nLab
Virasoro algebra

Context

Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Algebra

Contents

Idea

The Virasoro algebra or Virasoro Lie algebra is the nontrivial central extension of the Witt Lie algebra? (the Lie algebra of the group of diffeomorphisms of the circle). It is of central importance in some questions of complex analysis, in conformal field theory and the study of affine Lie algebras.

Definition

The generators L nL_n of the Virasoro algebra are indexed by an integer nn \in \mathbb{Z}, and they satisfy the commutation relation

[L m,L n]=(mn)L m+n+c12(m 3m)δ m+n,0. [L_m, L_n] = (m - n) L_{m+n} + \frac{c}{12}(m^3 - m) \delta_{m+n,0}.

Here, cc denotes the element of the algebra known as the central charge; it commutes with each generator,

[L n,c]=0n. [L_n, c] = 0 \forall n.

The factor of 1/12 is conventional and chosen for normalisation purposes.

Revised on September 12, 2013 01:19:08 by Urs Schreiber (145.116.131.249)