# Adjoint cohomology of Lie algebra commutes with direct sum?

The Witt algebra (denoted by $W$) is an infinite dimensional Lie algebra as:

$[L_{m},L_{n}]=(m-n)L_{m+n}; \,\,\,\ m, n\in \mathbb{Z}$.

I am looking for second adjoint cohomology $H^{2}(W_{1}\oplus W_{2};W_{1}\oplus W_{2})$ and the question is: Is there any theorem about decomposition of this formula? For instance, is it possible to decompose this relation so that $$H^{2}(W_{1}\oplus W_{2};W_{1}\oplus W_{2})$$ $$=H^{2}(W_{1};W_{1})\oplus H^{2}(W_{1};W_{2})\oplus H^{2}(W_{2};W_{1})\oplus H^{2}(W_{2};W_{2})\;?$$

The commutation relations of $W_{1}\oplus W_{2}$ are

$$[L_{m},L_{n}]=(m-n)L_{m+n};$$

$$[L_{m},M_{n}]=0;$$

$$[M_{m},M{n}]=(m-n)M_{m+n}.$$

• Yes I think so. One should see this with an explicit resolution, or more abstractly like this: Let A be the category of W modules. The direct sum is a functor from A x A to A which is exact and maps injectives to injectives. Now apply Grothendieck's spectral sequence. – user1688 Jan 14 '18 at 16:16
• Does it also work for infinite dimensional Lie algebras such as $W$? – Hamidreza Safari Jan 15 '18 at 9:53
• Yes, why not. The decisive point is that the direct sum is also the categorical direct product. Then injectivity survives and the argument applies. – user1688 Jan 15 '18 at 11:37