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Let $\mathfrak{g}$ be a solvable lie algebra and $\lambda\in (\mathfrak{g}/[\mathfrak{g},\mathfrak{g}])^*$ be the character of $\mathfrak{g}$. How to compute homology for $\mathbb{C}_\lambda$, the simple $\mathfrak{g}-$module with character $\lambda$?

I have faced strange behavior of homology, while calculating them manually for 2 and 3-dimensional algebras and cannot solve it in full generality. Thanks in advance!

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