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.

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

See Also

• Topic:Affine Lie algebras

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: