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Let $R$ be a commutative ring and $\mathfrak{g}$ a Lie $R$-algebra that has an $R$-module basis with $n$ elements. What is the relationship between $H_k(\mathfrak{g};R)$, $H_{n-k}(\mathfrak{g};R)$, $H^k(\mathfrak{g};R)$, $H^{n-k}(\mathfrak{g};R)$?

In Algebra, Geometry, and Software Systems by Joswig & Takayama on p.200 it says that $H_k(\mathfrak{g};R)\cong H_{n-k}(\mathfrak{g};R)$ when $R$ is a field of characteristic $0$.

Does Poincare duality $H^k(\mathfrak{g};R)\cong H_{n-k}(\mathfrak{g};R)$ hold over any $R$ (or at least any PID such as $\mathbb{Z}$)?

Also, the literature for (co)homology of Lie algebras seems scarce, please list any book that deals with this topic. So far, I have Weibel (Homological Algebra), Azcarraga & Izquierdo (Lie Groups, Lie Algebras, Cohomology), Hilgert & Neeb (Structure and Geometry of Lie Groups).

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  • $\begingroup$ Also on MathOF. $\endgroup$ – Leon Oct 16 '14 at 13:21

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