SIMETRIAS & REPRESENTAÇÕES

Past talks

2013

9 de dezembro, 2014, 11:00 sala 221

Local Weyl modules for truncated current algebras: recent results and conjectures

Ghislain Fourier

Abstract

I will introduce the truncated current algebra of a simple complex Lie algebra and explain why their local Weyl modules are important finite-dimensional modules. Little is known about these local Weyl modules; so far there is no character formula or dimension formula. I will provide a conjecture on the structure of these modules as tensor products of simple modules. For this I will have to introduce a partial order on partitions of a given dominant weight and explain the relation to Schur positivity results and conjectures of symmetric functions.
The conjecture on the tensor product structure has been proved in certain "extremal" cases during the last 15 years, and I will explain several approaches on how to attack this conjecture in general.

2 de dezembro, 2014, 11:00 sala 221

An introduction to geometric representation theory

Rollo Jenkins

25 de novembro, 2014, 11:00 sala 221

Finite dimensional representations of rational Cherednik algebras

Stephen Griffeth

Abstract

We will explain part of our joint work with A. Gusenbauer, D. Juteau, and M. Lanini on the classification of finite dimensional representations for the rational Cherednik algebras of complex reflection groups. Though we have not achieved a complete classification, we obtain necessary conditions for finite dimensionality that seem to be quite close to the truth in practice. The key point is the calculation of the eigenvalues of monodromy for certain systems of differential equations analogous to the Knizhnik-Zamolodchikov equations. For the Cherednik algebra of the symmetric group, our ideas can be simplified quite a bit to give a new, nearly elementary, proof of the classification in this case (originally due to Berest-Etingof-Ginzburg).

18 de novembro, 2014, 11:00 sala 221

Degenerescência da sequência espectral de Hodge–de Rham em característica positiva, II

Nuno Filipe de Andrade Cardoso

11 de novembro, 2014, 11:00 sala 221

Degenerescência da sequência espectral de Hodge–de Rham em característica positiva, I

Nuno Filipe de Andrade Cardoso

4 de novembro, 2014, 11:00 sala 221

Interactions between the representation theory of the symmetric group Sn, the ring of symmetric polynomials An, and the current algebra sl2[t]

Matthew Bennett

28 de outubro, 2014, 11:00 sala 221

Deformations of moduli stacks of vector bundles

Severin Barmeier

21 de outubro, 2014, 11:00 sala 221

Mutations of potentials and B-model Laurent phenomenon

John Alexander Cruz Morales

14 de outubro, 2014, 11:00 sala 221

Givental's mirror theorem

Elizabeth Gasparim

7 de outubro, 2014, 11:00 sala 221

Decomposição celular de variedades flag reais

Jordan Lambert

30 de setembro, 2014, 11:00 sala 221

Symplectic Lefschetz fibrations from a Lie-theoretical viewpoint

Brian Callander

23 de setembro, 2014, 11:00 sala 221

Recent developments of representation theory of nonrelativistic conformal algebras

Naruhiko Aizawa

Abstract

Nonrelativistic conformal algebras are a particular class of non-semisimple Lie algebras. A member of the class is a finite or an infinite dimensional Lie algebra. The semisimple part of the finite dimensional algebras is the direct sum of sl2 and sod, while the Virasoro algebra is the semisimple part of the infinite dimensional algebras.
This class of Lie algebras appears in various kinds of problems in theoretical and mathematical physics. For instance, one can find them in connection with fluid dynamics, gravity theory, the AdS/CFT correspondence and vertex operator algebras. This motivates us to study representations of the nonrelativistic conformal algebras.
In the beginning of this talk, I introduce various nonrelativistic conformal algebras. Then I pick up some of physical interest and study them in some more detail. Our first problem is the central extensions of the algebras. The list of possible central extensions is given. Our second problem is the irreducible representations of highest (lowest) weight type.
We start with the Verma module and study its irreducibility. This is done by calculating the Kac determinant and giving an explicit construction of singular vectors. If the Verma module is reducible, then it will be shown that how to obtain the irreducible modules.

16 de setembro, 2014, 11:00 sala 221

Mirror symmetry in the Hitchin system

Emilio Franco Gomez

9 de setembro, 2014, 11:00 sala 221

Symplectic and Koszul duality, II

Rollo Jenkins

2 de setembro, 2014, 11:00 sala 221

Symplectic and Koszul duality, I

Rollo Jenkins

Abstract

Symplectic duality (in the sense of Braden-Licata-Proudfoot-Webster) is a new construction that marries together a diverse range of subjects in Lie theory, non-commutative representation theory, algebraic and symplectic geometry, differential operator theory and theoretical physics. It is conjectured to give an algebraic formulation of mirror dualities in physics.
I'll give an overview of the construction of geometric and algebraic category O's with three examples. Commit now for the one-time price of fifty minutes* and get an introduction to Koszul duality absolutely free!
*price excludes ten minutes of question time.

26 de agosto, 2014, 11:00 sala 221

Lagrangian skeleton and mirror symmetry

Lino Grama

19 de agosto, 2014, 11:00 sala 221

A realization of tilting modules for sl2[t]

Matthew Bennett

12 de agosto, 2014, 11:00 sala 221

Higher symmetries of Laplacian via quantization

Jean-Philippe Michel

Abstract

We first review the seminal results of M. Eastwood on the so-called higher symmetries of the Laplacian [2]. In particular, in dimension n, they form an algebra isomorphic to the quotient of the universal enveloping algebra of o(n+2, C) by the Joseph ideal [3]. We propose a new method to classify those symmetries [4], relying on the conformally equivariant quantization [1]. This allows to compute the Joseph ideal and provide a geometric interpretation for it. Moreover, in signature (p, q), we provide links between: higher symmetries of the Laplacian, the minimal representation of O(p+1, q+1) on the kernel of the Laplacian and the invariant star-product on the minimal coadjoint orbit of O(p+1, q+1).
References.
[1] C. Duval, P.B.A. Lecomte & V. Yu. Ovsienko, "Conformally equivariant quantization: existence and uniqueness", Ann. Inst. Fourier (Grenoble), 49(6): 1999-2029, 1999.
[2] M.G. Eastwood, "Higher symmetries of the Laplacian", Ann. Math., 161(3): 1645-1665, 2005.
[3] A. Joseph, "The minimal orbit in a simple Lie algebra and its associated maximal ideal", Ann. Sci. Ecole Norm. Sup., 9(1): 1-29, 1976.
[4] J.-Ph. Michel, "Higher symmetries of Laplacian via quantization", Ann. Inst. Fourier (to appear).

26 de junho, 2014, 11:00 sala 121

Trendy topics in Lie theory

Adriano Moura

11 de junho, 2014, 11:00 sala 321

Fibrações de Lefschetz em órbitas adjuntas

Elizabeth Gasparim

3 de junho, 2014, 11:00 sala 121

Lie algebras, modular forms and elliptic curves

Reimundo Heiluani

Abstract

Ever since Frobenius it has been noticed that characters of representations are special functions. Every known family of orthogonal polynomials, Bessel functions, trigonometric functions, spherical functions and such appear this way. A new phenomenon started to unveil in the late 70's and 80's when it was discovered that characters of modules of certain infinite dimensional Lie algebras and groups where modular invariant. At the time there were plenty of examples both coming from physics and representation theory, but no good explanation of this phenomenon. It was in the mid 90's that Zhu settled this question putting in a rigorous framework some ideas from string theory: these characters are naturally defined (flat) sections of certain bundles on the moduli space of elliptic curves. This immediately shows the modular invariance since a coarse version of this moduli space is simply H (the upper half plane) divided by SL(2,Z) (the modular group).
In later years many examples have arisen involving super Lie algebras instead of simply Lie algebras. In all these examples some extended version of modularity is found. We show that under certain conditions the characters of modules for these Lie algebras are Jacobi modular forms (that is invariant for the group SL(2,Z) \ltimes Z²) by naturally constructing them as sections of some flat bundles on the moduli space of elliptic supercurves. The reduced part of this moduli space parametrizes an ellptic curve and a line bundle over it, so it is simply the universal elliptic curve (or rather its Jacobian), but some very involved subtleties arise when working on families over supercommutative rings.
This is joint work with Jethro Van Ekeren

29 de maio, 2014, 11:00 sala 121

Mirror symmetry as duality between LG models

Elizabeth Gasparim

22 de maio, 2014, 11:00 sala 121

Classes de Chern de variedades flag

Ailton Ribeiro de Oliveira

15 de maio, 2014, 11:00 sala 121

Grassmannianas de espaços de Hilbert

Jordan Lambert

29 de abril, 2014, 11:00 sala 121

The moduli stack of G-bundles, II

Emilio Franco Gomez

24 de abril, 2014, 11:00 sala 121

The moduli stack of G-bundles, I

Emilio Franco Gomez

Abstract

We will start by recalling the notions of sheaves, G-bundles and families of G-bundles. Introducing the classification problem of G-bundles, we will define the moduli functors on sets and grupoids. This motivates the definition of stacks as sheaves of grupoids on Grothendieck topologies. Later we will see an equivalent notion of stacks as categories fibered on groupoids. Finally (if we still have time) we will describe certain stacks of G-(Higgs-)bundles over elliptic curves and some nice properties of them.

16 de abril, 2014, 10:00 sala 221

Quantum differential operators on the quantum torus

Uma N. Iyer

Abstract

Following the definition of the algebra of quantum differential operators by Lunts and Rosenberg, we present concrete examples of these algebras and their properties.

10 de abril, 2014, 11:00 sala 121

Tame representations of quantum affine algebras of type B

Matheus Brito

8 de abril, 2014, 10:30 sala 121

Representações de álgebras de Hopf copontuais

Bárbara Pogorelsky

3 de abril, 2014, 11:00 sala 121

Tame representations of quantum affine algebras of type A

Matheus Brito

27 de março, 2014, 11:00 sala 121

An introduction to category O for Cherednik algebras and the KZ functor, II

Rollo Jenkins

20 de março, 2014, 11:00 sala 121

An introduction to category O for Cherednik algebras and the KZ functor, I

Rollo Jenkins