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I tried to guess what is a cohomology group of an algebra. I would like to find the correct definition of this. I know what is a cohomology group of a group, but I don't know how connect the second cohomology group to the central extension of a Lie-algebra. Could someone help me to tell the definition or explain to me this? Thanks!

Alíz
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The notion that you probably want is that of Hochschild (co)homology. This is a way of getting (co)homology groups for associative algebras, but it actually specializes to group cohomology and Lie algebra cohomology. The reason this is possible is because modules over a finite group $G$ are the same thing as modules over the group algebra $k[G]$, and similarly, modules over a Lie algebra $\mathfrak g$ are the same thing as modules over the universal enveloping algebra $\mathcal U\mathfrak g$. It turns out that (co)homology groups really only care about the category of modules, and so the framework of associative algebras subsumes that of finite groups and modules or lie algebras and their modules.

Of course, while this general framework gives you a unifying picture for everything, don't forget that you might have intepretations of cohomology groups in special cases that don't generalize.

Aaron
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    In fact, no. If the OP is interested in cohomology of Lie algebras Hochschild cohomology is probably not what he is after but Lie algebra cohomology (which is certainly related to Hochschild cohomology but not quite the same) – Mariano Suárez-Álvarez Mar 15 '15 at 19:26
  • Apologies, I hadn't read the comment where OP clarified algebra to mean Lie algebra. I still think that anybody dealing with algebraic cohomology theories should learn about Hochschild cohomology, though, so I will leave my answer. – Aaron Mar 15 '15 at 19:29
  • (Hochschild cohomology can be computed of a group algebra and of a Lie algebra, but in neither case the result coincides with the group cohomology of the group or the Lie algebra cohomology of the Lie algbra.) – Mariano Suárez-Álvarez Mar 15 '15 at 19:29
  • That's not true. If I remember the results correctly, you have an isomorphism if, for example, you view a group representation as a bi-module with the induced structrue on the left and the trivial structure on the right. – Aaron Mar 15 '15 at 19:31
  • Yes, for specific choices you can do that. But Hochschild cohomology of both group algebras and of enveloping algebras of Lie algebras is distinctly different from group cohomology and Li algebra cohomology as a theory. Trust me, I find Hochschild cohomology very much pleasing (it is the subject I work on, in fact) but to the OP going through the route of Hochschild cohomology would be a detour too long to make much sense. – Mariano Suárez-Álvarez Mar 15 '15 at 19:36
  • For one thing, the concrete interpretations of low degree groups —like the second one in terms of central extensions of Lie algebras and such— are going to be rather painful in terms of Hochschild cohomology. – Mariano Suárez-Álvarez Mar 15 '15 at 19:40
  • I agree that at a superficial level, the theories are different (they are computed using different standard complexes), and so the initial interpretations of low dimensional groups will be different, but all of this is just Ext and Tor anyway, and I personally find it comforting when two similar looking theories can both be subsumed into a larger unifying theory. – Aaron Mar 15 '15 at 19:45
  • It's all Exts and Tors, sure —that is the point of the book of Cartan-Eilenberg— and you could frame the whole thing in terms of cohomology of operads, if you want unification, or even more elaborate frameworks. Almost none of all that, though, will likely help the person who asked the question you are answering. – Mariano Suárez-Álvarez Mar 15 '15 at 19:49
  • Notice that it is not that the difference lies in that the theories use different standard complexes (indeed, the other point of Cartan and Eilenberg is that there is no need to worry about standard complexes at all...) but that they are Exts and Tors in different categories. – Mariano Suárez-Álvarez Mar 15 '15 at 19:54
  • Thank you for your answers, but I didn't deal with cohomology theories before, I just wanted to know specially what is the connection the Witt-algebra and the Virasoro-algebra through the second chomology group. – Alíz Mar 15 '15 at 19:55
  • @HajnalkaKorka, then do go read Weibel: he does not do the example of the Witt algebra to construct Virasoro, but the one used to construct affine Lie algebras, which is exactly analogous. – Mariano Suárez-Álvarez Mar 15 '15 at 19:57