkin diagrams Abstract.
In recent joint work with Vera Serganova we found that acyclic Dynkin diagrams with loops parametrize (equivalence classes of) certain embeddings of twisted generalized Weyl algebras (TGWAs) into the n:th Weyl algebra. This construction gives new solutions to the consistency relations for TGWAs. Examples include primitive quotients of enveloping algebras related to completely pointed simple weight modules in types $A$ and $C. We also have some partial results in the super algebra case.

October 25, 2012  
12 p.m.  Surge 284 
Jacob West
An introduction to almost split sequences Abstract.
Almost split sequences are primary objects of study in
AuslanderReiten theory. In this talk, I will give the definition of
an almost sequence and discuss various properties these sequences
enjoy.

November 6, 2012  
12 p.m.  Surge 284 
Liping Li
AuslanderReiten theory: a continuation Abstract.
AuslanderReiten theory is a central topic of algebraic representation theory. Following the talk
given by Jacob West, we continue to explore more aspects of this theory. In this talk I will
introduce irreducible morphisms and AuslanderReiten quivers. They have been proved to be very
useful and have a lot of important applications. Topological and combinatorial structures of
AR quivers, explored by Riedtmann, Gabriel and others, will be described. Some current
developments (ARtriangles, interaction with tilting theory, etc.) will be mentioned as well.

November 8, 2012  
12 p.m.  Surge 284 
Bernhard Keller (Université Paris 7, France)
Quiver mutations and quantum dilogarithm identities Abstract.
Quiver mutation is an elementary operation on quivers which appeared in physics in
Seiberg duality in the 1990s and in mathematics in FominZelevinsky's definition of
cluster algebras in 2002. In this talk, I will show how, by comparing sequences of quiver
mutations, one can construct identities between products of quantum dilogarithm
series. These identities generalize FaddeevKashaevVolkov's classical pentagon
identity and the identities obtained by Reineke. Morally, the new identities
follow from KontsevichSoibelman's theory of DonaldsonThomas invariants.
They can be proved rigorously using the theory linking cluster algebras to quiver
representations.

November 13, 2012  
12 p.m.  Surge 284 
Tomoki Nakanishi (Nagoya University, Japan)
Tropicalization method in cluster algebras Abstract. In cluster algebras, after making several mutations of sends, you may sometimes end up with the initial seed. That is the periodicity phenomenon in cluster algebras. Periodicity is a rare event, but once you have it, you can also get the associated dilogarithm identity, plus its quantum version, for free! There are two basic questions for periodicity: How to find it and how to prove it? The answer to the second question is given by the tropicalization method, which I explain in this talk by several examples. The first question is more difficult, and I do not know the answer. However, we are lucky to have several (infinitely many) conjectured periodicities from the Bethe ansatz method in 90's, even before the birth of cluster algebras, and they are recently proved by the tropicalization method. There is always some root system behind the scene.
The talk is based on the work with R. Inoue, O. Iyama, B. Keller, and A. Kuniba.

November 20, 2012  
12 p.m.  Surge 284 
Matt Highfield
Introduction to PI algebras

November 29, 2012  
12 p.m.  Surge 284 
Robert Marsh (University of Leeds, United Kingdom)
Reflection group presentations arising from cluster algebras Abstract.
We give a presentation of a finite crystallographic reflection
group in terms of an arbitrary seed in the corresponding cluster algebra
of finite type and interpret the presentation in terms of companion bases
in the associated root system. (This is joint work with Michael Barot.)

December 4, 2012  
12 p.m.  Surge 284 
John Dusel
Folding crystals and finding subcones Abstract. We will discuss a realization of the crystal $B(\infty)$ adapted for the action of an admissible diagram automorphism, and our progress towards a combinatorial description of a natural subcrystal for folded Cartan datum. 
April 10, 2012  
12 p.m.  Surge 284 
Matt Lunde
Prime Representations of Quantum Affine Algebras Abstract.
The study of the structure of irreducible representations of a
quantum affine algebra can be reduced to the so called prime
representations, those which cannot be written as a tensor product of two
nontrivial simple representations. In their recent paper, Prime
Representations from a Homological perspective, V. Chari, A. Moura and C.
Young work to understand these prime representations via self extensions.
Namely, they conjecture that an irreducible finite dimensional
representation $V$ is prime if and only if the space of self extensions has
dimension 1. I will be presenting some of the results of this paper that
prove the conjecture for the $\mathfrak{sl}_2$ case, and give partial
evidence in the case for general $\mathfrak g$.

April 17, 2012  
12 p.m.  Surge 284 
Irfan Bagci
Whittaker Categories and Whittaker Modules for Lie Superalgebras Abstract.
B. Kostant introduced a class of modules for finite dimensional complex semisimple Lie algebras.
He called them Whittaker modules because of their connection with the Whittaker equations that
arise in the study of the associated Lie group. Since then, a number of others have further
developed the idea of Whittaker modules for Lie algebras. Recently, in a joint work with
K. Christodoulopoulou and E. Wiesner, we have adapted some of these ideas to the setting of
Lie superalgebras.

April 24, 2012  
12 p.m.  Surge 284 
Matthew Highfield
Twisted Graded Hecke Algebras Abstract.
I will give an overview of the 2007 paper by S.
Witherspoon, "Twisted Graded Hecke Algebras." Given a finite group
$G$ acting on a finite dimensional complex vector space $V$ and a
2cocycle $\alpha$, we may form the twisted crossed product algebra
$TV \#_\alpha G$. The main theorem gives conditions for a PBW basis
for certain quotients of this algebra. A twisted graded Hecke algebra
is one that satisfies these conditions. One can think of these
algebras as certain deformations of $SV \#_\alpha G$. I will describe
several examples, including symplectic reflection algebras and the
case $G \cong (\mathbb{Z}/m \mathbb{Z})^n$.

May 3, 2012  
12 p.m.  Surge 284 
Adriano de Moura (UNICAMP, Brazil)
Extensions of finitedimensional representations of quantum affine algebras and prime representations Abstract.
Given an abelian category, one of the natural questions to be addressed is that of understanding the space of extensions between its simple objects. For the category of finitedimensional
representations of an affine KacMoody algebra, this question has been answered in the last few years. The quantum version of this category is far more complicated and the answer to this
question remains open. We shall discuss some ideas towards the answer with the help of the
concept of $q$characters and show, via examples, that the quantum answer is different from the classical one in an essential way. An interesting feature of the category of
finitedimensional representations of a quantum affine is that it has simple objects which
are not prime, i.e., which are isomorphic to a tensor product of two nontrivial simple objects.
It is then natural to try to classify the prime ones. Although this classification is also
unknown, the amount of known examples of prime modules has been growing.
In the main part of this talk we shall present results from a joint paper with V. Chari
and C. Young relating the study of prime representations to that of the space of extensions
between simple modules. In particular, we show that, if the underlying simple Lie algebra
is $\mathfrak{sl}(2)$, then a simple representation is prime if and only if the space of
its self extensions is onedimensional. It is tempting to conjecture that this is true
in general and we construct a large class of prime representations satisfying this
homological property.

May 8, 2012  
12 p.m.  Surge 284 
Jonathan Kujawa (University of Oklahoma)
Computing complexity Abstract.
Complexity is an established invariant of modules in nonsemisimple settings.
For a given module $M$ it is defined as the rate of growth of the minimal projective
resolution of $M$. Hence, in a sense, it is a measure of how far the module is from being
projective. I'll introduce the complexity and its geometric interpretation in the setting
of finite groups as an example. I'll also give the results of the recent calculation of the
complexity of the simple modules for the complex Lie superalgebra $\mathfrak{gl}(mn)$
(along with an intriguing geometric interpretation). I'll explain all the necessary
background as we go, so all are welcome. Our work is joint with Brian Boe and Dan Nakano.

May 10, 2012  
12 p.m.  Surge 284 
John Dusel
Folding $B(\infty)$ Abstract.
We will discuss a realization of
the crystal $B(\infty)$ adapted for the action of an admissible
diagram automorphism, with the aim of describing a natural subcrystal
for folded Cartan datum.

May 15, 2012  
12 p.m.  Surge 284 
Deniz Kuz (Universität zu Köln, Germany)
Demazure and Weyl modules for twisted current algebras Abstract.
We study finitedimensional representations of twisted current
algebras, especially Demazure and (twisted) Weyl modules. First we identify
these Weyl modules with corresponding affine Demazure modules, then we give an
explicit construction from untwisted Weyl modules which generalize the fusion

May 17, 2012  
12 p.m.  Surge 284 
Jonas Hartwig (Stanford University)
Quantized enveloping algebras, Galois orders and applications Abstract.
I will talk about recent joint work with V. Futorny in which we prove
that the quantized enveloping algebra $U_q(\mathfrak{gl}_n)$ has the structure of
a Galois order, a certain subring of invariants in a skew group
algebra. As an application we describe explicitly the structure of the
division ring of fractions of $U_q(\mathfrak{gl}_n)$, in particular obtaining a new proof
of the quantum GelfandKirillov conjecture for $\mathfrak{gl}_n$. Secondly we prove
that the GelfandTsetlin subalgebra of $U_q(\mathfrak{gl}_n)$ is maximal commutative, and
obtain a parametrization of irreducible GelfandTsetlin modules over
$U_q(\mathfrak{gl}_n)$.

May 1820, 2012  
Workshop "Algebraic and Combinatorial approaches to representation theory"
 
May 22, 2012  
12 p.m.  Surge 284 
Alexandre Bouayad (Université Paris VII, France)
Generalized quantum enveloping algebras and Langlands interpolating quantum groups Abstract.
We will define and study deformations along several parameters of the enveloping algebra of a semisimple finite dimensional Lie algebra, called the Generalized Quantum Enveloping (GQE) algebras.
We will see how GQE algebras can be used to define Langlands Interpolating Quantum (LIQ) groups and solve conjectures motivated by the geometric Langlands program. These conjectures have been suggested by E. Frenkel and D. Hernandez in an original work on different LIQ groups.

May 24, 2012  
12 p.m.  Surge 284 
Sachin Sharma (Institute of Mathematical Sciences, Chennai, India)
The $t$analog of the basic string function for twisted affine KacMoody algebras Abstract.
The Kostant partition function can be used to determine the weight
multiplicities associated to irreducible representations of KacMoody
algebras. Its $t$analog was used by Lusztig to define a $t$analog of
weight multiplicity.
We study Lusztig's $t$weight multiplicities associated to the level one
representation of twisted affine KacMoody
algebras. We will derive a closed form expression for the corresponding
$t$string
function using constant term identities of Macdonald and Cherednik. We
describe
how generalized exponents of certain representations of the underlying
finite dimensional simple
Lie algebra enter the picture.

May 29, 2012  
12:402 p.m.  Surge 284 
Matthew Bennett
Thesis defense: Tilting modules for the current algebra associated to a simple Lie algebra 
May 31, 2012  
12:402 p.m.  Surge 284 
John Dusel
Oral exam: folding $B(\infty)$ 
June 1, 2012  
12:402 p.m.  Surge 284 
Nathan Manning
Thesis defense: global Weyl modules for twisted and untwisted loop algebras 
January 17, 2012  
12 p.m.  Surge 284 
Nathan Manning
An introduction to vertex algebras 
January 24, 2012  
12 p.m.  Surge 284 
Matt Bennett
An introduction to vertex algebras 
January 26, 2012  
12 p.m.  Surge 284 
Matthew Bennet, Nathan Manning
Lattice vertex algebras 
February 2, 2012  
12 p.m.  Surge 284 
Arkady Berenstein (University of Oregon Eugene)
Quantum Hankel algebras Abstract. In my talk (based on a joint work with David Kazhdan) I will introduce a class of quantum Hankel algebras which are flat deformations of polynomial algebras and admit a number of automorphisms and same number of derivations. The simplest example is the quadratic algebra $H_1$ generated by $\{X_n\}$, where $n$ runs over integers, with a single relation $X_1X_0=qX_0X_1$, where $q$ is not a root of unity and the remaining relations coming from an automorphism and a derivation of $H_1$ both sending $X_n$ to $X_{n+1}$. Quite surprisingly, $H_1$ is a flat deformation of polynomials in infinitely many variables and:
I will also define the "$k$dimensional" quantum Hankel algebra $H_k$ whose
generators are labeled by
the $k$dimesnional lattice $\mathbb Z^k$ and whose relations are determined by
some basic ones and by
$k$ automorphisms and $k$ derivations; and will demonstrate that these
algebras share many properties of $H_1$.
Ultimately, I will explain that the flatness of $H_k$ and its
generalizations follows from the
(no less surprising) observation that Hecke algebras "look like" Hopf
algebras, which allows to produce
many new solutions of the quantum YangBaxter equation (QYBE) out of a
given initial one.

February 7, 2012  
12 p.m.  Surge 284 
Mathew Lunde
An introduction to vertex algebras 
February 9, 2012  
12 p.m.  Surge 284 
Katsuyuki Naoi (University of Tokyo, Japan)
Generalized Demazure module and the restricted classical limit of a tensor product of KR modules Abstract.
Among finite dimensional modules of a quantum affine algebra,
there is a distinguished family called KR modules. It is known that, by
taking the restricted classical limit, a KR module becomes isomorphic to
a certain Demazure module.
In this talk, I will generalize this result to a tensor product of KR
modules. In this case its restricted classical limit becomes isomorphic
to a certain generalized Demazure module defined using Joseph functor.
If time permitted, I will introduce some application of this result.

February 21, 2012  
12 p.m.  Surge 284 
Matthew Bennett
Tilting modules for current algebras Abstract.
The Lie algebra $\mathfrak g[t] = \mathfrak g\otimes \mathbb C[t] $ associated to a simple
Lie algebra $\mathfrak g$ is called the current algebra, and has connections to
the representation theory of the loop and affine Lie algebras associated to
$\mathfrak g$. Of particular interest is the graded representation theory of
$\mathfrak g[t]$. In a joint work with V. Chari and N. Manning we proved a BGGtype reciprocity formula for the graded
representations of $\mathfrak{sl}_2[t]$ which suggested the presence of a
highest weight category theory and of tilting modules. In this talk I will
outline a construction of tilting modules for $\mathfrak {sl_2}[t]$ and explain
a sufficient condition for the construction to work in general.

February 23, 2012  
12 p.m.  Surge 284 
Katsuyuki Naoi (University of Tokyo, Japan)
Loewy series of Weyl modules and the Poincare polynomials of quiver varieties Abstract.
A Loewy series of a module of finite length is by definition a
semisimple filtration whose length is smallest, and it is a fundamental
problem to determine the Loewy series of a module.
In this talk, we study this problem for Weyl modules of a current
algebra for ADE type. In this case, we can show the Loewy series is
unique and coincides with the natural filtration given by its grading.
As an application of this result, we can show that the Weyl module is
isomorphic as graded modules to a standard modules, which are defined as
the homology groups of quiver varieties. Hence we can study the Poincare
polynomials of quiver varieties using Weyl modules.
This talk is based on the joint work with Ryosuke Kodera.

September 29, 2011  
12 p.m.  Surge 284 
Jiarui Fei
General presentations of algebras Abstract.
For any finite dimensional basic associative algebra, we study the presentation spaces and their relation to the representation spaces. We prove two propositions about a general presentation, one on its
subrepresentations and the other on its canonical decomposition. As a special case, we consider rigid presentations. We show how to complete a rigid presentation and study the number of nonisomorphic
direct summands and different complements. Based on that, we construct a simplicial complex governing the canonical decompositions of rigid presentations and provide some examples.

October 4, 2011  
12 p.m.  Surge 284 
Charles Young (University of York, UK)
Extended $T$systems Abstract.
I will present some systems of short exact sequences in the categories
of finitedimensional representations of quantum affine algebras of
types $A$ and $B$. These systems contain the $T$system of relations among
KirillovReshetikhin modules, and extend it to include, for example,
all minimal affinizations. I will outline the proofs, which use the
theory of $q$characters, and comment on what can be expected in other
types. This is joint work with E. Mukhin.

October 6, 2011  
12 p.m.  Surge 284 
Charles Young (University of York, UK)
Extended $T$systems (cont.)

October 11, 2011  
12 p.m.  Surge 284 
Adam Katz
Cluster algebras

October 13, 2011  
12 p.m.  Surge 284 
Matthew Highfield
Cluster algebras

October 18, 2011  
12 p.m.  Surge 284 
Wee Liang Gan
Cluster algebras

October 20, 2011  
12 p.m.  Surge 284 
Nathan Manning
Cluster algebras

October 25, 2011  
12 p.m.  Surge 284 
Jacob Greenstein
Cluster algebras

October 27, 2011  
12 p.m.  Surge 284 
Jacob Greenstein
Cluster algebras

November 1, 2011  
12 p.m.  Surge 284 
Jiarui Fei
Cluster algebras: CalderoChapoton formula in the general case

November 3, 2011  
12 p.m.  Surge 284 
Matthew Bennett
Cluster algebras

November 22, 2011  
12 p.m.  Surge 284 
Christian Korff (University of Glasgow, United Kingdom)
Cylindric Macdonald functions and a deformation of the Verlinde algebra Abstract.
We define cylindric generalisations of skew Macdonald
functions when one of their parameters is set to zero. We define these
functions as weighted sums over cylindric skew tableaux, which are
periodic continuations of ordinary skew tableaux, employing a statistical lattice model
and nonintersecting paths. We show that the cylindric
Macdonald functions appear in the coproduct of a commutative Frobenius
algebra, which can be interpreted as a oneparameter deformation of
the $\mathfrak{sl}(n)$ Verlinde algebra, i.e. the structure constants of the
Frobenius algebra are polynomials in a variable t whose constant terms
are the WessZuminoNovikovWitten fusion coefficients. The latter are known to coincide
with dimensions of moduli spaces of generalized thetafunctions and multiplicities of tilting
modules of quantum groups at roots of unity. Alternatively, the
deformed Verlinde algebra can be realised as a commutative subalgebra
in the endomorphisms over a KirillovReshetikhin module of the quantum
affine $\mathfrak{sl}(n)$ algebra. Acting with special elements of this subalgebra
on a highest weight vector, one obtains Lusztig's canonical basis.

April 7, 2011  
12:402 p.m.  Surge 284 
Matthew Bennett
Representations of quivers

April 12, 2011  
12 p.m.  Surge 284 
Samuel Chamberlin
Representations of quivers

April 14, 2011  
12 p.m.  Surge 284 
Matthew Highfield
Hochshild cohomology of infinitesimal symplectic reflection algebras Abstract.
I will give a brief introduction to (infinitesimal)
symplectic reflection algebras. Hochschild cohomology provides
information about the deformation theory of an associative algebra.
Etingof and Ginzburg have computed the
Hochschild cohomology of symplectic reflection algebras. In the hope
of obtaining an analogous result for the infinitesimal case,
I will make a first step of computing the Hochschild cohomology for
the undeformed algebra $SV \rtimes U(\mathfrak{sl}_2)$.

April 26, 2011  
12 p.m.  Surge 284 
Samuel Chamberlin/Eliana Zoque
Representations of quivers

May 5, 2011  
12 p.m.  Surge 284 
Nicolas Guay (University of Alberta, Canada)
Twisted affine quantized enveloping superalgebra of type $Q$ Abstract.
We consider a twisted loop superalgebra built from a Lie superalgebra
of type $Q$. After presenting some of its properties, we will introduce a
quantization of a certain bisuperalgebra structure and we will explain how this
new quantized enveloping algebra is related to affine HeckeClifford algebras.
This is a $q$version of previous work of M. Nazarov about the Yangian attached
to Lie superalgebras of type $Q$.

May 10, 2011  
12 p.m.  Surge 284 
Mathew Lunde
Representations of quivers and preprojective algebras

May 17, 2011  
12 p.m.  Surge 284 
Emilie Wiesner (Ithaca college)
Whittaker Categories and the Virasoro Algebra Abstract.
Complex semisimple Lie algebras, as well as a variety of
other Lie algebras including the Virasoro algebra, possess a
triangular decomposition: $\mathfrak g=\mathfrak n^ \oplus \mathfrak h \oplus
\mathfrak n^+$ where $\mathfrak h$
is a Cartan subalgebra and $\mathfrak n^{\pm}$ are maximal nilpotent
subalgebras. Whittaker modules are defined in terms of this
decomposition and sit naturally inside a larger category of modules
that I refer to as a Whittaker category. I'll discuss some of the
historical development of these ideas, as well as my own work on the
Whittaker category for the Virasoro algebra.

May 2122, 2011  
Workshop on Lie Groups, Lie Algebras and their Representations
 
May 24, 2011  
12 p.m.  Surge 284 
Erhard Neher (University of Ottawa, Canada)
Equivariant map algebras: extensions and blocks

May 26, 2011  
12 p.m.  Surge 284 
Wee Liang Gan
TBA

May 31, 2011  
12 p.m.  Surge 284 
Ghislain Fourier (Universität zu Köln, Germany)
TBA

January 25, 2011  
12:402 p.m.  Surge 284 
Konstantina Christodoulopoulou
Quantized enveloping algebras

January 27, 2011  
12:402 p.m.  Surge 284 
Nathan Manning
Lusztig's braid group action on $\mathbf U_q(\mathfrak g)$

February 1, 2011  
12:402 p.m.  Surge 284 
Jacob Greenstein
PoincaréBirkhoffWitt bases of $\mathbf U_q^$ of finite type

February 8, 2011  
12:402 p.m.  Surge 284 
Samuel Chamberlin
Lusztig's bilinear form on $\mathbf U_q^$

February 10, 2011  
12:402 p.m.  Surge 284 
Eliana Zoque Lopez
The canonical basis of $\mathbf U_q^$ of finite type

March 3, 2011  
12:402 p.m.  Surge 284 
Adam Katz
TBA

March 10, 2011  
1:002 p.m.  Surge 284 
Dmytro Chebotarov (USC)
Vertex algebroids and localization of $\widehat{\mathfrak g}$modules. Abstract.
Vertex algebroids can be regarded as distant relatives of rings of twisted
differential operators (TDO) on smooth varieties. The latter are employed in
the classical BeilinsonBernstein equivalence result that relates $\mathfrak g$modules
to twisted $D$modules on the flag variety of $\mathfrak g$.
I will make a quick introduction to vertex algebroids and show how they can
be used to construct a version of BeilinsonBernstein localization for a
class of modules over affine Lie algebras at the critical level.

September 30, 2010  
12 p.m.  Surge 284 
Matthew Bennett
Homomorphisms between Global Weyl Modules Abstract.
Global Weyl modules, for generalized loop algebras $\mathfrak g\otimes A$,
where $\mathfrak g$ is a simple finite dimensional Lie algebra and $A$ is an associative commutative algebra, have been defined and studied for any dominant integral weight $\lambda$. We show that the space of morphisms between global Weyl modules shares some properties with the space of morphisms between Verma modules.
 
October 12, 2010  
12 p.m.  Surge 284 
Vyjayanthi Chari
An application of global Weyl modules to invariant theory  
October 14, 2010  
12 p.m.  Surge 284 
Apoorva Khare (Yale University)
Koszulity of blocks in category $\mathscr O$ over generalized Weyl algebras Abstract.
Generalized Weyl algebras (GWAs) include wellknown examples such as the Weyl
algebra and classical and quantum ${\mathfrak{sl}}(2)$. At the same time, they contain
"nonNoetherian examples" such as continuous Hecke algebras (defined by
Etingof, Gan, and Ginzburg).
We study blocks of the BGG category $\mathscr O$ over a GWA, with finitely many simple
objects. We compute the Extquiver (with relations) of the endomorphism algebra
of the projective generator. We also show that this algebra is Koszul and
satisfies the Strong KazhdanLusztig condition.
 
October 19, 2010  
12 p.m.  Surge 284 
Eric Friedlander (University of Southern California)
TBA  
October 19, 2010  RICHARD E BLOCK DISTINGUISHED LECTURE IN MATHEMATICS  
4:105 p.m.  Surge 284 
Eric Friedlander (University of Southern California)
Elementary modular representation theory  
October 21, 2010  
12 p.m.  Surge 284 
Samuel Chamberlin
Integral Bases for the Universal Enveloping Algebra of $\mathfrak g\otimes A$ Abstract.
Given a finite dimensional simple Lie algebra $\mathfrak g$ over $\mathbb C$ and
a commutative associative $\mathbb C$algebra with unity $A$, we exhibit a $\mathbb Z$form
for the universal enveloping algebra of $\mathfrak g\otimes A$ and an explicit
$\mathbb Z$basis for this $\mathbb Z$form. We also produce explicit commutation formulas in the universal enveloping algebra of $\mathfrak{sl}_2\otimes A$ that allow us to write certain elements in PoincaréBirkhoffWitt order.
 
October 28, 2010  
12 p.m.  Surge 284 
Wee Liang Gan
Necklace Lie bialgebra  
November 2, 2010  
12 p.m.  Surge 284 
Akaki Tikaradze (University of Toledo)
Modular representations of almost commutative algebras Abstract.
Examples of almost commutative algebra are abundant in
representation theory. In positive characteristic, these algebras tend to be
finite over their centers. In this talk I will discuss KacWeisfeiler type
estimates for dimensions of irreducible modules of an almost commutative
algebra in terms of dimensions of symplectic leaves of the corresponding
Poisson variety. Applications to symplectic reflection algebras will be
discussed.
 
November 4, 2010  
12 p.m.  Surge 284 
Irfan Bagci
Cohomology of Restricted Lie Superalgebras  
November 9, 2010  
12 p.m.  Surge 284 
Eliana Zoque Lopez
Kostka polynomials in Lie theory Abstract.
In this talk I will present combinatorial definitions of Kostka numbers and Kostka polynomials, their connection to the algebra of invariant polynomials and some applications to Lie Theory.
 
November 16, 2010  
12 p.m.  Surge 284 
Christopher Walker
Hopf algebra structures for Hall algebras Abstract.
One problematic feature of Hall algebras is the fact that the standard multiplication and comultiplication maps
do not satisfy the bialgebra compatibility condition in the underlying symmetric monoidal category $\rm{Vect}$.
In the past this problem has been resolved by working with a weaker structure called a "twisted" bialgebra. In this talk
we will present a different solution by first switching to a new underlying category ${\rm Vect}^K$ of vector spaces
graded by a group $K$ called the Grothendieck group. We equip this category with a nontrivial braiding which depends on
the $K$grading. With this braiding, we find that the Hall algebra does satisfy the bialgebra condition exactly for the
standard multiplication and comultiplication in this category, and can also be equipped with an antipode, making it a Hopf
algebra object in ${\rm Vect}^K$.

May 20, 2010  
12 p.m.  Surge 284 
Samuel Chamberlin
A generalization of results of H. Garland Abstract.
Given a finitedimensional, simple Lie algebra $\mathfrak g$ of rank $n$ with a Chevalley basis
$$
\{x_\alpha^\pm,\,h_i:\alpha\in R^+,i\in\{1,\dots,n\}\}
$$
H. Garland in 1978 found formulas in the universal enveloping algebra of the loop algebra of $\mathfrak g$ for
$$
(x_\alpha^+\otimes 1)^{(kr)}(x_\alpha^\otimes t)^{(k)}
$$
for all $k,r\in\mathbb Z_{\ge 0}$ with $r\le k$. We have generalized this result for $r\in\{0,1\}$ modulo $\mathbf U(\mathfrak g\otimes A)(\mathfrak n^+\otimes A)$ where $A$ is any commutative algebra with unity. I will discuss this generalization and some applications to Weyl modules.

May 27, 2010  
12 p.m.  Surge 284 
Jacob Greenstein
On quantum foldings Abstract. A classical result in Lie theory stipulates that a simple finite dimensional Lie algebra of type BCFG can be constructed as the subalgebra of a Lie algebra of type ADE fixed by an admissible diagram automorphism of the latter. This construction is known as folding and extends to KacMoody Lie algebras. Although foldings do not admit direct quantum analogues, it can be shown that there exists an embedding of crystals for the corresponding Langlands dual Lie algebras. The aim of this talk is to introduce algebraic analogues and generalizations of foldings in the quantum setting which yield new flat quantum deformations of nonsemisimple Lie algebras and of Poisson algebras (joint work with A. Berenstein). 
January 26, 2010  
12 p.m.  Surge 284 
Irfan Bagci
On cohomology and support varieties for Lie superalgebras Abstract.
We discuss finite generation of the relative cohomology rings for Lie superalgebras.
We formulate a definition for detecting subalgebras and also discuss realizability of support varieties.
As an application we compute the relative cohomology ring of the Lie superalgebra $\bar S(n)$
relative to the graded zero component $\bar S(n)_0$
and show that this ring is finitely generated. We also compute support varieties of all simple modules in the category of
finite dimensional $\bar S(n)$modules which are completely reducible over
$\bar S(n)_0$.

January 28, 2010  
12 p.m.  Surge 284 
Matt Bennett
The Catalan numbers and representation theory of current algebras Abstract.
We will discuss representation theory of $\mathfrak{sl}_2$, the current algebra, and a conjecture of Chari and Greenstein.

February 2, 2010  
12 p.m.  Surge 284 
Konstantina Christodoulopoulou
On blocks and modules for Whittaker pairs (following P. Batra and V. Mazorchuk) Abstract.
This talk will be based on the paper On Blocks and Modules for Whittaker Pairs by P. Batra and V. Mazorchuk, in which they describe a general framework for the study of Whittaker modules. We will give an overview of their main results.

February 9, 2010  
12:401:40 p.m.  Surge 284 
Eliana Zoque Lopez
Principal nilpotent pairs in a semisimple Lie algebra (following Ginzburg)
Abstract.
This talk is based on a paper in which V. Ginzburg defines certain
pairs of commuting elements in a semisimple Lie algebra. I will present
some results with special attention to the case of $\mathfrak{sl}_n$.

February 16, 2010  
12 p.m.  Surge 284 
Wee Liang Gan
Whittaker vectors and associated varieties Abstract.
I will speak on Matumoto's theorem that
gives a necessary condition for existence of
nonzero Whittaker vectors in terms of the associated variety.

January 21, 2010  
12 p.m.  Surge 284 
Adriano de Moura (UNICAMP, Brazil)
Characters of minimal affinizations of quantum groups Abstract.
In this talk I will present a few results regarding the structure
(characters) of minimal affinizations of quantum groups. The results are
obtained by applying a strategy developed by Chari and the speaker which
consists of comparing the classical limit of the minimal affinizations with
certain graded modules for the underlying current algebra. I will explain
part of this method. The concept of minimal affinizations was introduced by
Chari motivated by the impossibility of defining the concept of evaluation
modules in the quantum setting (in case the underlying simple Lie algebra is
not of type A). An important subclass of minimal affinizations is that of
KirilovReshetikhin modules which appears in the mathematicalphysics
literature. About 4 years ago, Chari and the speaker obtained several
character formulas for KirilovReshetikhin modules using this strategy. The
present talk will focus mostly on a new paper by the speaker which focuses
on extending the method to more general minimal affinizations.

February 25, 2010  
12 p.m.  Surge 284 
Irfan Bagci
An introduction to cohomology and representation theory of modular Lie (super)algebras Abstract.
I will briefly summarize what is known and what is not known about the cohomology and representation theory of modular Lie (super)algebras.

March 2, 2010  
12 p.m.  Surge 284 
Ghislain Fourier (Universität zu Köln, Germany)
Another basis for $\mathfrak{sl}_n$modules and its applications. Abstract.
We give generators and relations for the associated graded module of an
irreducible $\mathfrak{sl}_n$module with respect to the PBW filtration. As an application
we obtain a new basis and pattern for the $\mathfrak{sl}_n$module.

March 9, 2010  
12 p.m.  Surge 284 
Matt Bennett
TBA 
March 11, 2010  
12 p.m.  Surge 284 
Emanuel Stoica (MIT)
Unitary Representations of Rational Cherednik Algebras and Hecke Algebras Abstract.
In this talk I will explain the classification of unitary irreducible
representations in the highest weight category of the rational Cherednik
algebra of the symmetric group and how unitarity is preserved by the KZ
functor, that maps highest weight modules to modules over the corresponding
Hecke algebra.

September 29, 2009  
12:402 p.m.  Surge 284  Irfan Bagci
Cohomology and support varieties I Abstract.
The talk will start with an introduction to cohomology, support varieties for finite groups over a field of positive characteristic. After that we will discuss support variety theories for algebraic structures other than finite groups such as restricted Lie algebras, algebraic groups, quantum groups and Lie superalgebras.
 
October 1, 2009  
12:402 p.m.  Surge 284  Irfan Bagci
Cohomology and support varieties II Abstract.
In this second talk we will discuss cohomology and support varieties for Lie superalgebras over the field of complex numbers. I will focus on examples and present some results joint with Jonathan Kujawa and Daniel Nakano.
 
October 6, 2009  
12 p.m.  Surge 284  Christian Kassel (CNRS, Institut de Recherche
Mathématique Avancée, Strasbourg, France)
Drinfeld twists and finite groups  
October 13, 2009  
12:402 p.m.  Surge 284 
Eliana Zoque Lopez
On the variety of almost commuting nilpotent matrices Abstract.
We study the variety of $n\times n$ matrices with commutator of rank at most one. We describe its irreducible components; two of them correspond to the pairs of commuting matrices, and $n2$ components of smaller dimension corresponding to the pairs of rank one commutator. In our proof we define a map to the zero fiber of the Hilbert scheme of points and study the image and the fibers.
 
October 15, 2009  
12:402 p.m.  Surge 284 
Eliana Zoque Lopez
On the variety of almost commuting nilpotent matrices  
October 20, 2009  
12:402 p.m.  Surge 284 
Jacob Greenstein
Quivers, Hall algebras and quantum groups I  
October 22, 2009  
12:402 p.m.  Surge 284 
Jacob Greenstein
Quivers, Hall algebras and quantum groups II  
October 27, 2009  
12:402 p.m.  Surge 284 
Jacob Greenstein
Quivers, Hall algebras and quantum groups III  
November 5, 2009  
12 p.m.  Surge 284 
Qingtao Chen (University of Southern California)
Quantum invariants of links Abstract.
The colored HOMFLY polynomial is a quantum invariant of oriented links in $S^3$ associated with a collection of irreducible representations of each quantum group $U_q(\mathfrak{sl}_N)$ for each component of the link. We will discuss in detail how to construct these polynomials and their general structure. Then we will discuss the new progress, LabastidaMarinoOoguriVafa conjecture. The LMOV conjecture also gives the application of LichorishMillet type formula for links.
The corresponding theory of colored Kauffman polynomials associated to quantum group $U_q(\mathfrak{so}_{2N+1})$ and orthogonal version of the LMOV conjecture can also be developed in a same fashion by using more complicated algebra structures.
 
November 12, 2009  
12 p.m.  Surge 284 
Fedor Malikov (University of Southern California)
What is a chiral algebra? Abstract.
This talk is intended as an elementary and informal introduction to the
BeilinsonDrinfeld notion of a chiral algebra.
 
November 24, 2009  JOINT WITH ALGEBRAIC GEOMETRY SEMINAR  
12:401:30 p.m.  Surge 284 
Andrei Caldararu (University of Wisconsin, Madison)
The Duflo conjecture and the $\text{Ext}$algebra of branes Abstract.
The Duflo theorem is a statement in Lie theory which allows
us to compute the ring structure of the center of the universal
enveloping algebra of a finitedimensional Lie algebra. A categorical
version of it was used by Maxim Kontsevich to give a spectacular
proof of the socalled "Theorem on complex manifolds," which computes
the multiplicative structure of Hochschild cohomology of a complex
manifold in terms of the algebra of polyvector fields. In Lie theory
there are also more general Duflotype statements (mostly
conjectural), which study the case of a pair (Lie algebra, Lie
subalgebra). I will explain how these translate into conjectures
about the multiplicative structure of the Extalgebra of the
structure sheaf of a complex submanifold of a complex manifold, and
how from this interaction we can hope to gain new insights into both
algebraic geometry and Lie theory. (Based on discussions with Damien
Callaque.)
 
December 1, 2009  
12 p.m.  Surge 284 
Nathan Manning
TBA  
December 3, 2009  
12 p.m.  Surge 284 
David Jordan (MIT)
Quantum $\mathscr D$modules and higher genus braid groups Abstract.
One motivation for studying quantum groups and braided tensor
categories is that they provide a method for constructing representations of
the braid groups of type $A_n$. It is natural to ask what extra structure on a
braided tensor category is required to yield back representations of higher
genus braid groups, for example the socalled double affine braid groups, which
are $\pi_1$ of the configuration space of points on an elliptic curve.
In this talk we explain that in types $A_n$ and $BC_n$, the algebra $\mathscr D$ of quantum
differential operators provides this extra structure; more precisely, for any
(quantum) $\mathscr D$module, we construct representations of elliptic braid groups of
types $A_n$ and $BC_n$. Connections to classical Lie theory are provided via the
theory of double affine Hecke algebras and their degenerations. The $BC_n$
constructions we describe are joint work with Xiaoguang Ma.

January 13, 2009  
1:002:00 p.m.  Surge 284 
Wee Liang Gan
Introduction to linear algebraic groups Abstract.
This will be a series of introductory talks on linear algebraic groups.
The goal is to acquire a working knowledge of the subject rather than a
systematic development of the theory. Prerequisites will be minimal and I
will recall anything from algebraic geometry which we need.

January 15, 2009  
1:002:00 p.m.  Surge 284 
Wee Liang Gan
Introduction to linear algebraic groups (cont.) 
January 20, 2009  
1:002:00 p.m.  Surge 284 
Wee Liang Gan
Introduction to linear algebraic groups (cont.) 
January 22, 2009  
1:002:00 p.m.  Surge 284 
Wee Liang Gan
Introduction to linear algebraic groups (cont.) 
January 27, 2009  
12:401:40 p.m.  Surge 284 
Christof Geiss (Instituto de Matemáticas, UNAM, Mexico)
Preprojective algebras and cluster structures for unipotent cells Abstract.
We discuss total positivity for matrices and use this as an motivation
for the definition of cluster algebras which we will recall.
We also introduce some more recent concepts: $g$vectors and
$F$polynomials.

January 29, 2009  
1:002:00 p.m.  Surge 284 
Christof Geiss (Instituto de Matemáticas, UNAM, Mexico)
Preprojective algebras and cluster structures for unipotent cells (cont.) We introduce preprojective algebras and basic properties of their representation theory. 
February 3, 2009  
1:002:00 p.m.  Surge 284 
Christof Geiss (Instituto de Matemáticas, UNAM, Mexico)
Preprojective algebras and cluster structures for unipotent cells (cont.) Abstract.
We recall (our version) of Lusztig's construction of the corresponding
envelopping algebra $U(\mathfrak n)$ and of Verma and Irreducible modules for
the corresponding simple Lie algebra. This is fundamental for the
construction of our cluster character.

February 5, 2009  
1:002:00 p.m.  Surge 284 
Christof Geiss (Instituto de Matemáticas, UNAM, Mexico)
Preprojective algebras and cluster structures for unipotent cells (cont.) Abstract.
We explain how certain subcategories of the representations of a
preprojective algebra categorify the cluster algebra structure for
the unipotent cells of the corresponding Lie group  where we come
back to total positivity.

February 12, 2009  
12:402:00 p.m.  Surge 284 
Konstantina Christodoulopoulou
The bosonfermion correspondence Abstract.
I will describe some classical results for the bosonfermion
correspondence in the context of affine Lie algebras.

February 17, 2009  
1:002:00 p.m.  Surge 284 
Nicolas Guay (University of Edinburgh, UK)
Double affine quantum algebras Abstract.
I will introduce new families of quantum algebras of double affine type
which can be seen as Lie algebra analogs of certain algebras of Hecke type
which have become of interest in the past ten years, namely Cherednik
algebras, symplectic reflection algebras and deformed preprojective
algebras. Those quantum algebras are deformations of the enveloping
algebra of $\mathfrak{sl}_{n}$ over either $\mathbb C[u,v] \rtimes \Gamma$,
where $\Gamma$ is a finite subgroup of $SL_{2}(\mathbb C)$, or over
$\Pi(Q)$ (the preprojective algebra of a quiver $Q$). They
are related to Yangians.

February 19, 2009  
1:002:00 p.m.  Surge 284 
Wee Liang Gan
Introduction to linear algebraic groups (cont.) 
February 24, 2009  
1:002:00 p.m.  Surge 284 
Wee Liang Gan
Introduction to linear algebraic groups (cont.) 
February 26, 2009  
1:002:00 p.m.  Surge 284 
Wee Liang Gan
Introduction to linear algebraic groups (cont.) 
March 3, 2009  
1:002:00 p.m.  Surge 284 
Wee Liang Gan
Introduction to linear algebraic groups (cont.) 
March 5, 2009  
1:002:00 p.m.  Surge 284 
Wee Liang Gan
Introduction to linear algebraic groups (cont.) 
September 30, 2008  
12:402:00 p.m.  Surge 284 
Wee Liang Gan
On Whittaker vectors and representation theory (following Kostant) Abstract. The talk will be based on Kostant's 1978 paper in Inventiones. 
October 2, 2008  
12:402:00 p.m.  Surge 284 
Jacob Greenstein
An introduction to crystals 
October 7, 2008  
1:002:00 p.m.  Surge 284 
Thomas Bliem (Universität zu Köln, Germany)
Generalized GelfandTsetlin patterns, vector partition functions and weight multiplicities Abstract.
I will give an interpretation of Littelmann's generalized GelfandTsetlin
patterns as saying that computing weight multiplicities for semisimple
complex Lie algebras is equivalent to counting points with integral
coordinates in certain families of polytopes. The notion of a "chopped
and sliced cone" formalizes this kind of families.
Using the BlakleySturmfels theorem on vector partition functions I obtain
properties of functions described by chopped and sliced cones, notably a
version of the DuistermaatHeckman theorem in this context. When applied
to semisimple complex Lie algebras this generalizes ideas of
BilleyGuilleminRassart for $\mathfrak{sl}_{k}(\mathbb C)$.
I will present some general results as well as an explicit calculation of
the complete character table for
$\mathfrak{so}_{5}(\mathbb C)$.

October 9, 2008  
1:002:00 p.m.  Surge 284 
Jacob Greenstein
An introduction to crystals (cont.) 
October 14, 2008  
1:002:00 p.m.  Surge 284 
Konstantina Christodoulopoulou
On modules induced from Whittaker modules I Abstract.
I will review some results by MilicicSoergel on modules induced from
Whittaker modules (in the sense of Kostant) in the setting of complex
semisimple Lie algebras. Then I will describe some extensions of these
results in the context of affine Lie algebras.

October 16, 2008  
1:002:00 p.m.  Surge 284 
Konstantina Christodoulopoulou
On modules induced from Whittaker modules II Abstract.
I will describe the irreducible Whittaker modules for the Lie algebra
formed by adjoining a degree derivation to an infinitedimensional
Heisenberg Lie algebra. I will use these modules to construct a new class
of modules for nontwisted affine Lie algebras and I will describe an
irreducibility criterion for them.

October 21, 2008  
1:002:00 p.m.  Surge 284 
Sebastian Zwicknagl
Equivariant Quantizations of Symmetric Algebras Abstract.
In this talk I will introduce coPoisson module algebras and their
quantizations as a framework for studying quantizations of module algebras.
Then I will give some examples, ranging from quantizations of semidirect
Lie bialgebra structures to quantized symmetric algebras derived from
subalgebras of the adfinite part of quantized enveloping algebra
$U_{q}(\mathfrak g)$.

October 23, 2008  
1:002:00 p.m.  Surge 284 
Apoorva Khare
Infinitesimal Hecke algebras Abstract.
We study a family of infinitedimensional algebras that are similar to
semisimple Lie algebras as well as symplectic reflection algebras.
Infinitesimal Hecke algebras over $\mathfrak{sl}(2)$ have a triangular
decomposition and a nontrivial center, which yields an analogue of Duflo's
Theorem (about primitive ideals), as well as a block decomposition of the
BGG category $\mathscr O$. These algebras also have a quantized version,
with similar representation theory; in particular, category $\mathscr O$ has
a block decomposition, even though the center is trivial. Finally, we
discuss some questions about the higher rank cases. (Joint with
A.Tikaradze, and also with W.L.Gan.)

October 28, 2008  
12:401:20 p.m.  Surge 284 
Nathan Manning
Introduction to Tits Systems: Part I Abstract.
We study a class of structures in groups which provide a powerful tool in
classifying the finite simple groups of Lie type.
In this talk, we shall discuss the definitions and some basic properties of
these systems, provide some examples, and provide further motivation,
including a proof that any group with a Tits system admits a Bruhat
decomposition.

1:202:00 p.m.  Surge 284 
Paul Oeser
Introduction to Tits Systems: Part II Abstract.
We use some results on Coxeter systems to prove properties of subgroups of
$G$ containing $B$ (for the Tits system
$(G,B,N,S)$). We define parabolic
subgroups of $G$ and show some of their basic properties.

October 30, 2008  
12:401:20 p.m.  Surge 284 
Matthew Bennet
Introduction to Tits Systems: Part III Abstract.
Using the results established by the other talks, I will work up to a
theorem which establishes a simplicity condition on certain subgroups of a
group with a Tits system, and exhibit an example of its use.

1:202:00 p.m.  Surge 284 
Tim Ridenour
Finite order automorphisms of simple Lie algebras. Abstract.
In this talk, we will discuss some general facts about the group of automorphisms,
$\text{Aut}(\mathfrak g)$, of a simple Lie algebra $\mathfrak g$ over $\mathbb C$,
including the classification of all finiteorder elements
in $\text{Aut}(\mathfrak g)$. In particular, we will focus on automorphisms of order 2, also known as involutions,
and the associated $\mathbb Z_2$gradings of $\mathfrak g$ that they induce.

November 4, 2008  
1:002:00 p.m.  Surge 284 
Xiaoguang Ma (MIT)
Lietheoretic construction of representations of the degenerate affine and double affine Hecke algebras Abstract.
In this talk, we begin with the definitions of the degenerated affine
Hecke algebra (dAHA) and the degenerated double affine Hecke algebra
(dDAHA). We will describe a Lietheoretic construction of representations
of the dAHA of type $A_{n}$ (given by T. Arakawa and T.
Suzuki), the dDAHA of type $A_{n}$ (given by D. Calaque, B.
Enriquez and P. Etingof) and the dAHA and dDAHA of type
$BC_{n}$ (given by P. Etingof, R. Freund and X.Ma). Then we
will talk about what kinds of dAHA modules we get from above Lietheoretic
constructions.

November 6, 2008  
1:002:00 p.m.  Surge 284 
Ting Xue (MIT)
Nilpotent orbits in characteristic 2 and the Springer correspondence Abstract. Let $G$ be an adjoint algebraic group of type $B$, $C$ or $D$ defined over a field $\mathbb k$ of characteristic 2 and $\mathfrak g$ be the Lie algebra of $G$. Let $\mathfrak g^{*}$ be the dual vector space of $\mathfrak g$. We classify the nilpotent orbits in $\mathfrak g$ over a finite field $\mathbb k$ and construct Springer correspondence for the nilpotent variety in $\mathfrak g$. The correspondence would be a bijective map between the set of isomorphism classes of irreducible representations of the Weyl group of $G$ and the set of all pairs $(c,F)$ where $c$ is a nilpotent $G$orbit in $\mathfrak g$ and $F$ is an irreducible $G$equivariant local system on $c$ (up to isomorphism). We also classify the nilpotent orbits in $\mathfrak g^{*}$ over an algebraically closed or a finite field $\mathbb k$ and construct Springer correspondence for the nilpotent variety in $\mathfrak g^{*}$. 
November 13, 2008  
12:402:00 p.m.  Surge 284 
Apoorva Khare
Quivers, with a view toward Gabriel's theorem Abstract. 
November 18, 2008  
12:402:00 p.m.  Surge 284 
Apoorva Khare
Quivers, with a view toward Gabriel's theorem (cont.) 
November 20, 2008  
12:402:00 p.m.  Surge 284 
Apoorva Khare
Quivers, with a view toward Gabriel's theorem (cont.) 
November 25, 2008  
12:402:00 p.m.  Surge 284 
Jacob Greenstein
Quivers with relations arising from algebras of $\mathfrak g$invariants 
December 2, 2008  
12:402:00 p.m.  Surge 284 
Apoorva Khare
Quivers, with a view toward Gabriel's theorem (cont.) 
December 4, 2008  
12:402:00 p.m.  Surge 284 
Apoorva Khare
Quivers, with a view toward Gabriel's theorem (cont.) 
December 12, 2008  SPECIAL SEMINAR  
1:002:30 p.m.  Surge 284 
Pavel Etingof (MIT)
Orbifold Hecke Algebras Abstract.
To a group $G$ acting discretely on a simply connected complex
manifold $X$, I will attach a Hecke algebra
$\mathcal H_{q}(G,X)$, which is a deformation of
the group algebra of $G$. We will see that if
$H^{2}(X,\mathbb C)=0$ then this
deformation is flat. We will also see that this setting unifies many known
types of Hecke algebras  usual (finite), affine, double affine
(Cherednik), Hecke algebras of complex reflection groups
(BroueMalleRouquier), and many others. In particular, there are orbifold
Hecke algebras which provide quantization of Del Pezzo surfaces and their
Hilbert schemes.

April 3, 2008  
1:002:00 p.m.  Surge 284  Michael Lau (University of Windsor, Canada)
Forms of Conformal Superalgebras Abstract.
Conformal superalgebras describe symmetries of superconformal field theories and come equipped with an infinite family of products. They also arise as singular parts of the vertex operator superalgebras associated with some wellknown Lie structures (e.g. affine, Virasoro, NeveuSchwarz).
In joint work with Arturo Pianzola and Victor Kac, we classify forms of conformal superalgebras using a nonabelian Cechlike cohomology set. As the products in scalar extensions are not given by linear extension of the products in the base ring, the usual descent formalism cannot be applied blindly. As a corollary, we obtain a rigourous proof of the pairwise nonisomorphism of an infinite family of N=4 conformal superalgebras appearing in mathematical physics.

April 8, 2008  
1:002:00 p.m.  Surge 284  Bernard Leclerc (Université de Caen, France)
Introduction to cluster algebras Abstract.
I will give a quick introduction to the theory of
cluster algebras introduced by Fomin and Zelevinsky.
I will illustrate it by examples like coordinate rings
of unipotent groups and flag varieties.

April 10, 2008  
1:002:00 p.m.  Surge 284  Bernard Leclerc (Université de Caen, France)
Monoidal categorifications of cluster algebras Abstract.
I will introduce the notion of monoidal categorification
of a cluster algebra, and will give examples coming
from the representation theory of quantum affine algebras.

April 17, 2008  
1:002:00 p.m.  Surge 284  Bernhard Keller (Université Paris 7, France)
Generalized cluster categories, after C. Amiot Abstract.
Fomin and Zelevinsky invented cluster algebras in 2000. Soon, it
became clear that these new algebras were intimately related to quiver
representations. Cluster categories, introduced in 2004, have provided
a beautiful framework for making this relation precise. However,
cluster categories are only defined for quivers without oriented
cycles. Building on DerksenWeymanZelevinsky's fundamental work on
quivers with potentials Claire Amiot has recently been able to extend
the construction of the cluster category to a large class of quivers
admitting oriented cycles and endowed with a potential, namely the
socalled Jacobifinite quivers with potential. I will report on her
results and their links to previous work, due notably to
GeissLeclercSchroer and BuanIyamaReitenScott.

April 22, 2008  
1:002:00 p.m.  Surge 284  Tim Ridenour
On abelian ideals in root systems of simple Lie algebras Abstract.
It is a well known result due to D. Peterson that the number of abelian ideals in the positive roots of a
simple Lie algebra of rank $n$ is $2^{n}$. In this talk, I will discuss general results for ideals in simple Lie algebras
including generalizations to $k$nilpotent ideals. Furthermore, I will give the details of a simple proof of Peterson's theorem and
give a method for explicitly defining all such ideals.

April 29, 2008  
1:002:00 p.m.  Surge 284  Henning Haahr Andersen (University of Aarhus, Denmark)
Some applications of tilting modules for quantum groups Abstract.
Let $U_{q}$ denote a quantum group associated to a finite dimensional
semisimple complex Lie algebra.The RingelDonkin theory of tilting modules
gives for each dominant weight $\lambda$ a unique indecomposable tilting
module $T(\lambda)$ with highest weight $\lambda$. In the generic case these
modules are just the finite dimensional irreducible modules but when $q$
is a root of unity we get new interesting modules for $U_{q}$. We shall show
that they play a crucial role for instance in the theory of quantum
invariants for 3manifolds, in the theory of (quantum) Schur algebras at
roots of unity.

May 6, 2008  
1:002:00 p.m.  Surge 284 
Sergei Loktev (ITEP, Russia)
Representations of mutlivariable currents and a generalization of the Catalan and Narayana numbers Abstract.
For each partition we construct a natural representation of the
Lie algebra of matrixvalued polynomials. We discuss
universality properties of these repreresntations as well as combinatorics
of their characters. We present explicit answers for
currents in up to three variables.

May 8, 2008  
1:002:00 p.m.  Surge 284 
Dmitriy Boyarchenko (University of Chicago)
Character sheaves on unipotent groups in characteristic $p>0$ (joint work with Vladimir Drinfeld) Abstract.
Let $G$ be a connected unipotent group over an algebraically closed
field $k$ of characteristic $p>0$. We define a collection of irreducible
conjugationequivariant perverse sheaves on $G$, which we call character sheaves.
The set of all character sheaves naturally decomposes as a disjoint union of
finite subsets, called $L$packets of character sheaves.

May 13, 2008  
1:002:00 p.m.  Surge 284 
Eugene Mukhin (Indiana University  Purdue University Indianapolis)
Bethe Ansatz and around Abstract.
The Bethe Ansatz is a method to find eigenvectors of a certain
family of commutative matrices. This method is often more complicated than
the standard methods of linear algebra, moreover, sometimes it fails to
produce the complete set of the eigenvectors. However, the attempts to
understand it lead to a number of interesting connections with
surprisingly many areas of mathematics  and to new results in those
areas. In this talk I will try to give an introduction to the Bethe
Ansatz method.

May 27, 2008  
1:002:00 p.m.  Surge 284 
Charles Conley (University of North Texas)
Extensions of tensor density modules Abstract.
We will survey some results on tensor density modules of
$Vec(\mathbb R)$, $Vec(S^{1})$, and the Virasoro Lie algebra, beginning with the work of KaplanskySantharoubane,
ChariPressley, MartinPiard, and Mathieu on Kac' conjecture concerning irreducible modules of the Virasoro Lie algebra,
and the work of Goncharova and FeiginFuchs on cohomology. The focus will be on more recent results of CohenManinZagier,
Duval, Lecomte, Ovsienko, Roger, and others on modules of differential operators between tensor density modules. We will
conclude with a brief look at similar problems over higher dimensional manifolds which are still open.

May 29, 2008  
1:002:00 p.m.  Surge 284 
Tanusree Pal (HarishChandra Research Institute, India)
Integrable Representations of Graded Multiloop Lie Algebras Abstract.
Let $gA$ be the graded multiloop Lie algebra and $gA(\mu)$ be the graded
twisted multiloop Lie algebra, associated with the simple finite
dimensional Lie algebra $g$ over $\mathbb C$. In this talk, we describe the isomorphism
classes of irreducible integrable $gA$modules with finite dimensional weight
spaces. We also describe the isomorphism classes of irreducible integrable
$gA(\mu)$ modules which are obtained from the above $gA$modules by considering
the restriction action.
The talk is based on a joint work with Punita Batra

June 3, 2008  
1:002:00 p.m.  Surge 284 
Sebastian Zwicknagl
Inhomogeneous Quantum Groups and Lie Bialgebras Abstract.
Inhomogeneous Lie groups and algebras play an important role in physics,
and so do some inhomogeneous quantum groups. In this talk I will introduce
the notions of a inhomogeneous Lie algebra and Lie bialgebra and show how
one can obtain classification results for the inhomogeneous quantum groups
by studying inhomogeneous Lie bialgebras. If time permits I shall explain
how one obtains quantum symmetric algebras from inhomogeneous quantum
groups.

June 5, 2008  
1:002:00 p.m.  Surge 284 
R. J. Dolbin
TBA 
January 15, 2008  
1:002:00 p.m.  Surge 284  Adriano de Moura (UNICAMP, Brazil)
FiniteDimensional Representations of Hyper Loop Algebras over non algebraically closed fields Abstract. Title:
The talk will focus on finitedimensional representations of hyper loop algebras
over arbitrary fields. Hyperalgebras are certain Hopf algebras related to
algebraic groups. When the field is of characteristic zero, a given hyper
loop algebra coincide with the universal enveloping algebra of a certain "classical" loop algebra.
The main results we will discuss are: the classification of the irreducible
representations, construction of the Weyl modules, a study of base change
(forms), and tensor products of irreducible modules. Some of these results
are more interesting when the field is not algebraically closed and are beautifully
related to the study of irreducible representations of polynomial algebras
and field theory.

January 17, 2008  
1:002:00 p.m.  Surge 284  Benjamin Wilson (University of Sydney, Australia/Universidade de São Paulo, Brazil)
Representations of Polynomial Lie Algebras Abstract.
Let $\mathfrak g$ denote a Lie algebra over a field of characteristic zero, and let
$P(\mathfrak g)$ denote the tensor product of g with a ring of truncated polynomials.
The Lie algebra $P(\mathfrak g)$ is called a polynomial Lie algebra, a truncated
current Lie algebra, or a generalized Takiff algebra. In this talk, we
develop a highestweight theory for $P(\mathfrak g)$ when the underlying Lie algebra $\mathfrak g$
possesses a triangular decomposition. We describe a reducibility criterion
for the Verma modules of $P(\mathfrak g)$ for a wide class of Lie algebras $\mathfrak g$, including
the symmetrizable KacMoody Lie algebras, the Heisenberg algebra, and the
Virasoro algebra.

January 24, 2008  
1:002:00 p.m.  Surge 284  Anthony Licata (Stanford)
Representations of affine lie algebras in type $A$ and sheaves on $\mathbb C P^{2}$ Abstract.
In this talk we discuss two parallel geometric constructions of the basic representation
of the affine Lie algebra $\widehat{\mathfrak{gl}(r)}$, one using the Hilbert scheme of points on the
ALE space $\widetilde{\mathbb C^2/\mathbb Z_r}$ (the original construction of Nakajima and Grojnowski) and the
other using the moduli space of rank $r$ framed torsionfree sheaves on $\mathbb CP^2$. These
constructions give a geometric interpretation of levelrank duality in the representation
theory of the affine Lie algebra $\widehat{\mathfrak{gl}(r)}$.

January 31, 2008  
1:002:00 p.m.  Surge 284  David Hernandez (CNRS, Université de Versailles, France)
On the structure of minimal affinizations of representations of quantum groups Abstract. Minimal affinizations of representations of quantum groups
introduced
by Chari are relevant modules for quantum integrable
systems. We present
new results on their structure: we prove that all minimal
affinizations in
types $A$, $B$, $G$ are "special" in the sense of monomials (an
analog property
is also proved for a large class in types $C$, $D$, $F)$. As an
application, the
FrenkelMukhin algorithm works for these modules, and then
we prove
previously predicted explicit $q$character formulas.

February 7, 2008  
1:002:00 p.m.  Surge 284  Benjamin Jones (University of Georgia)
Singular Chern Classes of Schubert Varieties Abstract.
Schubert varieties and their singularities are important in
the study of representation theory and algebraic groups. In this talk
I will describe one aspect of this story which involves singular Chern
classes, characteristic cycles, and (small) resolutions of
singularities. For concreteness, I'll focus on the case of Schubert
varieties in the Grassmannian. In this context there is an open
"positivity conjecture" which is interesting from both the geometric
and combinatorial points of view.

February 12, 2008  
1:002:00 p.m.  Surge 284  Rinat Kedem (University of Illinois at UrbanaChampaign)
The combinatorial KirillovReshetikhin conjecture and fusion products Abstract.
I will give an overview of the various statements which are
called the
KirillovReshetikhin conjecture. These describe the structure of
special modules of the Yangian of a Lie algebra $\mathfrak g$ or the
associated
quantum affine algebra. I'll explain how to prove that all these
conjectures are equivalent (and hence are now proven), and why it
implies the FeiginLoktev conjecture for the fusion product of the
corresponding modules defined by Chari for the algebra of
polynomials
with coefficients in $\mathfrak g$, $\mathfrak g[t]$.

February 14, 2008  
1:002:00 p.m.  Surge 284  Vyacheslav Futorny (IME  USP, Brazil)
GelfandTsetlin modules over Yangians Abstract.
We will discuss the classification problem of irreducible
GelfandTsetlin modules for Yangians and
finite $W$algebras associated with the Lie algebra $\mathfrak{gl}(n)$.

February 21, 2008  
1:002:00 p.m.  Surge 284  Yiqiang Li (Yale)
Geometric Realization of Irreducible Representations of Quantum Groups and their canonical basis Abstract.
Let $U$ be a quantum group. In this talk, I will discuss a geometric
realization of certain simple $U$modules and their canonical bases, via
certain perverse sheaves on open subvarieties of the representation
spaces of a quiver.

February 26, 2008  
1:002:00 p.m.  Surge 284  Vasiliy Dolgushev
A curious $L_\infty$morphism for negative cyclic chains Abstract.
For an associative algebra $A$ negative cyclic
chains $CC^{}(A)$ form a module over the
DG Lie algebra $C(A)$ of Hochschild cochains.
In recent preprint arXiv:0802.1706 A. Cattaneo
and G. Felder consider this DG Lie algebra module
for $A$ being the algebra of functions on a smooth
real manifold equipped with a volume form.
Using an interesting modification of the Poisson sigma
model the authors construct a curious Linfinity
morphism (not a quasiisomorphism!) from the
DG Lie algebra module $CC^{}(A)$ to a DG Lie algebra module
modeled on polyvector fields using the volume form.
The authors also apply this result to a construction
of a specific trace on the deformation quantization
algebra of a unimodular Poisson manifold. Although this
trace can be constructed using the formality quasi
isomorphism
for Hochschild chains, the relation of the
Linfinity morphism of A. Cattaneo and G. Felder to the
formality quasiisomorphism is a mystery.

February 28, 2008  
1:002:00 p.m.  Surge 284  Travis Schedler (University of Chicago)
CalabiYau Frobenius Algebras Abstract. In this talk, we will explore a generalization of symmetric Frobenius algebras (i.e., where the inner product is symmetric) to the case where the pairing is symmetric after some homological shift. We will explain how this property closely resembles the CalabiYau property for infinitedimensional algebras, and will call such algebras "CalabiYau Frobenius algebras". It turns out that the Hochschild (co)homology of such algebras has a very nice structure, and is best described by a $\mathbb Z$graded version of Hochschild (co)homology, which is a Hochschild analogue of Tate cohomology. The Hochschild cohomology is then a Frobenius algebra. In the case of periodic algebras (algebras which have a periodic bimodule resolution), we obtain a BatalinVilkovisky structure on Hochschild cohomology, which is conjecturally selfadjoint with respect to the Frobenius structure. We will explain these results in detail in the case of preprojective
algebras of Dynkin quivers, giving a full computation of their
Hochschild (co)homology over the integers.

March 6, 2008  
1:002:00 p.m.  Surge 284  Reimundo Heluani (UC Berkeley)
Supersymmetry of the Chiral de Rham complex Abstract.
The "chiral de Rham complex" of MalikovShechtmanVaintrob
is a sheaf of vertex superalgebras associated to any manifold $X$. We
will show how, in the smooth context, extra geometric data on $X$
(e.g. having special holonomy) translates into extra symmetries of
the corresponding vertex superalgebras of global sections.

October 2, 2007  
1:002:00 p.m.  Surge 284  Vasiliy Dolgushev
The proof of the multiplicative part of Caldararu's conjecture Abstract. I am going to talk about recent preprint arXiv:0708.2725 "Hochschild cohomology and Atiyah classes" by D. Calaque and M. Van den Bergh. In this paper they proved a multiplicative version of Caldararu's conjecture which describes the Hochschild cohomology of a smooth algebraic variety as a graded ring. I will formulate the result of Calaque and Van den Bergh and explain how they proved it using Kontsevich's formality quasiisomorphism. 
October 9, 2007  
12:402:00 p.m.  Surge 284  Vyjayanthi Chari
Current algebras, highest weight categories and quivers 
October 11, 2007  
12:402:00 p.m.  Surge 284  Vyjayanthi Chari
Current algebras, highest weight categories and quivers 
October 16, 2007  
12:402:00 p.m.  Surge 284  Vyjayanthi Chari
Current algebras, highest weight categories and quivers 
October 18, 2007  
12:402:00 p.m.  Surge 284  Jacob Greenstein
KirillovReshetikhin modules and finite dimensional algebras 
October 23, 2007  
12:402:00 p.m.  Surge 284  Jacob Greenstein
KirillovReshetikhin modules and finite dimensional algebras (cont.) 
October 25, 2007  
1:002:00 p.m.  Surge 284  Arkady Berenstein (University of Oregon, Eugene)
Lie algebras and Lie groups over noncommutative rings Abstract. In my talk (based on the joint paper with Vladimir Retakh) I will introduce a version of Lie algebras and Lie groups over noncommutative rings. For any Lie algebra $\mathfrak g$ sitting inside an associative algebra $A$ and any associative algebra $F$, I will define a Lie algebra $(\mathfrak g, A)(F)$ functorially in $F$ and $A$. In particular, if $F$ is commutative, the Lie algebra $(\mathfrak g, A)(F)$ is simply the loop Lie algebra of $\mathfrak g$ with coefficients in $F$. In the case when $\mathfrak g$ is semisimple or KacMoody and $F$ is noncommutative, I will explicitly compute $(\mathfrak g, A)(F)$ in terms of commutator ideals of $F$ (surprisingly, these ideals have previously emerged as building blocks in M. Kapranov's approach to noncommutative geometry). Furthermore, to each Lie algebra $(\mathfrak g, A)(F)$ one associates a "noncommutative algebraic" group which naturally acts on $(\mathfrak g, A)(F)$ by conjugations. I will conclude my talk with examples of such groups and with the description of "noncommutative root systems" of rank 1. 
October 30, 2007  
12:402:00 p.m.  Surge 284  Apoorva Khare
The BGG Category $\mathscr O$ over tensor products and skew group rings Abstract. I study the Category $\mathscr O$ over the wreath product of $\mathfrak{sl}(2,\mathbb C)$ (15 copies of $U(\mathfrak{sl}(2))$ times $S_{15}$). Complete reducibility and block decomposition hold here, because they hold for $\mathfrak{sl}(2)$. Next, I tensor this algebra 77 times, and ask whether complete reducibility and block decomposition hold in its category $\mathscr O$. Finally, I draw a "commuting cube" involving (sets of) simple modules in various categories $O$, where the "duality functor", "tensor product", and "wreath product induction" form the edges in the $X,Y,Z$directions. (The numbers 2,15,77 above can be generalized to any $n,m,k > 0$  and more.) 
November 6, 2007  
1:002:00 p.m.  Surge 284  Vasiliy Dolgushev
An algebraic index theorem for Poisson manifolds Abstract.
The formality theorem for Hochschild chains of the algebra
of functions on a smooth manifold gives us a version of the
trace density map from the zeroth Hochschild homology of a
deformation quantization algebra to the zeroth Poisson
homology. I will talk about my recent paper
with V. Rubtsov in which we propose a version of the
algebraic index theorem for a Poisson manifold based
on this trace density map.

November 8, 2007  
12:402:00 p.m.  Surge 284  Vasiliy Dolgushev
An algebraic index theorem for Poisson manifolds (cont.) 
November 15, 2007  
1:002:00 p.m.  Surge 284  Sebastian Zwicknagl
Crystal Commutors and the unitarized $R$matrix Abstract. In my talk I will report on commutors for crystals which were
introduced and studied by Kamnitzer and Henriques. We will define the
commutors associated to tensor products of crystal bases of modules over
quantized enveloping algebras. Then, we will lift them to the modules and
show how they are related to Drinfeld's unitarized $R$matrix, as shown
recently by Kamnitzer and Tingley.

November 20, 2007  
12:402:00 p.m.  Surge 284  Sebastian Zwicknagl
Crystal Commutors and the unitarized $R$matrix (cont.) 
November 29, 2007  
1:002:00 p.m.  Surge 284  Farkhod Eshmatov (University of Michigan)
Deformed preprojective algebras and the CalogeroMoser correspondence Abstract. In this talk we discuss the relation between the following objects: rank 1 projective modules (ideals) over the first Weyl algebra $A_{1}(\mathbb C)$, simple modules over deformed preprojective algebras $\Pi_{\lambda}(Q)$, and simple modules over the rational Cherednik algebras $H_{0,c}(S_{n})$ associated to symmetric groups. The isomorphism classes of each type of these objects can be geometrically parametrized by the same space, the CalogeroMoser algebraic varieties. We will give a conceptual explanation of this bijection by constructing a natu ral functor between the corresponding module categories. This is joint work with Y. Berest and O. Chalykh. 
April 3, 2007  
12:402:00 p.m.  Surge 284 
Vyjayanthi Chari
Categorification and Representation theory 
April 5, 2007  
1:002:00 p.m.  Surge 284 
Rajeev Walia (Michigan State University)
Tensor factorization and Spin construction for KacMoody algebras Abstract. We will discuss the "Factorization Phenomenon" which occurs when a representation of a Lie algebra is restricted to a subalgebra, and the result factors into smaller representations of the subalgebra. The original Lie algebra may be any symmetrizable KacMoody algebra (including finitedimensional, semisimple Lie algebras). We will provide an algebraic explanation for such a phenomenon using "Spin construction". We will present a few Factorization results for any embedding of a symmetrizable KacMoody algebra into another, using Spin construction and give some combinatorial consequences of it. We will extend the notion of Spin from finitedimensional to symmetrizable KacMoody algebras which requires a very delicate treatment. We will introduce a category of "$d$finite, Orthogonal Level zero" representations for which, surprisingly, the Spin gives a representation in the BernsteinGelfand Gelfand category $\mathscr O$. We will give the formula for the character of Spin for the above category and refine the factorization results in the case of affine Lie algebras. Finally, we will discuss classification of "Coprimary representations" i.e those representations whose Spin is irreducible. 
April 10, 2007  
12:402:00 p.m.  Surge 284 
Vyjayanthi Chari
Categorification and Representation theory (cont.) 
April 12, 2007  
12:402:00 p.m.  Surge 284 
Vyjayanthi Chari
Categorification and Representation theory (cont.) 
April 24, 2007  
12:402:00 p.m.  Surge 284 
Vyjayanthi Chari
Categorification and Representation theory (cont.) 
April 26, 2007  
12:402:00 p.m.  Surge 284 
Vyjayanthi Chari
Categorification and Representation theory (cont.) 
May 1, 2007  
12:402:00 p.m.  Surge 284 
Vyjayanthi Chari
Categorification and Representation theory (cont.) 
May 3, 2007  
12:402:00 p.m.  Surge 284 
Vyjayanthi Chari
Categorification and Representation theory (cont.) 
May 15, 2007  
12:402:00 p.m.  Surge 284 
Wee Liang Gan
On quantization of Slodowy slices 
May 17, 2007  
12:402:00 p.m.  Surge 284 
Wee Liang Gan
Khovanov homology. 
May 22, 2007  
12:402:00 p.m.  Surge 284 
Apoorva Khare
Categorification of the Khovanov algebra by projectiveinjective modules in the parabolic category $\mathscr O$. Abstract. I will talk about recent work by Stroppel (0608234), which relates two algebras. The first is the endomorphism ring of a "minimal" projectiveinjective progenerator in the principal block of the parabolic BGG Category $\mathscr O$, for $\mathfrak{gl}(2n)$ and the maximal parabolic subalgebra for the partition $(n,n)$. (This will occupy most of the talk.) The second is the Khovanov algebra obtained by considering the $2d$ TQFT associated to the Frobenius algebra of dual numbers. Stroppel establishes an isomorphism of both of these, as graded $\mathbb C$algebras. 
January 11, 2007  
1:002:00 p.m.  Surge 284  Dimitar Grantcharov (CS SanJose)
On the category of modules with bounded weight multiplicities Abstract. Let $\mathfrak g$ be a finite dimensional simple Lie algebra. In this talk we will focus on the category $\mathscr B$ of all bounded weight $\mathfrak g$modules, i.e. those that are direct sum of their weight spaces and have uniformly bounded weight multiplicities. A result of Fernando implies that bounded weight $\mathfrak g$modules exist only for $\mathfrak g= \mathfrak{sl}(n)$ and $\mathfrak g= \mathfrak{sp}(2n)$. In the second case we show that the category $\mathscr B$ has enough projectives if and only if $n$>1 and is wild if and only if $n>2$. The case $\mathfrak g= \mathfrak{sl}(n)$ is much more complicated as the description of each block $\mathscr B^\chi$ of $\mathscr B$ depends on the type of the central character $chi$. 
January 25, 2007  
1:002:00 p.m.  Surge 284  Mark Colarusso (UCSD)
GelfandZeitlin algebras and the polarization of generic adjoint orbits for $\mathfrak{gl}(n)$ and $\mathfrak{so}(n)$. Abstract. Let $\mathfrak g_{n}$ be either the $n\times n$ complex general linear Lie algebra $\mathfrak{gl}(n)$ or the $n\times n$ complex orthogonal Lie algebra $\mathfrak{so}(n)$ and let $G_{n}$ be the corresponding adjoint group. Let $P(\mathfrak g_{n})$ denote the algebra of polynomials on $\mathfrak g_{n}$. The associative commutator on the universal enveloping algebra of $\mathfrak g_{n}$ induces a Poisson structure on $P(\mathfrak g_{n})$. Let $J(\mathfrak g_{n})$ be the commutative Poisson subalgebra of $P(\mathfrak g_{n})$ generated by the invariants $P(\mathfrak g_{m})^{G_{m}}$ for $m=1,\dots n$. $J(\mathfrak g_{n})$ gives rise to a commutative Lie algebra of Hamiltonian vector fields on $\mathfrak g_{n}$; $V=\{ \xi_{f}\,:\, f \in J(\mathfrak g_{n})\}$. Choosing an appropriate set of generators for $J(\mathfrak g_{n})$ gives rise to a subalgebra $V'\subset V$. This subalgebra integrates to an action of a commutative, simply connected complex analytic group isomorphic to $\mathbb C^{d/2}$ on $\mathfrak g_{n}$, where $d$ is the dimension of a generic adjoint orbit in $\mathfrak g_{n}$. This fact should then allow one to polarize open submanifolds of generic adjoint orbits. We will discuss the orbit structure of the action of this group on $\mathfrak g_{n}$. In the case of $\mathfrak g_{n}= \mathfrak{gl}(n)$, we will give a description of the work of KostantWallach in the most generic case in such a form that can be used to establish a formalism for dealing with the less generic orbits studied by the speaker. In the case of $\mathfrak g_{n}=\mathfrak{so}(n)$, the speaker will discuss his work on analyzing the orbit structure of the group on certain sets of semisimple elements. 
February 1, 2007  
1:002:00 p.m.  Surge 284 
Stephen Griffeth (University of Minnesota)
Finite dimensional representations of rational Cherednik algebras Abstract. The rational Cherednik algebras are an interesting family of algebras that can be attached to any complex reflection group. In this talk, I will show how to study finite dimensional modules for the rational Cherednik algebras attached to the infinite family of complex reflection groups $G(r,p,n)$ via eigenspace decompositions. Our approach allows us to construct finite dimensional irreducible modules of dimension $m^{n}$ for each integer $m$ coprime to the "Coxeter" number of $G(r,p,n)$, to build "BGG" resolutions of these modules, and to construct a basis of the coinvariant ring for $G(r,p,n)$ generalizing the GarsiaStanton "descent monomial" basis for the coinvariant ring for the symmetric group $G(1,1,n)$. 
February 8, 2007  
1:002:00 p.m.  Surge 284  Adriano de Moura (UNICAMP, Brazil)
FiniteDimensional Representations of Hyper Loop Algebras Abstract. This talk is based on a joint work with D. Jakelic where we study finitedimensional representations of hyper loop algebras, i.e., the hyperalgebras over a field of positive characteristic associated to nontwisted affine KacMoody algebras. In the talk we will go over the classification of the irreducible modules, a version of Steinberg's Tensor Product Theorem, and the construction of positive characteristic analogues of the Weyl modules as defined by Chari and Pressley in the characteristic zero case. We will also discuss reduction modulo $p$ and a Conjecture regarding reduction modulo $p$ of Weyl modules. 
February 15, 2007  
1:002:00 p.m.  Surge 284  Adriano de Moura (UNICAMP, Brazil)
On Applications Of Geometric Invariant Theory to Representation Theory Abstract. A. King has proposed a method for organizing the representation theory of wild algebras by using the concept of stability which originally arose in the context of Mumfors's geometric invariant theory. I will talk about a joint work with V. Futorny and M. Jardim where we explore some applications of these ideas to certain categories of modules for Lie algebras. 
February 22, 2007  
1:002:00 p.m.  Surge 284  Ghislain Fourier (Universität zu Köln, Germany)
Demazure modules for current algebras Abstract. This talk is based on joint works with P. Littelmann. We study finite dimensional modules for current algebras, starting with Demazure modules for (twisted) affine KacMoody algebras. A few useful properties of these Demazure modules are found and proved. With these properties one can prove that Weyl modules as well as KirillovReshetikhin modules are (in the simplylaced case) in fact isomorphic to Demazure modules as modules for the current algebra. As some applications of this isomorphism one can prove the conjectured dimension formula for Weyl modules, some conjectures about fusion products, that the conjectural KirillovReshetikhin crystals are unique etc 
March 1, 2007  
1:002:00 p.m.  Surge 284  Ghislain Fourier (Universität zu Köln, Germany)
Demazure modules for current algebras Abstract. This talk is based on joint works with P. Littelmann. We study finite dimensional modules for current algebras, starting with Demazure modules for (twisted) affine KacMoody algebras. A few useful properties of these Demazure modules are found and proved. With these properties one can prove that Weyl modules as well as KirillovReshetikhin modules are (in the simplylaced case) in fact isomorphic to Demazure modules as modules for the current algebra. As some applications of this isomorphism one can prove the conjectured dimension formula for Weyl modules, some conjectures about fusion products, that the conjectural KirillovReshetikhin crystals are unique etc 
March 8, 2007  
1:002:00 p.m.  Surge 284  Wee Liang Gan
Symplectic reflection algebras Abstract. I will give an introduction to symplectic reflection algebras. These algebras are closely related to quotient singularities of the form $V/G$, where $V$ is a symplectic vector space, and $G$ is a finite group of automorphisms of $V$. 
March 15, 2007  
1:002:00 p.m.  Surge 284  Apoorva Khare
A deformationtheoretic proof of the PoincaréBirkhoffWitt Theorem for quadratic Koszul algebras Abstract. I will present a theorem by Braverman and Gaitsgory that characterizes what Koszul algebras generated by "quadratic" relations, have a PBWtype theorem. The usual PBW theorem for Lie algebras is an example, as are Weyl and Clifford algebras. 
October 3, 2006  
1:002:00 p.m.  Surge 284  Jacob Greenstein
Quotient categories and Directed Categories Abstract. Quotient categories were introduced to extend to the categorical setting the notion of a quotient module and to provide a formal way of "getting rid of" subquotients in module categories. In this talk we will discuss an application of quotient categories to structural properties of a special class of highest weight categories. 
October 5, 2006  
1:002:00 p.m.  Surge 284  Joel Kamnitzer (UC Berkeley/AIM)
Knot Homology via Derived Category of Coherent Sheaves Abstract. We will give a construction of a knot homology theory using the derived category of coherent sheaves of a certain variety arising in geometric representation theory. We conjecture that our knot homology is related to Khovanov homology (joint work with Sabin Cautis). 
October 10, 2006  
1:002:00 p.m.  Surge 284  Jacob Greenstein
Quotient categories and Directed Categories (cont.) 
October 12, 2006  
1:002:00 p.m.  Surge 284  Apoorva Khare
The BGG category $\mathscr O$ Abstract. The BGG (after BernsteinGelfandGelfand) category $\mathscr O$ is an important category of modules over a complex semisimple Lie algebra, that contains all finitedimensional and all Verma modules. We show how category $\mathscr O$ decomposes into a direct sum of subcategories, and explore some of their properties. We further use category $\mathscr O$ to obtain certain wellknown formulae, such as Weyl character formula and BGG reciprocity formula. 
October 17, 2006  
1:002:00 p.m.  Surge 284  Apoorva Khare
The BGG category $\mathscr O$ (cont.) 
October 19, 2006  
1:002:00 p.m.  Surge 284  Hans Wenzl (UC SanDiego)
Restriction coefficients and Brauer Algebras Abstract. We give a fairly simple proof for formulas for multiplicities for restricting representations of $GL(N)$ to $O(N)$ using Brauer algebras and fusion categories. The result is stated in tems of certain reflection groups. This formalism and results about tilting modules of quantum groups also suggest a description of the nonsemisimple Brauer algebra in terms of certain parabolic KazhdanLusztig polynomials. 
October 24, 2006  
1:002:00 p.m.  Surge 284  Andrew Linshaw (UC SanDiego)
Chiral equivariant cohomology Abstract. The equivariant cohomology ring $H_G(M)$ is an algebraic invariant one can attach to a smooth manifold M equipped with an action of a compact Lie group G. The chiral equivariant cohomology is a "chiralization" of $H_G(M)$, that is, a vertex algebra which contains $H_G(M)$ as the subspace of conformal weight zero. I will give a brief introduction to vertex algebras, and then discuss the construction of the new cohomology and some of the basic results and examples. This a joint work with Bong Lian and Bailin Song 
October 26, 2006  
1:002:00 p.m.  Surge 284  Prasad Senesi
Spectral Characters of FiniteDimensional Representations of Twisted Affine Lie Algebras Abstract. A block decomposition of the category $\mathscr C$ of finitedimensional representations of a twisted affine Lie algebra is examined. The result is an extension of one given by Chari and Moura for the untwisted affine Lie algebras, in which the blocks of $\mathscr C$ are shown to be in bijective correspondence with the spectral characters of the Lie algebra. 
October 31, 2006  
1:002:00 p.m.  Surge 284  Prasad Senesi
Spectral Characters of FiniteDimensional Representations of Twisted Affine Lie Algebras (cont.) Abstract. A block decomposition of the category $\mathscr C$ of finitedimensional representations of a twisted affine Lie algebra is examined. The result is an extension of one given by Chari and Moura for the untwisted affine Lie algebras, in which the blocks of $\mathscr C$ are shown to be in bijective correspondence with the spectral characters of the Lie algebra. 
November 2, 2006  
1:002:00 p.m.  Surge 284  Victor Ostrik
(University of Oregon, Eugene)
Tensor categories attached to cells in finite Weyl groups Abstract. For every two sided cell in a Weyl group Lusztig attached a tensor category (via suitably truncated convolution of perverse sheaves on the corresponding flag variety). Moreover, Lusztig proposed a precise conjecture which describes this category in elementary terms. In this talk we report on recent joint work with R.Bezrukavnikov and M.Finkelberg where this conjecture was proved for almost all two sided cells. On the other hand the conjecture fails in the remaining cases. 
November 7, 2006  
1:002:00 p.m.  Surge 284  Wee Liang Gan
Introduction to Dunkl operators Abstract. Dunkl operators are differentialdifference operators introduced by Charles Dunkl in 1989. Due to the work of Opdam, Heckman and others, they are now a key tool in the theory of multivariable orthogonal polynomials. A major development was Cherednik's discovery of their intimate connection with degenerate affine Hecke algebras. I will give an introduction to Dunkl operators and some of their applications. 
November 9, 2006  
1:002:00 p.m.  Surge 284  Sebastian Zwicknagl (University of Oregon, Eugene/UC Riverside)
Equivariant Poisson structures and quantum symmetric algebras Abstract.
M. Kontsevich showed that one can deform any Poisson algebra whose
spectrum is a manifold or a variety. His results, however, leave the
following natural questions:

November 14, 2006  
1:002:00 p.m.  Surge 284  Kobi Kremnizer (MIT)
Proof of the de ConciniKacProcesi conjecture Abstract. I will introduce the quantum flag variety and quantum $D$modules on it. These are noncommutative algebrogeometric objects. In roots of unity they localize to the Springer resolution and allow for a computation of dimensions of modules over the quantum group. 
December 7, 2006  
1:002:00 p.m.  Surge 284  Alexei Oblomkov (Princeton)
Quantum cohomology of Hilbert scheme of points of ADE resolution and loop algebras Abstract. Joint with D. Maulik. Let $X$ be a resolution of the ADE singularity $\mathbb C^2/\Gamma$. We formulate the conjectural description of the structure of the ring of quantum equivariant cohomology of $Hilb_n(X)$. In particular the generators of the ring are given in terms of the loop algebra of the corresponding type. In the case of $A_n$ singularity the conjecture is a theorem. In my talk I will mostly discuss the case of $A_1$ singularity. All necessary geometric definitions unfamiliar to the audience will be reminded. 