Central extensions. Many times I've seen the term "a Lie algebra has a central extension given by" and I got used to it. However, when a Lie algebra has a central extension? Is it unique in some sense?
 A: A Lie algebra $\mathfrak g$ always has many central extensions. Indeed, one way to view central extensions simply as Lie algebra surjections $\phi:\mathfrak h\to\mathfrak g$ such that the kernel of $\phi$ is contained in the center of $\mathfrak h$.
One can always consider, for example, $\mathfrak h=\mathfrak g\oplus\mathfrak a$ with $\mathfrak a$ an arbitrary abelian Lie algebra, and the obvious map $\phi:\mathfrak h\to\mathfrak g$.
It is a standard part of the cohomology theory of Lie algebras to classify extensions of Lie algebras and, in particular, those that are central. This is explained —at least— in Hilton-Stambach's book on homological algebra and, if I recall correctly, in Weibel's book on the same subject. 
There is one special case in which there is a distinguished central extension: if $\mathfrak g$ is a perfect Lie algebra, then there is a universal central extension of $\mathfrak g$ in some sense (and this is the one used usually to construct affine algebras and friends) This is done more or less in detail in Weibel's book.
