This was a sub-question in my previous post that I ask separately now.

In Introduction to Conformal Field Theory by Blumenhagen and Plauschinn (springer link) the Virasoro algebra is introduced the central extension of the Witt algebra. They give the central extension $$\widetilde{\mathfrak{g}} = \mathfrak{g}\oplus \mathbb{C}$$ of a Lie algebra $\mathfrak{g}$ by $\mathbb{C}$ is characterised by the following commutations:

\begin{align}[\widetilde{x},\widetilde{y}]_\widetilde{\mathfrak{g}} &= [x,y]_\mathfrak{g} + c p(x,y),\tag{2.14a}\\ [\widetilde{x}, c]_\widetilde{\mathfrak{g}} &= 0,\tag{2.14b}\\ [c,c]_\widetilde{\mathfrak{g}} &= 0,\tag{2.14c}\end{align}

with $\widetilde{x},\widetilde{y}\in \widetilde{\mathfrak{g}}$, $x,y\in\mathfrak{g}$, $c \in \mathbb{C}$ and $p:\mathfrak{g} \times \mathfrak{g} \rightarrow \mathbb{C}$.

Later on, they write the elements of the central extension as $L_n$, and write the Lie bracket as:

$$[L_m,L_n] = (m-n)L_{m+n} + cp(m,n)\tag{2.15}$$

In the previous section they derived the Witt algebra commutator as: $$[l_m,l_n] = (m-n)l_{m+n}.\tag{2.12}$$

I don't understand why they can write $(m-n)L_{m+n}$ instead of $(m-n)l_{m+n}$ in (2.15). Doesn't (2.14a) tell us that $[x,y]$ is a Lie bracket in $\mathfrak{g}$, while $L$ is in $\widetilde{\mathfrak{g}}$?

  • $\begingroup$ Following your previous post, are you sure it is not a problem of notation?: If $(2.15)$ was written $[\widetilde{L}_m,\widetilde{L}_n] = (m-n)L_{m+n} + cp(m,n)$ then it follows immediately from $(2.12)$ and $(2.14\text{a})$. Otherwise your thought looks correct. $\endgroup$ – Quantumness Jun 19 at 3:28

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