




Infinite Dimensional Lie Theory and Representation Theory / Théorie
de Lie en dimension infinie et théorie des représentations (Stephen Berman, Organizer)
 BRUCE ALLISON, University of Alberta, Edmonton, Alberta
Extended affine Lie algebras of nonreduced type

Extended affine Lie algebras are higher nullity generalizations of
affine KacMoody Lie algebras. The structure of extended affine Lie
algebras when the associated finite root system has reduced type is now
fairly well understood. In this talk we report on recent progress in
the case when the root system has nonreduced type.
 GEORGIA BENKART, University of Wisconsin, Madison, Madison, Wisconsin 53706,
USA
Classical Lie (super)algebras over associative algebras

>From matrices over simple associative domains and their onesided
ideals, we construct families of simple Lie algebras and Lie
superalgebras. Our construction intermixes the structure of the
associative algebra and the structure of the matrix algebra. The
resulting Lie (super)algebras have a grading by a finite root system.
This is joint work with X. Xu and K. Zhao.
 YULY BILLIG, Carleton
To be announced

 DAN BRITTEN, University of Windsor, Windsor, Ontario
On the lattice of submodules of generalized Verma modules

Let G be a finite dimensional simple Lie algebra over the
complex numbers with semisimple regular subalgebra [^(G)]. Let F be a torsion free [^(G)]module and P be the parabolic subalgebra of
G generated by [^(G)], the Cartan subalgebra of
G and all positive root vectors of G. Fix a
basis of h_{1},..., h_{l} of ann_{H}([^(G)])={h Î H  [h,[^(G)]]=0} and
define a Pmodule structure on F by
X_{a}·F=0 for all positive root vectorsX_{a} Ï 
^ G


 h_{i}·F=a_{i} with a_{i} arbitrary complex numbers. 


Our aim is to study the lattice of submodules of the generalized Verma
module, GVM, over U(G) define by parabolic induction,
M(F:a_{1},...,a_{l})=U(G) 
Å
U(P)

F. 

In particular, the lattice of submodules of M(F:a_{1},...,a_{l}) is related to the lattice of submodules within a quotient
module of a Verma modulo a sum of its Verma submodules. Through this
relationship an analogue of the BGG criterion is obtained for GVMs.
(Joint work with V. M. Futorny and F. W. Lemire)
 CHONGYING DONG, California, Santa Cruz
To be announced

 TERRY GANNON, Alberta
To be announced

 YUN GAO, York
Double affine algebras

 MARIA GOLENISHCHEVAKUTUZOVA, University of Florida, Department of Mathematics, Gainesville,
Florida 326118105, USA
Gconformal algebras

Gconformal algebra is an axiomatic description of the operator
product expansion of chiral fields with simple poles at finitely many
points. We classify these algebras and their representations interms
of Lie algebras and their representations with an action of the group
G. To every Gconformal algebra and a character of
G we associate a Lie algebra generated by fields with the OPE
with simple poles. Examples include twisted affine KacMoody algebras,
the sinalgebra (which is a ``Gconformal'' analogue of the
general linear algebra) and the algebra of pseudodifferential operators
on the circle.
 BOB GRIESS, Ann Arbor
To be announced

 FRANK LEMIRE, University of Windsor, Windsor, Ontario
A duality theorem for torsion free modules of symplectic Lie
algebras

Let L denote a semisimple complex Lie algebra with Cartan subalgebra
H and let P denote a parabolic subalgebra of L with P=[(L)\tilde]Å[(H)\tilde]ÅN ([(L)\tilde] is a regular semisimple
subalgebra of L, [(H)\tilde] is the complement of the Cartan
subalgebra of [(L)\tilde] orthogonal to [(L)\tilde] in H and N is the
nilpotent part of P). Let L denote the category of all
torsion free [(L)\tilde] modules of finite degree. Following [FKM] we
define O(P,L) to be the category of all finitely
generated, N finite L modules which when viewed as [(L)\tilde]
modules decompose into a direct sum of modules in L and the
kernels of all L module homomorphisms when viewed as [(L)\tilde]
modules decompose into a direct sum of modules in L. When
[(L)\tilde] is trivial this category reduces to the classical category
O.
In category O the classical BGG reciprocity theory
states that there is a bijective correspondence between indecomposable
projective modules and simple modules and further that, using Verma
modules as intermediaries, here is a reciprocity between the
multiplicity of a Verma module in an indecomposable projective module
and the multiplicity of the corresponding simple module in the Verma
module. In order to provide an analogous reciprocity theorem for the
category O(P,L) Futorny et. al. assumed a so called
duality propertyi.e. for any two simple modules V, W Î L there exists a constant i(V,W) such that for any finite
dimensional [(L)\tilde] module F we have [FÄV:W]=i(V,W)[FÄW:V]. They also establish that if [(L)\tilde] @ sl (2,C) the assumption is valid.
In this talk we show that this duality property holds whenever [(L)\tilde] @ sp (2n,C) with i(V,W)=1 for all simple modules V,W however
the duality property is not valid when [(L)\tilde] @ sl (n,C) for
n ³ 3.
References
[FKM ] V. Futorny, S. Konig and V. Mazorchuk, Categories
of Induced Modules and Projectively Stratified Algebra. 1999,
preprint.
 HAI SHENG LI, Rutgers, The State University of New Jersey, Camden,
New Jersey 08102, USA
A certain higher dimensional analogue of vertex algebras

A higher dimensional analogue of the notion of vertex algebra
is formulated in terms of formal variable language with Borcherds'
notion of Gvertex algebra as a motivation. Some examples are
given and certain analogous duality properties are proved.
Furthermore, it is proved that for any vector space W, any set
of mutually local multivariable vertex operators on W generates
in a certain canonical way a vertex algebra with W as a
natural module.
 KIYOKAZU NAGATOMO, Osaka University, Department of Mathematic, Graduate School of
Science, Osaka University, Toyonaka, Osaka 5600043, Japan
Finiteness of conformal blocks and C_{2} finiteness

We first give the definition of the conformal blocks over compact
Riemann surfaces based on vertex operator algebras. Then we will show
that if a vertex operator algebra V and their irreducible modules
satisfy C_{2} finiteness condition then the corresponding conformal
blocks are all finite dimensional. In particular this shows the
finiteness of the conforma blocks for both WZNW model and the minimal
model, which correspond to affine Lie algebras and the Virasoro
algebra, respectively.
 ERHART NEHER, University of Ottawa, Ottawa, Ontario K1N 6N5
Lie superalgebras graded by 3graded root systems

Lie superalgebras graded by root systems are Lie superalgebras L with
a grading L=Å_{a Î RÈ{0}} L_{a}, [L_{a},L_{b}] Ì L_{a+ b} for a,b Î RÈ{0},
which is induced in a suitable sense by a split simple subalgebra G
of L_{[`0]}, the even part of L. Here R is the root system of
G, and this may be finite or only locally finite, in which case G
is finitedimensional respectively locallyfinite.
The corresponding Lie algebras have recently been investigated by
several authors, among them Allison, Benkart, Berman, Gao, Moody,
Zelmanov and the speaker. One reason for studying these Lie algebras is
that they enter in the construction of extended affine Lie algebras. In
the case of infinite root systems special examples turn up in the
representation theory of affine Lie algebras. The superversions of
these algebras have recently been investigated by Cheng and Wang.
In the talk I will describe my recent joint work with Esther Garcia
(Oviedo) in which we describe Lie superalgebras graded by 3graded
root systems, i.e., R is not of type E_{8}, F_{4}
or G_{2}. In this case, the Lie superalgebras are central
extensions of the TitsKantorKoecher superalgebra of a certain class
of Jordan superpairs which I have recently classified. An important
ingredient in the description of these Lie superalgebras is their
supercoordinate system. It can for example be used to describe
simplicity or the GelfandKirillov dimension.
 ARTURO PIANZOLA, University of Alberta, Edmonton, Alberta T6G 2G1
A geometric interpretation of affine KacMoody Lie algebra

We will review Kac's realization of the affine algebras as twisted loop
algebras. The main idea is then to interpret twisted loop algebras in
general, as principal homogeneous spaces over the ring of Laurent
polynomials.
 YOJI YOSHII, University of Alberta, Edmonton Alberta T6G 2G1
A simple characterization of the core of an EALA

It is important to consider affine Lie algebras as Zgraded
algebras. In this spirit we consider the core of an extended affine
Lie algebra (EALA) as a Z^{n}graded algebra. We call a
certain Z^{n}graded Lie algebra a Lie ntorus or
simply a Lie torus. A Lie torus is a natural generalization of
the core of an EALA. It turns out that there exists a nontrivial
graded form on any Lie torus, and this gives us a simple
characterization of the core of an EALA. Namely, the core of any EALA
is a Lie torus and any Lie torus is centrally isogeneous to some core
of an EALA. For example, a Lie 1torus is exactly a (twisted) loop
algebra or its universal covering.
 MIKHAIL ZAICEV, Moscow
To be announced


