


Lie Algebras and Moonshine / Algèbres de Lie et Moonshine (Org: Abdellah Sebbar and/et Erhard Neher)
 BRUCE ALLISON, Department of Mathematical and Statistical Sciences,
University of Alberta, Edmonton T6G 2G1
Isomorphism of loop algebras
[PDF] 
The (twisted) loop algebra L(g,s) of a Lie algebra
g relative to an automorphism s of finite order is
an important construction in the theory of infinite dimensional Lie
algebras. When the base algebra g is a finite dimensional
simple Lie algebra, loop algebras provide explicit constructions of
affine KacMoody Lie algebras. When the base algebra is affine, loop
algebras are used to construct extended affine Lie algebras of nullity
two. In this talk, based on joint work with Stephen Berman and Arturo
Pianzola, we describe necessary and sufficient conditions for two loop
algebras of symmetrizable KacMoody Lie algebras to be isomorphic. Our
approach is to view loop algebras as forms of untwisted loop algebras.
 YURI BAHTURIN, Memorial University, St. John's, Newfoundland A1C 5S7
Locally finite simple Lie algebas
[PDF] 
We discuss the possibility of describing some simple locally finite Lie
algebras L in the language used in the theory of root graded algebras
and its generalizations. This is at least possible in the case of the
so called ``pure'' and ``onesided'' direct limits of
finitedimensional simple Lie algebras of type A (terminology due to
BaranovZhilinski). The coordinate algebra in this case is a locally
finite simple associative algebra. Some recent results of
BaranovZalesski on so called finitedimensional plain Lie algebras
form a basis for representing some locally finite simple Lie algebras
(of which our algebras are a particular case) as the Lie commutator
subalgebras of appropriate locally finite simple associative algebras.
Thus, our results and those of BaranovZalesski establish even closer
connection between locally finite simple Lie algebras and locally
finite simple associative algebras. One of the application of our
description is the possibility of constructing new Lmodules.
The results of this talk are joint with Georgia Benkart.
 GEORGIA BENKART, University of WisconsinMadison, Department of Mathematics,
Madison, Wisconsin 53706, USA
TemperleyLieb combinatorics
[PDF] 
TemperleyLieb algebras have appeared in many diverse settings in
connection with statistical mechanics, subfactors of von Neumann
algebras, and knot and link invariants. They are subalgebras of Brauer
algebras and are closely related to the representation theory of
orthogonal and symplectic Lie algebras. This talk will focus on some
representations of TemperleyLieb algebras, their centralizing
algebras, and their combinatorics. This is joint work with Dongho
Moon.
 NANTEL BERGERON, York University, Toronto, Ontario M3J 1P3
Temperley Lieb invariants and covariants
[PDF] 
Hivert defined an action of the symmetric group on polynomials in n
variables for which the quasisymmetric polynomials correspond to the
invariants of the action. The symmetric group algebra mod out by the
kernel of this action can be identified with the temperleyLieb algebra
TL_{n}, an algebra of dimension equal to the nth Catalan number
C_{n}. The quasisymmetric polynomials are thus identified as the
polynomial invariants of the algebra TL_{n}.
The action of Hivert is not compatible with multiplication and does not
preserve the ideal generated by nonconstant homogeneous
quasisymmetric polynomials. Yet we can still consider the quotient
R_{n} of the polynomial ring by this ideal. This yields some striking
facts related to TL_{n}invariants: The quasisymmetric functions are
closed under multiplication in particular they form a subring of
polynomials. Moreover, if we let n go to infinity, there is a
graded Hopf algebra structure on quasisymmetric functions that is free
and cofree with cogenerators in every degree. Moreover, the space
R_{n} of TL_{n}covariants has dimension equal to C_{n}, the dimension
of TL_{n}.
These facts are very similar to the classical theory of group
invariants. Unfortunately the analogy is incomplete as Hivert's action
does not induce an action on R_{n}. This raises new open questions for
future investigation...
 STEPHEN BERMAN, University of Saskatchewan, Saskatchewan
Some Factorizations of U.E.A.'s of 3 dimensional Lie
algebras and some generalizations
[PDF] 
This is a report of joint work with J. Morita and Y. Yoshi. We say a
Lie algebra L has a plusminus pair if it has two subalgebras P,
M whose sum is not all of L which satisfy U(L)=U(P)U(M)U(P). We
show a 3 dimensional Lie algebra over a field of characteristic zero
has a plusminus pair if and only if it is two generated and then use
this to show there are only two 3 dimensional Lie algebras which do
not have a plusminus pair. Related results for more general Lie
algebras are discussed.
 YULY BILLIG, Carleton University, Ottawa, Ontario
Representations of the full toroidal Lie algebras
[PDF] 
This talk will be an introduction to the theory of the toroidal Lie
algebras and their representations. Toroidal Lie algebras are the
natural multivariable generalizations of the affine KacMoody
algebras. We will present two approaches to the representation theory
of these algebras  an abstract construction using the Verma module
technique, and an approach based on the theory of the vertex operator
algebras, which allows us to give explicit realizations for the
irreducible modules and obtain their characters. For the subalgebra of
the toroidal Lie algebra, corresponding to the divergencefree vector
fields, we obtain a result that has a striking resemblance to the
formula for the critical dimension in bosonic string theory. Let
V_{aff}(c) be an affine vertex algebra at level c, V^{+}_{hyp} be
a subalgebra of the hyperbolic lattice vertex algebra and let
V_{Vir}(c_{1}) be a Virasoro vertex algebra of rank c_{1}. Then the
tensor product
V_{aff}(c) ÄV^{+}_{hyp} ÄV_{Vir}(c_{1}) 

has a structure of an irreducible module for an (N+1)toroidal Lie
algebra when

cdim(g) c + h\check

+ 2(N+1) + c_{1} = 26. 

 CHRIS CUMMINS, Concordia
Congruence subgroups of the modular group
[PDF] 
In recent years modular invariance has played an increasing important
role in many areas of mathematics and physics. In this talk I shall
discuss the computation and classification of congruence subgroups of
the modular group of small genus. This is joint work with Sebastian
Pauli.
 MARCELO PEREIRA DE OLIVEIRA, Department of Mathematics, University of Toronto, Toronto,
Ontario M5S 3G3
A universal Bergman element
[PDF] 
We present new developments in 3graded Lie algebras, more precisely,
results in which such a Lie algebra is generated by a pair: a
construction of the free case over a field of characteristic zero, and
also over a field of characteristic p > 3 (following suggestion of
E. Zelmanov). A comparison with KKT algebras is provided. Finally, we
introduce the formal canonical kernel and explain why this is a natural
candidate for a universal Bergman element, to be defined on the formal
completion of the universal enveloping algebra of the above free
3graded Lie algebra. Its expression reduces to the usual Bergman
operator for the adjoint representation; all of this is also motivated
by recent analytic expressions for Bergman kernels of holomorphic
discrete series representations obtained in terms of the canonical
kernel function.
 DRAGOMIR DJOKOVIC, University of Waterloo, Waterloo, Ontario
The closure diagram for nilpotent orbits of the split
real form of E_{8}
[PDF] 
We have constructed the Hasse diagram for the partially ordered set
whose elements are the nilpotent adjoint orbits of the split real form
of the simple complex Lie algebra of type E_{8}. The partial order is
defined via the closure relation. The construction is based on the fact
that the KostantSekiguchi correspondence preserves the closure
relation. The proof uses in an essential way two kinds of
prehomogeneous vector spaces which are naturally attached to each of
the above orbits.
 CHONGYING DONG, University of California, Santa Cruz, California 95064, USA
The monstrous moonshine and permutation orbifolds
[PDF] 
This talk will discuss a connection between the monstrous moonshine and
permutation orbifolds.
 TERRY GANNON, University of Alberta, Edmonton, Alberta
CFT and subfactors
[PDF] 
Conformal field theory (CFT) has motivated some very pretty
questions. In this talk I will focus on the ones which have also arisen
naturally in the subfactor theory of von Neumann algebras, as developed
by Ocneanu and others. The simplest case not yet fully understood is
related to affine (3). The affine (3) NIMreps (i.e.
fusion ring representations) arising in subfactor theory have recently
been classified by Ocneanu, and the natural question is to what extent
is his classification mirrored by CFT. It appears that if one adds
some additional structure to the CFT, then Ocneanu's classification
should be recovered. But is this extra structure necessary or even
desirable? I try to probe this question by addressing the affine
(3) NIMrep classification in its purest form, i.e.
without imposing additional (perhaps spurious) conditions.
 YUN GAO, York University, Toronto, Ontario M3J 1P3
A `quantized TitsKantorKoecher' algebra
[PDF] 
We will discuss a TitsKantorKoecher algebra arising from the extended
affine Lie algebra of type A_{1}. Then we propose a quantum anologue
of the TitsKantorKoecher algebra by looking at the vertex operator
construction over a Fock space. This is a joint work with Naihuan
Jing.
 HAISHENG LI, Rutgers UniversityCamden, New Jersey, USA
Vertex algebras and vertex poisson algebras
[PDF] 
In this talk, we shall review the definitions of vertex Lie algebras
and vertex poisson algebras, which are ``stringy'' analogues of Lie
algebras and poisson algebras while vertex algebras are ``stringy''
analogues of associative algebras. We shall introduce the notion of
filtered vertex algebra and then associate a vertex poisson algebra to
each filtered vertex algebra. We shall discuss how to determine and
construct filtrations for a vertex operator algebra. At the end, we
shall discuss formal deformation of vertex poisson algebras and present
some basic results.
 ADRIAN OCNEANU, The Pennsylvania State University, University Park,
Pennsylvania 16802, USA
Quantum subgroups, lattices and canonical bases of Lie groups
[PDF] 
We show that from a quantum subgroup of SU(2), or the corresponding
subfactor, one can construct in a canonical manner the quantum simple
Lie group with the same Coxeter graph. The construction yields in a
simple and elementary way the roots, weights, the quantum universal
enveloping algebra with a canonical basis and the irreducible
representations of the quantum Lie group. The basis, which is shown to
be as canonical as possible, does not depend on a choice of a simple
basis for the Lie group.
From the quantum subgroups of other simple Lie groups we construct new
finite dimensional Euclidean unimodular lattices of weights and roots.
Even in the simplest cases, these lattices appear to be new.
We provide the classification of the quantum subgroups of SU(2),
SU(3) and SU(4). While the number of exceptional usual subgroups
grows rapidly, the number of exceptional quantum subgroups is small:
2 for SU(2), 3 for SU(3) and 3 for SU(4).
 ARTURO PIANZOLA, University of Alberta, Edmonton, Alberta
Automorphisms of Chevalley Lie algebras
[PDF] 
We start with a commutative ring k (the base), a Chevalley Lie
algebra over k, a (commutative unital associative) kalgebra R,
and the Lie algebra g(R) obtained from g by base ring extension.
The talk will focus on the description of the group of automorphisms of
g(R) viewed as a Lie algebra over k (for example if k is the
complex numbers and R is Laurent polynomials we are dealing with
untwisted affine KacMoody Lie algebras).
We will also also compare our results and methods (which exploit that
conjugacy holds locally for the étale topology of Spec(R)) with
what is known in the case of Chevalley groups.
 MATTHEW SZCZESNY, University of Pennsylvania, Philadelphia, Pennsylvania, USA
Twisted modules, conformal blocks, and Prym varieties
[PDF] 
We give a geometric approach to twisted vertex operators on an
algebraic curve. This allows us to define a notion of conformal block
valued in twisted modules for a vertex algebra. In certain cases these
conformal blocks can be realized as Prym theta functions. They also
give rise to localization functors from representations of twisted
affine Lie algebras to twisted Dmodules on Prymlike varieties.
This is joint work with Edward Frenkel.
 YOJI YOSHII, WisconsinMadison
Lie tori and structurable tori
[PDF] 
We report some recent progress on Lie tori. Lie ntori are certain
Z^{n}graded Lie algebras which coincide with the cores of
extended affine Lie algebras. Lie 1tori of type A_{1} and BC_{1}
are exactly affine KacMoody Lie algebras of type A_{1}^{(1)} and
A_{2}^{(2)} respectively. We discuss Lie 2tori of type A_{1} and
BC_{1} which are coordinatized by analogs of the algebra of Laurent
polynomials in 2 variables. It turns out that the coordinate
algebras for type A_{1} include one infinite family containing the
algebra of Laurent polynomials in 2 variables and precisely one other
algebra. For type BC_{1} the situation is rather different. There are
precisely 5 different coordinate algebras in that case.
 NORIKO YUI, Queen's University
The modularity of CalabiYau threefolds with K3 fibrations
[PDF] 
We consider certain CalabiYau threefolds which are fibered by
nonconstant semistable K3 surfaces. These are nonrigid CalabiYau
threefolds, reaching the ArakelovYau bound. We address the modularity
of these CalabiYau threefolds.

