The Affine Lie Algebra sl(n)(C) and its Z-algebra Representation.

摘要

At the end of the 1960s, Victor Kac and Robert V. Moody independently discovered the infinite dimensional analogs of finite semisimple Lie algebras which we now refer to as Kac-Moody Lie algebras. These algebras have provided many avenues of interesting research, especially in the case of the affine Kac-Moody Lie algebras, which have led to important developments in vertex operators and other areas of mathematics and physics.;James Lepowsky and Robert L. Wilson introduced, in 1981, Z-algebras associated with any integrable module of an affine Lie algebra. Z-algebras are generated by Z-operators which centralize the action of the Heisenberg subalgebra and hence act on the vacuum space of the module. Their work was especially significant for providing Lie theoretic proofs of Roger-Ramanujan identities. In 1986, Minoru Wakimoto published his family of vertex operator realizations of sl(2). This family offered many realizations of sl(2) at arbitrary level, via creation and annihilation operators acting on an infinite-dimensional Fock space of Laurent polynomials. Subsequently, Boris Feigin and Edward Frenkel generalized Wakimoto's realizations for the affine Lie algebra sl(n). In particular, they gave explicit formulas for the action of the simple root vectors. Later, they extended these results to all affine Lie algebras.;In Chapter 2, we revisit Wakimoto's family of representations for sl(2). Through this explicit realization, we construct the associated Lepowsky-Wilson Z-algebra. Chapter 3 extends Wakimoto's representation to sl(n), following the work of Feigin and Frenkel. Next, we give a general Wakimoto-style formula for the action of all positive root vectors, which is followed by a brief remark on the construction of the Z-algebras associated with sl(n). We then give the explicit Wakimoto realization for sl(3) in Chapter 4 and offer the defining relations for its associated Z-algebra. In the last chapter, we give a new realization, at the critical level, of the Z-algebra associated with the Wakimoto modules of sl(2) acting on a Clifford-type algebra and calculate the character of the associated vacuum space.