Robert Griess (University of Michigan). Title: Thinking about the sporadic simple
groups. (Plenary talk of TMS Annual Meeting to be held in the Activity Center) ...
Workshop on Algebraic Aspects of Lie Theory December 20-23, 2007
Titles and Abstracts
Institute of Mathematics, Academia Sinica Taipei, Taiwan
Jon Brundan (University of Oregon) Title: Cellularity of Khovanov's algebra and related algebras Abstract: I will discuss Khovanov's diagram algebras and their quasi-hereditary covers. As can be seen directly from their diagrammatic definitions, the former are cellular, symmetric algebras, the latter are cellular, quasi- hereditary, Koszul algebras, and the two sides are related by a Schur-Weyl duality. I will try to explain along the way some of the known connections between these algebras and Grassmannians, parabolic category O, degenerate cyclotomic Hecke algebras of level 2, .... This is joint work with Catharina Stroppel.
Vyjayanthi Chari (University of California, Riverside) Title: Kirillov Reshetikhin modules and finite dimensional algebras Abstract: We discuss a relationship between the representation theory of current and finite-dimensional algebras. This allows us to prove that the Kirillov-Reshetikhin modules for classical Lie algebras are projective objects in a suitable category.
Yun Gao (University of York) Title: Vertex operators and a quantum Tits-Kantor-Koecher algebra Abstract: We will discuss a Tits-Kantor-Koecher algebra arising from the extended affine Lie algebra of type $A_1$. Then we propose a quantum anologue of the Tits-Kantor-Koecher algebra by looking at the vertex operator construction. This is a joint work with Naihuan Jing.
Robert Griess (University of Michigan) Title: Thinking about the sporadic simple groups (Plenary talk of TMS Annual Meeting to be held in the Activity Center)
Seok-Jin Kang (Seoul National University) Title: Perfect Crystals and Young Walls Abstract: We will discuss the connection between the theory of perfect crystals and combinatorics of Young walls. The Young walls arise as combinatorial model for a certain family of perfect crystals. Some possible generalization of Young walls will be given.
Alexander Kleshchev (University of Oregon) Title: Representations of W-algebras Abstract: In this talk we describe highest weight theory for finite W-algebras developed recently jointly with Brundan and Goodwin. We then give more details in type A case. We will also discuss some applications, for example to primitive ideals in universal enveloping algebras.
Ching Hung Lam (National Cheng Kung University) Title: On McKay's E6, E7 and E8 observations Abstract: I will discuss McKay's E6, E7 and E8 observations on the Monster, Babymonster and Fisher group $Fi_{24}$ using the theory of vertex operator algebras.
George Lusztig (MIT) Title: Canonical bases and the construction of Chevalley groups over the integers Abstract: In the 1950's Chevalley showed that any semisimple adjoint group G over the complex numbers has an analogue over any field. This has led him to some new finite simple groups (for example $E_8$ over a finite field). Later Chevalley gave a construction of a Z-form of G and more generally a form of G over any commutative ring. He also gave an analogous construction in the case where G is not assumed to be adjoint. Kostant proposed a more elegant definition of Chevalley's group over Z. But several statements of Kostant have remained unproved. In this talk I will describe how to put Kostant's proposal on a firm foundation using the theory of canonical bases in modified enveloping algebras.
Jun Morita (University of Tsukuba) Title: Group presentations, coverings and Schur multipliers in Lie theory Abstract: We will mainly discuss group presentations, coverings, and Schur multipliers for Chevalley groups, Kac-Moody groups and EALA groups. The talk consists of three parts -- introductory part, reviewing part and development part.
Shu-Yen Pan (National Tsing Hua University) Title: Langlands Functoriality and Local Theta Correspondence Abstract: The preservation principle of local theta correspondences of even orthogonal groups and symplectic groups predicts the existence of a sequence of irreducible supercuspidal representations of these groups. Adams/Harris-Kudla-Sweet have a conjecture about the Langlands parameters for the sequence of supercuspidal representations. In this talk, we want to discuss their conjecture for the case of supercuspidal representations with unipotent reductions.
Nicolai Reshetikhin (University of California, Berkeley) Title: Quantum groups at roots of unity and invariants of links. Abstract: One of the important applications of quantized universal enveloping algebras and their representation theory is the construction of invariants of knots and other low-dimensional manifolds. In this talk I will remind some facts about representations of quantized universal enveloping algebras at roots of unity, and will demonstrate how they can be used to construct invariants of links with flat connections in the complement.
Catharina Stroppel (University of Glasgow) Title: The combinatorics of 2-block Springer fibres Abstract: The cohomology rings of Springer fibres was used by Springer to construct in a geometric way irreducible and induced representations of the symmetric group. On the other hand Springer fibres occur in connection with categorification of link invariants. In this talk we first give some overview about such connections and then describe the geometry for 2-block Springer fibres explicitly. The combinatorics of the torus action (attracting sets, fixed points etc.) on the Springer fibre describes decomposition multiplicities for the corresponding parabolic category O. We use this geometry to define a positively graded algebra which, on the one hand, defines a natural subcategory of coherent sheaves on a resolution of the slice to the corresponding nilpotent orbit, and on the other hand looks like the algebra describing the corresponding parabolic category O.
Minoru Wakimoto (Kyushu University) Title: Some problems in representation theory of W-algebras Abstract: W-algebras provide very interesting and important class of vertex algebras. Associated to a finite-dimensional simple Lie algebra $\frak{g}$ and its nilpotent element $f$ and a comple number $k$, a vertex algebra $W(\frak{g},f)_k$ is constructed by the method of quantum reduction, which is called the W-algebra of level $k$. Twisted W-algebras are also obtained by twisting this construction by using an automorphism of $\frak{g}$. Among them, most important are Ramond twisted W-algebras. In this lecture, we discuss on representation theory of Ramond W-algebras, in particular, exceptional nilpotent element and exceptional level for which all of characters of the W-algebra are holomorphic functions, and the convergence and modular properties of characters, and some of important problems which arise in our study of representation theory of W-algebras. This talk is based on the joint work with Victor Kac.
Jie Xiao (Tsinghua University) Title: Hall algebras associated to triangulated categories. Abstract: My talk based on a joint work with Fan Xu. By counting with triangles and the octohedral axiom, we find a direct way to prove the formula of Toen for a triangulated category with (left) homological-finite condition.
