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Let $\mathfrak{g}$ be a finite dimensional complex Lie algebra and let $H^k(\mathfrak{g},\mathfrak{g})$ denote the $k$-th cohomology group of $\mathfrak{g}$ with coefficients in the adjoint representation.

Let $\mathfrak{h}$ and $\mathfrak{a}$ be two finite dimensional complex Lie algebras, and let $\mathfrak{g} = \mathfrak{h} \times \mathfrak{a}$. Is there any formula for $H^k(\mathfrak{g},\mathfrak{g})$ in terms of $H^p(\mathfrak{h},\mathfrak{h})$ and $H^q(\mathfrak{a},\mathfrak{a})$? I am interesting in the case when $\mathfrak{a}=\mathbb{C}$ and $\mathfrak{h}$ is a Lie algebra such that $H^p(\mathfrak{h},\mathfrak{h})=0$ for all $p\geq 0$.

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I only know Künneth formulas for the trivial module $\Bbb C$. But one could also use the Hochschild-Serre formula for the semidirect product $\mathfrak{g}=\mathfrak{h}\ltimes \mathfrak{a}$, which gives $$ H^k(\mathfrak{g},\mathfrak{g})\cong \bigoplus_{i+j=k}H^i(\mathfrak{h},\Bbb C)\otimes H^j(\mathfrak{a},\mathfrak{g})^{\mathfrak{h}}. $$

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  • $\begingroup$ Thank you so much !!! ❤️ $\endgroup$ Commented May 17 at 15:13
  • $\begingroup$ You are welcome! :) $\endgroup$ Commented May 17 at 15:45

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