Virasoro algebra

The orientation-preserving diffeomorphism group of the circle, Diff(S¹) admits a central extension called the Virasoro group. Its complexified Lie algebra is spanned by {L_i}_{i in Z} and c with L_n+L_-n and c being real elements. c is called the central charge. The algebra satisfies

[c,L_n]=0

[L_m,L_n]=(n-m)L_m+n+c/12 (m³-m)δ_m,-n.

The factor of 1/12 is merely a matter of convention.

Note that the Virasoro algebra generates both a centrally extended orientation-preserving diffeomorphism group and a centrally extended orientation-preserving homeomorphism group of the circle. The difference lies in the topology chosen.