SCIENTIFIC PROGRAMS AND ACTIVITIES

October 25, 2014

THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES

2013 - 2014
Quantum Algebra Seminar

Friday, 4:00 p.m.
Fields Institute, Stewart Library

Organizers: David Penneys, Peter Samuelson

Upcoming Seminars

  Seminars TBA
Past Seminars

March 28, 2014
BA B025 4-5pm

Dave Penneys
Quantum doubles of fusion categories

I'll go over the construction of the quantum double (or Drinfel'd center) of a fusion category. We'll then talk about some applications.

Friday, March 14,
Stewart Library, 4pm

Chia-Cheng Liu
An introduction to Koszul duality

Roughly speaking, the Koszul duality is a derived equivalence between (graded) module categories of a Koszul ring and its "Koszul dual". We will
give an introduction to the theory, and then briefly discuss its application in representation theory of Lie algebras. The main reference is the 1996 paper Koszul duality patterns in representation theory by Beilinson, Ginzburg, and Soergel.

 

Mar. 7

Alex Weekes
Shift of Argument Algebras, Moduli Spaces, and the Cactus Group

Shift of argument algebras are a certain family of maximal commutative algebras of U(g), where g is a semi-simple Lie algebra. They yield some interesting examples as limits, such as the Gelfand-Tsetlin subalgebras in type A. The moduli space of these subalgebras is also interesting and has the cactus group of g as its fundamental group. This naturally leads to two actions of the cactus group on crystals coming from representations of g, which we conjecture to be the same. We will describe these objects and the links between them.

 

Feb. 28

Iva Halacheva
Demazure and KR Crystals

After a general overview of crystals in representation theory, we will talk about Demazure and Kirillov-Reshetikhin modules and their crystals. We will then discuss results by Fourier-Schilling-Shimozono and Schilling-Tingley relating the two.

Feb. 14
Room 230

Peter Crooks
An Introduction to Quantum Cohomology

Quantum cohomology is a deformation of the cup product on the integral cohomology ring. I will describe the quantum cohomology ring in the special case that the space in question is a partial flag variety. This will require some consideration of 3-point Gromov-Witten invariants and enumeration problems in geometry. We will conclude with a computation of the quantum cohomology ring of complex projective space.

**Please note this semianr will take place in Room 230

Feb. 7

Jamie Thind
Hall Algebras and Quantum Groups

I'll give an introduction to Hall Algebras. After introducing the basics, I'll discuss how the Hall algebra of the category of representations of an A,D,E quiver can be used to construct (the positive half of) the corresponding quantum group.

Jan. 31

Peter Samuelson
Rational Cherednik algebras and KZ equations

The rational Cherednik algebras H_k had their origins in integrable systems and special functions. We give a very brief historical introduction, and then define Dunkl operators and the algebra H_k(g) associated to the root system of a Lie algebra g. We then discuss category O, which shares many nice properties with the category O for Lie algebras. Finally, we'll define the KZ functor from category O to the category of representations of the braid group of type g. By the end of the talk it may or may not be clear how (or if) the KZ functor is related to the KZ equations discussed by Chia Cheng last semester.


Nov. 22

Chia-Cheng
Knizhnik-Zamolodchikov equations, part II

In this talk I will briefly review the theory of quantum group associated to a Lie algebra g and its braided tensor category. Then I will define the Knizhnik-Zamolodchikov (KZ) equations and explain how to use the monodromy of KZ equations to define a non-trivial tensor structure on the category of finite-dimensional representations of g. The remarkable results of Drinfeld, Kazhdan-Lusztig show that the above two braided tensor categories are equivalent.

Nov. 15

Chia-Cheng
Knizhnik-Zamolodchikov equations

In this talk I will briefly review the theory of quantum group associated to a Lie algebra g and its braided tensor category. Then I will define the Knizhnik-Zamolodchikov (KZ) equations and explain how to use the monodromy of KZ equations to define a non-trivial tensor structure on the category of finite-dimensional representations of g. The remarkable results of Drinfeld, Kazhdan-Lusztig show that the above two braided tensor categories are equivalent.

Nov. 1

Dinakar Muthiah
On the Bernstein Presentation of the Affine Hecke Algebra

I will discuss two presentations of the affine Hecke algebra. The first presentation, which is perhaps more familiar, is the Coxeter (or Kac-Moody) presentation, which arises by viewing a loop group as a Kac-Moody group. The second presentation is the Bernstein presentation, which arises by viewing the loop group over a finite field as a type of p-adic group and studying its principal series representations.

Oct. 25

Zsuzsanna Dancso
Bipartite algebras and a categorification of the flow lattice of graphs

In this talk we will construct an algebra ("bipartite algebra") associated to a bipartite quiver, and discuss its representation theory and an application in which we categorify the flow lattice of graphs. This is joint work in progress with Anthony Licata.

Oct. 18

Dave Penneys
Temperley-Lieb

I will give a basic introduction to the Temperley-Lieb 2-category. We will then look at its idempotent completion (Karoubi envelope), and we will discuss when this 2-category is unitary. Time permitting, we will discuss how some quotient of Temperley-Lieb appears in every rigid tensor category.

 

Oct. 11

Iva Halacheva
Quasi-Hopf algebras

A quasi-Hopf algebra is a generalization of a Hopf algebra with the coassociativity condition weakened. We will introduce some of the theory behind them, including Drinfeld's motivation for first defining them, and the twisted quantum double as an example of particular interest.

Oct. 4

Stephen Morgan
W-algebras and their applications in representation theory

W-algebras are algebraic structures defined from semi-simple Lie algebras along with a specified nilpotent element. Though originally defined by and of interest to physicists, they also have applications in representation theory. In particular, the representation theory of W-algebras is closely related to the block decomposition of category O. We'll discuss W-algebras and how they arise through quantisation.

Sept. 27

 

Alex Weekes
Yangian and their applications

We will look at Yangians, which are quantizations Y(g) of the universal enveloping algebra of polynomials g[t], where g is a Lie algebra. We will define these, after reviewing some background, and look at (some of) their (many) applications.

Sept. 20

Peter Samuelson
Double affine Hecke algebras and Macdonald polynomials

We'll discuss Cherednik's double affine Hecke algebra H(g), which is a 2-parameter quantum algebra associated to a (nice) Lie algebra g. We'll give some basic properties and some indications of how Cherednik used this algebra to prove Macdonald's conjectures. (These involve several properties of Macdonald polynomials, which are a family of symmetric orthogonal polynomials associated to g). To keep things concrete, most of the talk will focus on the case g=sl_2.

 

 

 

 

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