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Given that I have an infinite basis for a Lie algebra $L$, and the information that $M$ is its Universal Central Extension, is $M$ unique? If so, what is the basis of $M$ in terms of that of $L$?

Vaibhav Karve
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  • Uniqueness follow from the universal property as usual. This is proved in every textbook which treats the subject. – Mariano Suárez-Álvarez Jul 21 '14 at 23:56
  • Af for what is a basis of the universal extension, well... it depends on the Lie álgebra! I cannot imagine what sort of answer you expect here. – Mariano Suárez-Álvarez Jul 21 '14 at 23:57
  • Oh! I did not know that since I am largely ignorant of anything Category theoretic. Can we say something about the basis of $M$ then? – Vaibhav Karve Jul 21 '14 at 23:59
  • I have a basis for $L$, so I know all the required Lie brackets. But when I look at $M$, there will be an additional central term right? To rephrase my question -- does the "universality" of $M$ as a central extension of $L$ impose any sort of condition on the cocyle that will appear as the coefficient of the central term? – Vaibhav Karve Jul 22 '14 at 00:02
  • The central extensión is completely determined in terms of the cohomology of the álgebra. You could do worse than read the relevant section in Weibel's book on homological algebra. – Mariano Suárez-Álvarez Jul 22 '14 at 00:23
  • In particular, there may be one or several central terms in the extension, and they are described by cohomology. – Mariano Suárez-Álvarez Jul 22 '14 at 07:48
  • Ok @MarianoSuárez-Alvarez. Thanks for your help. If you post this as an answer, I will happy to tag this question as solved! – Vaibhav Karve Jul 22 '14 at 15:32

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