Virasoro algebra

The Virasoro algebra, denoted Vir, is an infinite-dimensional Lie algebra, defined as central extension of the complexification of the Lie algebra of vector fields on the circle. One may think of it as a deformed version of the Lie algebra for the group of orientation-preserving diffeomorphisms of the circle. The representation theory of Virasoro algebra is rich, and has diverse applications in Mathematics and Physics.

Formal Definition
Vir is the Lie algebra over the field of complex numbers with the following generators: with the following relations: where $$\delta_{m+n}$$ is 1 when $$ m+n=0 $$ and is zero otherwise.
 * $$d_n $$ ,with n running through every integer,
 * $$c $$
 * $$[d_n, c] = 0 $$,
 * $$[d_m, d_n] = (m-n) d_{m+n} + \delta_{m+n} \frac{m^3 - m}{12} c $$, with m and n each running through every integer

Representation Theory

 * Oscillator representations
 * Verma modules
 * Unitary representations
 * Topic:Boson-Fermion correspondence
 * Topic:Schur polynomials
 * Kac determinant formula
 * Sugawara construction
 * Coset construction
 * Weyl-Kac character formula

Applications

 * Topic:KP hierarchy

Reference

 * Kac, V. G. and Raina, A. K.-- Highest Weight Representations of Infinite Dimensional Lie Algebras, ISBN 9971-50-396-4
 * Frenkel and ben-Zvi, Vertex algebras and algebraic curves, ISBN 0821828940, p.41(definition), p.326(geometric description)
 * Kac's article in Encyclopedia of Mathematics, Springer: