Virosoro Algebra

A virosoro algebra has various definitions:
[Ln,Lm] = (m-n)L[n+m] +(1/12)delta(m+n,0)(m^3-m)
We can write L(z) = sum Ln e^(in z)
[L(z),L(w)] = (2m-t)L[t]e^(i(t-m) z+ m w) + (1/12)(m^3-m) e^(i -m z+ m w)
[L(z),L(w)] = 2 L(z) delta'(w-z) - L'(z) delta(w-z) + (1/12) [ delta'''(w-z) - delta'(w-z)]
[L(f),L(g)] = L [ f g'- g f'] + We can write L(f) = sum(f_n Ln)?
[L[f],L[g]] = L[f g'-g f'] + (c/12) int[ f'' g' - g'' f']
Central extension $$\sqrt{1}$$