5 Virasoro algebra
Let \(\mathbb {K}\) be a field of characteristic zero. The Virasoro algebra \(\mathfrak {vir}\) is the Lie algebra over \(\mathbb {K}\) obtained as the central extension of the Witt algebra \(\mathfrak {witt}\) corresponding to the Virasoro cocycle \({\gamma }_{\mathfrak {vir}} \in C^2(\mathfrak {witt},\mathbb {K})\).
From the definition of the Virasoro algebra and the Virasoro cocycle, Definition 24 and 17, we directly obtain that \(\mathfrak {vir}\) has a basis of the following form.
The Virasoro algebra \(\mathfrak {vir}\) has a basis consisting of \((L_n)_{n \in \mathbb {Z}}\) and \(C\), with Lie brackets determined by the following
\begin{align*} [L_n, L_m] = \; & (n-m) \, L_{n+m} + \delta _{n+m,0} \frac{n^3 - n}{12} \, C , \end{align*}
\begin{align*} [C, L_n] = 0 , \qquad [C, C] = 0 , \end{align*}
for \(n,m \in \mathbb {Z}\).