10.8. Central extension Consider commutation relations given by

[Ln,Lm] = (n−m)Ln+m +f (n,m)C, where f (n,m)=−f (m, n) and [C,Ln] = 0.

a. Show that these commutation relations satisfy the Jacobi identities if and only if f (n,m) = ηn(n2 −1)δn+m,0 +(n−m)λ(n+m)

where η is a constant and λ(n) a function.

b. Show that the second term in f (n,m) can be absorbed by a redefinition of the generators Ln, i.e. Ln→Ln −Cg(n).