The purpose of this thesis is to give a new construction for central extensions of certain classes of infinite dimensional Lie algebras which include multiloop Lie algebras as motivating examples. The key idea of this construction is to view multiloop Lie algebras as twisted forms. This perspective provides a beautiful bridge between infinite dimensional Lie theory and descent theory and is cru...

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 establish results about the second cohomology with coefficients in trivial module, symmetric invariant bilinear forms and derivations of a Lie algebra extended over commutative associative without unit. These provide simple unified approach to number questions treated earlier completely separated ways: periodization semisimple algebras (Anna Larsson), derivation algebras, prescribed part, ni...

The classification of structurable tori with nontrivial involution, which was begun by Allison and Yoshii, is completed. New examples of struct-urable tori are obtained using a construction of structurable algebras from a semilinear version of cubic forms satisfying the adjoint identity. The classification uses techniques borrowed from quadratic forms over Z 2 and from the geometry of generaliz...

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...

A double algebra is a linear space V equipped with linear map V ⊗ V → V ⊗ V . Additional conditions on this map lead to the notions of Lie and associative double algebras. We prove that simple finite-dimensional Lie double algebras do not exist over an arbitrary field, and all simple finite-dimensional associative double algebras over an algebraically closed field are trivial. Over an arbitrary...

Generalizing the characters of compact simple Lie groups, Ian Macdonald introduced in [M1,M2] and other works remarkable symmetric trigonometric polynomials dependent on the parameters q, t. He came up with four main conjectures formulated for arbitrary root systems. A new approach to the Macdonald theory was suggested in [C1] on the basis of double affine Hecke algebras (new objects in mathema...