Current Algebras and Groups pp 171-191 | Cite as

# The Virasoro Algebra

## Abstract

In this chapter we shall study the Lie algebra *Vect S* ^{1} of vector fields on a circle and some of its generalizations. The Lie algebra *Vect S* ^{1} has a central extension, the *Virasoro algebra*. The representation theory of the Virasoro algebra is closely related to the representation theory of affine Lie algebras. In fact, through the Sugawara construction, to be defined below, a highest weight representation of an affine Lie algebra carries always a highest weight representation of the Virasoro algebra. All the irreducible highest weight representations of the Virasoro algebra are known and they can be exponentiated to representations of associated infinite-dimensional Lie groups. The representation theory of the algebra of vector fields on a higher dimensional manifold is much less understood; we shall discuss the extensions of these algebras in Section 7.6.

## Keywords

Central Charge High Weight Vector Virasoro Algebra High Weight Representation Finite Linear Combination## Preview

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