Vacuum Vector Representations of the Virasoro Algebra

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A lot of attention has been focused lately on certain infinite dimensional Lie algebras for their importance in some physical theories as well as the richness of their mathematical theories. One of these algebras is the Virasoro algebra. The Virasoro algebra is known to physicists in the theory of dual string models (cf. [25]). The first mathematical reference on the Virasoro algebra that is known to us is by Gelfand and Fuchs [9]. They proved that the second cohomology of the Lie algebra v of polynomial vector fields on the circle is one-dimensional. Using this one can show that the Virasoro algebra is the universal central extension \( \mathop{v}\limits^{ \wedge } \) of v (see §4 below). The Virasoro algebra was later realized as an algebra of operators on the representation space of a Kac-Moody algebra (cf. [5, 3, 11, 17]), in a way reminiscent of its earlier introduction in dual models.

Partially supported by National Science Foundation Grant MCS-8201260.