There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Sphere Packings, Lattices and Groups

* uschian groups of the first kind * Automorphic forms and functions * Hecke operators and the zeta-functions associated with modular forms * Elliptic curves * Abelian extensions of imaginary quadratic fields and complex multiplication of elliptic curves * Modular functions of higher level * Zeta-functions of algebraic curves and abelian varieties * The cohomology group assoicated with cusp forms * Arithmetic Fuschian groups

Introduction to the arithmetic theory of automorphic functions

General introduction to the special odd case General lemmas Theorem $C^*_2$: Stage 1 Theorem $C^*_2$: Stage 2 Theorem $C_2$: Stage 3 Theorem $C_2$: Stage 4 Theorem $C_2$: Stage 5 Theorem $C_3$: Stage 1 Theorem $C_3$: Stages 2 and 3 IV$_K$: Preliminary properties of $K$-groups Background references Expository references Glossary Index.

http://inis.jinr.ru/sl/M_Mathematics/MA_Algebra/MAtg_Group%20theory/Gorenstein%20D.,%20Lyons%20R.,%20Solomon%20R.%20Classification%20of%20finite%20simple%20groups%201%20(AMS%20survey%2040%20no.1,%201994,%202000)(176s).pdf

The Classification of the Finite Simple Groups

Introduction to lie algebras and representation theory

This paper gives an analogue of Ag(V) theory for a vertex operator superalgebra V and an automorphism g of finite order. The relation between the g-twisted V-modules and Ag(V)-modules is established. It is proved that if V is g-rational, then Ag(V) is finite-dimensional semi-simple associative algebra and there are only finitely many irreducible g-twisted V-modules.

Twisted representations of vertex operator algebras

The goal of the present paper is to provide a mathematically rigorous foundation to certain aspects of the theory of rational orbifold models in conformal field theory, in other words the theory of rational vertex operator algebras and their automorphisms.

Modular-Invariance of Trace Functions¶in Orbifold Theory and Generalized Moonshine

We provide a rigorous mathematical foundation to the study of strongly rational, holomorphic vertex operator algebras V of central charge c = 8, 16 and 24 initiated by Schellekens. If c = 8 or 16 we show that V is isomorphic to a lattice theory corresponding to a rank c even, self-dual lattice. If c = 24 we prove, among other things, that either V is isomorphic to a lattice theory corresponding to a Niemeier lattice or the Leech lattice, or else the Lie algebra on the weight one subspace V 1 is semisimple (possibly 0) of Lie rank less than 24.

/pdf/holomorphic-vertex-operator-algebras-of-small-central-charge-b43tz0ljq3.pdf

Holomorphic vertex operator algebras of small central charge

The following integrability theorem for vertex operator algebras V satisfying some finiteness conditions (C2-cofinite and CFT-type) is proved: the vertex operator subalgebra generated by a simple Lie subalgebra g of the weight one subspace V1 is isomorphic to the irreducible highest weight ˆ-module L(k,0) for a positive integer k, and V is an integrable ˆ-module. The case in which g is replaced by an abelian Lie subalgebra is also considered, and several consequences of integrability are discussed. 2000MSC:17B69

Integrability of C2-cofinite vertex operator algebras

Let V be a vertex operator algebra, and for k a positive integer, let g be a k-cycle permutation of the vertex operator algebra V

 ⊗

 k
 . We prove that the categories of weak, weak admissible and ordinary g-twisted modules for the tensor product vertex operator algebra V

 ⊗

 k
 are isomorphic to the categories of weak, weak admissible and ordinary V-modules, respectively. The main result is an explicit construction of the weak g-twisted V

 ⊗

 k
 -modules from weak V-modules. For an arbitrary permutation automorphism g of V

 ⊗

 k
 the category of weak admissible g-twisted modules for V

 ⊗

 k
 is semisimple and the simple objects are determined if V is rational. In addition, we extend these results to the more general setting of γg-twisted V

 ⊗

 k
 -modules for γ a general automorphism of V acting diagonally on V

 ⊗

 k
 and g a permutation automorphism of V

 ⊗

 k
 .

Geoffrey Mason

Papers

Twisted representations of vertex operator algebras

Modular-Invariance of Trace Functions¶in Orbifold Theory and Generalized Moonshine

Holomorphic vertex operator algebras of small central charge

Integrability of C2-cofinite vertex operator algebras

Twisted sectors for tensor product vertex operator algebras associated to permutation groups