We give a proof of the fact that the third cohomology group \(H^3_\alpha (L,M)\) of the Hom–Lie algebra \((L,\alpha _L)\) with coefficients in the Hom-L-module \((M,\alpha _M)\) is in bijection to the set \({{\,\mathrm{{\mathrm {Ext}}}\,}}^2_\alpha (L,M)\) of the equivalence classes of \(\alpha \)-crossed module extensions of \((L,\alpha _L)\) by \((M,\alpha _M)\).

中文翻译：

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第三上同调群和 $$\alpha $$ α -Hom-Lie 代数的交叉扩展

我们证明了 Hom-Lie 代数\((L,\alpha _L)\)的第三个上同调群\(H^3_\alpha (L,M)\)的系数在 Hom- L -module \((M,\alpha _M)\)与集合\({{\,\mathrm{{\mathrm {Ext}}}\,}}^2_\alpha (L,M)\ )的等价类的\(\alpha \)交叉模块扩展\((L,\alpha _L)\)由\((M,\alpha _M)\)。