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?
Central extensions.
1 Answers
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.
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0Incidentally, it is sort of interesting how the theory of central extensions of Lie algebras parallels the analog in group theory (as group cohomology and Lie algebra cohomology parallel each other). Do you know if there's some "high concept" explanation for this? – 2012-01-09
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0Do you mean "An introduction to homological algebra" by Weibel? (a very recent book) – 2012-01-09
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0Yes. But I would not call it *very recent*... – 2012-01-09
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0@Akhil, not really; apart from the psychological explanation that the Lie algebra version is the infinitessimal part of the Lie group version. – 2012-01-09