---

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 Kac-Moody 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 Kac-Moody 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 ``one-sided'' direct limits of finite-dimensional simple Lie algebras of type A (terminology due to Baranov-Zhilinski). The coordinate algebra in this case is a locally finite simple associative algebra. Some recent results of Baranov-Zalesski on so called finite-dimensional 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 Baranov-Zalesski 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 L-modules.

The results of this talk are joint with Georgia Benkart.

GEORGIA BENKART, University of Wisconsin-Madison, Department of Mathematics, Madison, Wisconsin  53706, USA
Temperley-Lieb combinatorics [PDF]

Temperley-Lieb 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 Temperley-Lieb 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 quasi-symmetric 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 temperley-Lieb algebra TL_n, an algebra of dimension equal to the nth Catalan number C_n. The quasi-symmetric 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 non-constant homogeneous quasi-symmetric 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 quasi-symmetric 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 quasi-symmetric 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 plus-minus 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 plus-minus 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 plus-minus 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 multi-variable generalizations of the affine Kac-Moody 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 divergence-free 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(c₁) be a Virasoro vertex algebra of rank c₁. Then the tensor product

Vaff(c) ÄV+hyp ÄVVir(c1)

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

cdim(g) c + h\check + 2(N+1) + c1 = 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 3-graded 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 3-graded 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 E₈ [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 E₈. The partial order is defined via the closure relation. The construction is based on the fact that the Kostant-Sekiguchi 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) NIM-reps (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) NIM-rep classification in its purest form, i.e. without imposing additional (perhaps spurious) conditions.

YUN GAO, York University, Toronto, Ontario  M3J 1P3
A `quantized Tits-Kantor-Koecher' algebra [PDF]

We will discuss a Tits-Kantor-Koecher algebra arising from the extended affine Lie algebra of type A₁. Then we propose a quantum anologue of the Tits-Kantor-Koecher algebra by looking at the vertex operator construction over a Fock space. This is a joint work with Naihuan Jing.

HAISHENG LI, Rutgers University-Camden, 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) k-algebra 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 Kac-Moody 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 D-modules on Prym-like varieties. This is joint work with Edward Frenkel.

YOJI YOSHII, Wisconsin-Madison
Lie tori and structurable tori [PDF]

We report some recent progress on Lie tori. Lie n-tori are certain Zⁿ-graded Lie algebras which coincide with the cores of extended affine Lie algebras. Lie 1-tori of type A₁ and BC₁ are exactly affine Kac-Moody Lie algebras of type A₁⁽¹⁾ and A₂⁽²⁾ respectively. We discuss Lie 2-tori of type A₁ and BC₁ which are coordinatized by analogs of the algebra of Laurent polynomials in 2 variables. It turns out that the coordinate algebras for type A₁ include one infinite family containing the algebra of Laurent polynomials in 2 variables and precisely one other algebra. For type BC₁ the situation is rather different. There are precisely 5 different coordinate algebras in that case.

NORIKO YUI, Queen's University
The modularity of Calabi-Yau threefolds with K3 fibrations [PDF]

We consider certain Calabi-Yau threefolds which are fibered by non-constant semi-stable K3 surfaces. These are non-rigid Calabi-Yau threefolds, reaching the Arakelov-Yau bound. We address the modularity of these Calabi-Yau threefolds.