# THEMATIC SEMESTER SEMINAR

**Saturday, June 7, 2014**

10:30 - 17:00

Room 4336-4384, Pavillon Aisenstadt, Université de Montréal

**“Learning seminar on Kac-Moody algebras” **

*Abstract*: Participants are expected to work on an exercise taken from the book by Kac on "Infinite dimensional Lie algebras.

Anybody interested in participating should send an email to Erhard Neher (neher@uottawa.ca).

**Saturday, May 17, 2014**

10:00 - 17:00

Room 4336-4384, Pavillon Aisenstadt, Université de Montréal

**“Learning seminar on Kac-Moody algebras” **

*Abstract*: Participants are expected to work on an exercise taken from the book by Kac on "Infinite dimensional Lie algebras.

Anybody interested in participating should send an email to Erhard Neher (neher@uottawa.ca).

**Wednesday, May 14, 2014**

16:00 - 17:00

Room 4336, Pavillon Aisenstadt, Université de Montréal

Speaker: Vyacheslav Futorny (Sao Paulo)

**“Representations of Weyl algebras”**

*Abstract*: We will discuss representations of infinite rank Weyl algebras and related representations of Affine Kac-Moody algebras. This is based on various joint results with G.Benkart, V.Bekkert, D.Grantcharov, I.Kashuba and V.Mazorchuk.

**Wednesday, May 14, 2014**

14:30 - 15:30

Room 4336, Pavillon Aisenstadt, Université de Montréal

Speaker: Ivan Dimitrov (Queen's University)

**“Integrable weight modules of $gl(\infty)$”**

*Abstract*: I will present a theorem classifying the irreducible integrable weight modules with finite dimensional weight spaces over the Lie algebra $gl(\infty)$ consisting of finitary infinite matrices. Every such module belongs to one of the following three classes: highest weight modules, infinite symmetric powers of the natural representations, and modules which are not highest weight but whose weights are dominated by a single weight. For the modules in the new third class I will present different realizations and will provide explicit parametrization. I will define all necessary terms and will state the problem and the main result.

**Wednesday, May 7, 2014**

16:00 - 17:00

Room 5340, Pavillon Aisenstadt, Université de Montréal

Speaker: Rajendran Venkatesh (CRM)

**“Fusion product structure of Demazure modules”**

*Abstract*: Let g be a finite--dimensional complex simple Lie algebra. The g-stable Demazure modules of the untwisted affine Lie algebra associated to g naturally become finite-dimensional graded modules for the current algebra g[t] by restriction. In this talk, I will discuss results on the fusion product structure of these g[t]--modules and its connection with several important conjectures. These results generalize previous work by Fourier and Littelmann.

**Wednesday, May 7, 2014**

14:30 - 15:30

Room 5340, Pavillon Aisenstadt, Université de Montréal

Speaker: B. Ravinder (Institute of Mathematical Sciences, Chennai, India)

**“Area maximization on the Gelfand-Tsetlin polytope”**

*Abstract*: The Gelfand-Tsetlin polytope GT(lambda, mu) is a convex polytope of triangular arrays with bounds lambda and weight mu. Recently we have defined a quadratic function A on the GT(lambda, mu), related to the Chari-Loktev basis for the local Weyl module W(lambda) of the current algebra sl_n[t]. We proved that, A attains its maximum at a unique point and further prove that this point is integral. In this talk we discuss about the Gelfand-Tsetlin patterns, the Chari-Loktev bases for the local Weyl modules, their relation and the quadratic function A. This talk is based on joint work with S. Viswanath and K.N. Raghavan.

**Wednesday, April 30, 2014**

16:00 - 17:00

Room 5340, Pavillon Aisenstadt, Université de Montréal

Speaker: Ryosuke Kodera (Kyoto University)

**“Ext^1 for simple modules over Uq(Lsl2)”**

*Abstract*: I give a report on calculation of the first extension groups for finite-dimensional simple modules over the quantum loop algebra Uq(Lsl2). Only limited cases have been done but I determined the simple modules that admit non-trivial extensions with the trivial module.

**Wednesday, April 30, 2014**

14:30 - 15:30

Room 5340, Pavillon Aisenstadt, Université de Montréal

Speaker: Ghislain Fourier (Universität zu Köln)

**“PBW filtration and some applications”**

*Abstract*: We will recall recent result on the PBW filtration on cyclic modules of simple complex Lie algebras and their associated graded modules. Generalizing these results make them applicable to fusion products of cyclic modules of the current algebra. We will further show some applications to conjectures on fusion product, conjectures on Schur positivity, as well as conjectures on Weyl modules of truncated current algebras.

**Wednesday, April 30, 2014**

13:00 - 14:00

Room 5340, Pavillon Aisenstadt, Université de Montréal

Speaker: Seok Jin Kang (Seoul National University)

**“Cyclotomic categorification theorem and 2-representation theory”**

Slide presentation

*Abstract*: In this talk, we will present the basic idea of cyclotomic categorification theorem for Khovanov-Lauda-Rouquier algebras and 2-representation theory. We will also discuss possible further developmemts in this direction.

**Saturday, April 26, 2014**

10:00 - 17:00

Room 4336-4384, Pavillon Aisenstadt, Université de Montréal

**“Learning seminar on Kac-Moody algebras”**

*Abstract*: Participants are expected to work on an exercise taken from the book by Kac on "Infinite dimensional Lie algebras".
Anybody interested in participating should send an email to Erhard Neher (neher@uottawa.ca).

**Wednesday, April 16, 2014**

16:00 - 17:00

Room 5340, Pavillon Aisenstadt, Université de Montréal

Speaker: Daniele Rosso (CRM)

**“Twisted Heisenberg Doubles”**

*Abstract*: Given a dual pair of graded Hopf algebras, we can define an associative algebra called the Heisenberg double, which has a natural representation called the Fock space. This construction gives a generalization of the classical Heisenberg algebra, and includes also the Weyl algebra as an example. We will give a similar construction for the case of pairs of twisted Hopf algebras that satisfy a twisted duality and a compatibility condition. We call the resulting algebra the twisted Heisenberg double, and we show that the quantum Weyl algebra is an example. We’ll mention briefly how these constructions are related to categorification arising from towers of algebras. This is joint work with Alistair Savage.

**Wednesday, April 16, 2014**

14:30 - 15:30

Room 5340, Pavillon Aisenstadt, Université de Montréal

Speaker: Erhard Neher (Ottawa)

**“Fundamental representations of sl_2(A), A associative”**

*Abstract*: We describe the category of fundamental (=natural) representations of the Lie algebra sl_2(A), the derived algebra of the Lie algebra of 2 x 2 matrices over A, where A is any associative algebra. We show that this category is equivalent to the module category of an associative algebra B and we determine B for some examples of present interest (A a quantum torus and A a generalized path algebra) using results from Jordan algebras. The talk is based on work in progress with Nathan Manning and Hadi Salmasian.

**Wednesday, April 9, 2014 **

16:00 - 17:00

Room 4336-4384, Pavillon Aisenstadt, Université de Montréal

Speaker: Laurent Manivel (Grenoble)

«**Quantum cohomology and the Satake isomorphism**»

*Abstract*: The quantum cohomology of homogeneous spaces has been thoroughly investigated starting from the early nineties, both from a combinatorial point of view and with more conceptual perspectives. For certain spaces there are unexpected
relationships with representations of Lie algebras, in particular through a twisted
version of the Satake isomorphism. I will explain how this is related with a
fundamental paper by Kostant from 1959.

**Wednesday, April 9, 2014**

14:30 - 15:30

Room 5340, Pavillon Aisenstadt, Université de Montréal

Speaker: Mathieu Mansuy (Université Paris-Diderot Paris 7)

**“Representations of quantum toroidal algebras”**

*Abstract*: We construct new integrable representations of quantum toroidal algebras (double affinization of quantum groups), called extremal loop weight representations. Their definition is given by Hernandez in 2009, following the one of extremal weight representations of quantum affine algebras by Kashiwara. The aim, like in the works of Kashiwara, is to construct finite-dimensional representations of the quantum toroidal algebras, but at roots of unity in this case.

**Saturday, March 29, 2014**

10:00 - 17:00

Room 4336-4384, Pavillon Aisenstadt, Université de Montréal

**“Learning seminar on Kac-Moody algebras”**

*Abstract*: Participants are expected to work on an exercise taken from the book by Kac on "Infinite dimensional Lie algebras".
Anybody interested in participating should send an email to Erhard Neher (neher@uottawa.ca).

**Wednesday, March 12, 2014**

13:30 - 14:30

Room 5340, Pavillon Aisenstadt, Université de Montréal

Speaker: Philippe Gille (IMAR, Bucharest)

“**Multiloop Lie algebras of exceptional type**”

*Abstract*: This is a report on joint work with A. Pianzola. We shall recall the notion of an anisotropic multiloop Lie algebra and discuss the case of exceptional types G2, F4 and E8 in small nullity.

**Saturday, March 8, 2014**

10:00 - 17:00

Room 4336-4384, Pavillon Aisenstadt, Université de Montréal

“**Learning seminar on Kac-Moody algebras**”

*Abstract*: Participants are expected to work on an exercise taken from the book by Kac on "Infinite dimensional Lie algebras".
Anybody interested in participating should send an email to Erhard Neher (neher@uottawa.ca).

**Thursday, March 6, 2014**

16:45 - 17:45

Room: 6214, Pavillon Aisenstadt, Université de Montréal

Speaker: Yoji Yoshii (Iwate University, Japan)

“**From affine Lie algebras to locally affine Lie algebras**”

*Abstract*: We review some graded structure of finite-dimensional split simple Lie algebras and affine Lie algebras. The main ingredients of such Lie algebras are so-called Cartan subalgebras and root systems. We generalize classical root systems and construct some new graded Lie algebras having such generalized root systems. In particular, we introduce locally affine Lie algebras and explain their properties and classification.