Robert V. Moody

Toroidal Lie algebras and vertex representations

The paper describes the theory of the toroidal Lie algebra, i.e. the Lie algebra of polynomial maps of a complex torus ℂ××ℂ× into a finite-dimensional simple Lie algebra g. We describe the universal central extension t of this algebra and give an abstract presentation for it in terms of generators and relations involving the extended Cartan matrix of g. Using this presentation and vertex operators we obtain a large class of integrable indecomposable representations of t in the case that g is of type A, D, or E. The submodule structure of these indecomposable modules is described in terms of the ideal structure of a suitable commutative associative algebra.

The Mathematics of long-range aperiodic order

Preface. Knotted Tilings C.C. Adams. Solution of the Coincidence Problem in Dimensions d smaller than or equal to 4 M. Baake. Self-Similar Tilings and Patterns Described by Mappings C. Bandt. Delone Graphs and Certain Species of Such L. Danzer, N. Dolbilin. What is the Long Range Order in the Kolakoski Sequence? F.M. Dekking. Topics in Aperiodicity: Penrose Tiling Growth and Quantum Circuits D.P. DiVincenzo. The Diffraction Pattern of Self-Similar Tilings F. Gahler, R. Klitzing. Pisot-Cyclotomic Integers for Quasilattices J.-P. Gazeau. Aperiodic Ising Models U. Grimm, M. Baake. Diffraction by Aperiodic Structures A. Hof. Aperiodic Schrodinger Operators T. Janssen. Symmetry Concepts for Quasicrystals and Non-Commutative Crystallography P. Kramer, Z. Papadopolos. Local Rules for Quasiperiodic Tilings T.T.Q. Le. Almost-Periodic Sequences and Pseudo-Random Sequences M.M. France. The Symmetry of Crystals N.D. Mermin. Meyer Sets and Their Duals R.V. Moody. Non-Crystallographic Root Systems and Quasicrystals J. Patera. Remarks on Tiling: Details of a (1+epsilon + epsilon2)-Aperiodic Set R. Penrose. Aperiodic Tilings, Ergodic Theory, and Rotations C. Radin. A Critique of the Projection Method M. Senechal. Index.

Vertex representations for

Vertex representations are obtained for toroidal Lie algebras for any number of variables. These representations afford representations of certainn-variable generalizations of the Virasoro algebra that are abelian extensions of the Lie algebra of vector fields on a torus.

n

Even when reduced to its simplest form, namely that of point sets in euclidean space, the phenomenon of genuine quasi-periodicity appears extraordinary. Although it seems unfruitful to try and define the concept precisely, the following properties may be considered as representative: discreteness; extensiveness; finiteness of local complexity; repetitivity; diffractivity; aperiodicity; existence of exotic symmetry (optional).

-toroidal Lie algebras and a generalization of the Virasoro algebra

This paper is about toroidal Lie algebras, certain intersection matrix Lie algebras defined by Slodowy, and their relationship to one another and to certain Lie algebra analogues of Steinberg groups. The main result of the paper is the identification of the intersection matrix algebras arising from multiply-affinized Cartan matrices of types A, D and E with certain Steinberg Lie algebras and toroidal Lie algebras (Propositions 5.9 and 5.10). A major part of the paper studies and classifies Lie algebras graded by finite root systems. These become the princi- pal tool in our analysis of intersection matrix algebras. Each Lie algebra graded by a simply-laced finite root system of rank > 2 has attached to it an algebra which, according to the type and rank, is either commutative and associative, only associative, or alternative. All these possibilities occur in our description of inter- section matrix algebras. Let R be any associative algebra with identity, not necessarily finite dimen- sional, over a field k of characteristic 0. For each positive integer n the associative algebra M,(R) of n “ n matrices with coefficients in R forms a Lie algebra over k under the commutator product. We denote this Lie algebra by ol,(R). Let Eis be the (i, j) matrix unit of M.(R) and assume that n > 2. The subalgebra e.(R) of OI.(R) generated by the elements rEis, r ~ R, i 4= j, is an ideal of 91,(R) and is perfect, i.e. it is its own derived algebra. Now any perfect Lie algebra O has a universal central extension, also perfect, called a universal covering algebra (u.c.a.) of O [Ga], so in particular, e.(R) has a u.c.a, that we will denote by ~t,(R). We define 112,.(R) by the exact sequence (0.1) 0 ~ f2,,(R) ~ ~t.(R) ~ e,(R) ~ 0 * Dedicated to our teacher Maria J. Wonenburger ** Both authors gratefully acknowledge the support of the Natural Sciences and Engineering Research Council of Canada

Model Sets: A Survey

2. Preliminaries. In this note, * will always denote a field of characteristic zero. An integral square matrix satisfying M l , M2, and M3 will be called a generalized Cartan matrix, or g.c.m. for short. Z will denote the integers, and in any Lie algebra we will use the symbol [lu h, , In] to denote the product [ • • • [[hh] • • • ]L]. 1 These results were obtained in my dissertation at the University of Toronto under the supervision of Professor M.J. Wonenburger.

Lie algebras graded by finite root systems and the intersection matrix algebras of Slodowy

Abstract. We show that for multi-colored Delone point sets with finite local complexity and uniform cluster frequencies the notions of pure point diffraction and pure point dynamical spectrum are equivalent.

Lie algebras associated with generalized Cartan matrices

It is shown how regular model sets can be characterized in terms of the regularity properties of their associated dynamical systems. The proof proceeds in two steps. First, we characterize regular model sets in terms of a certain map

Pure Point Dynamical and Diffraction Spectra

\beta

Characterization of model sets by dynamical systems

and then relate the properties of