| Central extensions are used to enrich and elucidate the representation theory of infinite-dimensional Lie algebras. Such extensions may be described homologically, and for each Lie algebra g over a field K , there is a bijection between cohomology classes in H 2( g,K ) and one-dimensional central extensions of g . Some of the most important central extensions can be constructed as spaces of quadratic operators acting on certain highest weight modules for Weyl or Clifford algebras. These modules, called Fock spaces, are spanned by monomials corresponding to the possible energy states of particles in some system. They are called bosonic or fermionic , according to the occupancy statistics that the particles satisfy.; We use the interplay between several cohomology theories to compute the dimension of the universal central extension of the Lie algebra gln (Ar,s) of n x n matrices over a family of localizations Ar,s of the Weyl algebra Ar. We then explicitly compute formulas for each nontrivial 2-cocycle in H2( g;K ), where g is the quotient of gln (Ar,s) by its center.; The remainder of the thesis considers Fock representations of central extensions. Our main result is a uniform construction of bosonic and fermionic modules for a one-dimensional central extension g&d5; of any Lie algebra g over a field of characteristic 0. The procedure is completely general and gives an explicit formula for the 2-cocycle defining the central extension. We illustrate these techniques with several concrete examples, including the Virasoro algebra, the affine Lie algebra A1n , some of the algebras mentioned in the previous paragraph.; Under mild conditions, our construction also gives an embedding of g&d5; into a central extension of an algebra of infinite matrices. We use this embedding to obtain an isomorphism, called a boson-fermion correspondence , between a bosonic representation B and a fermionic representation F for g&d5; . The action of g&d5; on B is described by vertex operators, functions originally studied in the context of dual resonance models. |
