In this paper explicit decompositions are provided of the Weyl reflections in affine Lie algebras, in terms of fundamental Weyl reflections. PACS: 02.20.Sv; 11.25.Hf

Kac-Moody algebras are a generalization of the finite-dimensional semisimple Lie algebras that have many characteristics similar to the finite-dimensional ones. These possibly infinite-dimensional Lie algebras have found applications everywhere from modular forms to conformal field theory in physics. In this thesis we give two main results of the theory of Kac-Moody algebras. First, we present ...

We provide a motivated introduction to the theory of categorical actions groups and local geometric Langlands program. Along way, we emphasize applications, old new, usual representation reductive affine Lie algebras.

1.1. Today, vertex algebras have become rather ubiquitous in mathematics. Among the subjects on which they bear direct influence, we can list: finite group theory, number theory (both through the classical theory of elliptic functions and the modern geometric Langlands theory), combinatorics, representation theory, and algebraic geometry. Before we discuss these relations, let us recall that de...

For all affine Toda field theories we propose a new type of generic boundary bootstrap equations, which can be viewed as a very specific combination of elementary boundary bootstrap equations. These equations allow to construct generic solutions for the boundary reflection amplitudes, which are valid for theories related to all simple Lie algebras, that is simply laced and non-simply laced. We ...

We study the spectra of G/G coset models by computing BRST cohomology of affine Lie algebras with coefficients in tensor product of two modules. One-to-one correspondence between the spectra of A1/A 1 1 and that of the minimal matter coupled to gravity (including boundary states of the Kac table) is observed. This phenomena is discussed from the point of hamiltonian reduction of BRST complexes ...

Starting from Borcherds’ fake monster Lie algebra we construct a sequence of six generalized Kac-Moody algebras whose denominator formulas, root systems and all root multiplicities can be described explicitly. The root systems decompose space into convex holes, of finite and affine type, similar to the situation in the case of the Leech lattice. As a corollary, we obtain strong upper bounds for...

The object of this work is the systematical study of a certain type of generalized Cartan matrices associated with the Dynkin diagrams that characterize CartanLie and affine Kac-Moody algebras. These generalized matrices are associated to graphs which arise in the study and classification of Calabi-Yau spaces through Toric Geometry. We focus in the study of what should be considered the general...

We study connections between the ring of symmetric functions and the characters of irreducible finite-dimensional representations of quantum affine algebras. We study two families of representations of the symplectic and orthogonal Lie algebras. One is defined via combinatorial properties and is easy to calculate; the other is closely related to the q = 1 limit of the “minimal affinization” rep...