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For a perfect Lie algebra $L$ over $C,$ the kernel of its universal central extension is isomorphic to $H_2(L,C),$ and its central extensions are in 1-1 correspondence to $H^2(L,C).$

Question (1): Can we know one of the $H_2$ or $H^2$ from the another directly?

I find in some book, they compute both of them.

In textbooks, the universal central extension ($\tilde{L}$, $\alpha$) of $L$ is defined as:

(a) $\tilde{L}$ is a central extension of $L$

(b) for a central extension $(L',\beta)$, there exists a unique $\gamma: \tilde{L}\rightarrow L'$, such that $\alpha=\beta\gamma$.

Question (2): In condition (b), can $L'$ be replaced by a "one dimensional central extension"?

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    $\begingroup$ The universal coefficient formula says that they are dual to each other. $\endgroup$ Commented Mar 16, 2011 at 5:09
  • $\begingroup$ In infinite dimension it's not possible to be dual to each other (since the bidual is bigger than the predual) $\endgroup$
    – YCor
    Commented Jun 18, 2016 at 18:15

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