I am currently reading Edward Frenkel's "Langlands Correspondence for Loop Groups", freely available here: https://math.berkeley.edu/~frenkel/loop.pdf.
In the appendix $A.4$ he describes central extensions of Lie algebras: let $\mathfrak{g}$ and $\mathfrak{a}$ be Lie algebras over the same field, with $\mathfrak{a}$ abelian. We say that $\hat{\mathfrak{g}}$ is an extension of $\mathfrak{g}$ by $\mathfrak{a}$ if there exists a short exact sequence of Lie algebras
We also require that $\mathfrak{a}$ is central in $\hat{\mathfrak{g}}$, that is $[a,x]=0$ for all $a\in \mathfrak{a}$ and $x \in \hat{\mathfrak{g}}$.
Now the part I don't understand, he says: suppose that $\mathfrak{a}$ is one dimensional, with generator $v$. As a vector space $\hat{\mathfrak{g}}$ is isomorphic to $\mathfrak{g}\oplus \mathbb{C}v$, and he declares that "the Lie bracket on $\hat{\mathfrak{g}}$ gives rise to a bracket on $\mathfrak{g}\oplus \mathbb{C}v$ such that $$[x,y]_{new}=[x,y]_{old}+c(x,y)v$$ where $x,y \in \mathfrak{g}$ and $c \colon \mathfrak{g}\otimes \mathfrak{g} \to \mathbb{C}$ is a linear map".
I can't see why the difference $[x,y]_{new}-[x,y]_{old}$ must lie in $\mathfrak{a}$, and thus be a multiple of $v$.
Can someone please enlighten me? This is not a homework question.
Thanks in advance
The value of $[x,y]_{new}$ is an element of $\mathfrak{g}⊕\mathfrak{a}$, not of $\mathfrak{g}$. It has two components. The fact that the projection $\mathfrak{g}⊕\mathfrak{a}→\mathfrak{g}$ is a Lie algebra map tells you that the first component coincides with $[−,−]_{old}$, but the second component may be nontrivial.