# 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 edit

*Vir* is the Lie algebra over the field of complex numbers with the following generators:

- ,with
*n*running through every integer,

with the following relations:

- ,
- , with
*m*and*n*each running through every integer

where is 1 when and is zero otherwise.

## Representation Theory edit

## Applications edit

## See Also edit

## Reference edit

**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: [1]