# 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:

• ${\displaystyle d_{n}}$ ,with n running through every integer,
• ${\displaystyle c}$

with the following relations:

• ${\displaystyle [d_{n},c]=0}$,
• ${\displaystyle [d_{m},d_{n}]=(m-n)d_{m+n}+\delta _{m+n}{\frac {m^{3}-m}{12}}c}$, with m and n each running through every integer

where ${\displaystyle \delta _{m+n}}$ is 1 when ${\displaystyle m+n=0}$ and is zero otherwise.