Topic:Virasoro algebra
From Wikiversity
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.
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[edit] Formal Definition
Vir is the Lie algebra over the field of complex numbers with the following generators:
- dn ,with n running through every integer,
- c
with the following relations:
- [dn,c] = 0,
![[d_m , d_n] = (m-n) d_{m+n} + \delta_{m+n} \frac{m^3 - m}{12} c](http://upload.wikimedia.org/math/3/6/0/360d1a5c01840bcb3591844f85d8062e.png)
, with m and n each running through every integer
where δm + n is 1 when m + n = 0 and is zero otherwise.
[edit] 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
[edit] Applications
[edit] See Also
[edit] 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)
