Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra (or the corresponding Lie group). For definiteness, I'll take $\mathfrak{g}$ to be of type $A\_n$, that is, $\mathfrak{g} = \mathfrak{sl}\_{n+1}(\mathbb{C})$, but my question applies to semisimple Lie algebras of arbitrary Lie type. Consider the Dynkin diagram for $\mathfrak{g}$. We can remove a node from the diagram to obtain a sub-diagram of type $A\_{n-1}$. The sub-diagram corresponds to a copy of the Lie algebra $\mathfrak{g}':=\mathfrak{sl}\_n(\mathbb{C})$ sitting inside $\mathfrak{g}$.


The structure of the Lie algebra cohomology rings $H^\bullet(\mathfrak{g},\mathbb{C})$ and $H^\bullet(\mathfrak{g}',\mathbb{C})$ are known, and are the same as the cohomology rings $H^\bullet(G,\mathbb{C})$ and $H^\bullet(G',\mathbb{C})$ for the corresponding complex Lie group. The computation of the cohomology rings is classical; for Lie algebras the computation is a result of Koszul.


In the specific case $\mathfrak{g} = \mathfrak{sl}\_{n+1}(\mathbb{C})$, we have $H^\bullet(\mathfrak{g},\mathbb{C}) = \Lambda(x\_3,x\_5,\ldots,x\_{2n+1})$, an exterior algebra on homogeneous generators of degrees $3,5,\ldots,2n+1$. Then $H^\bullet(\mathfrak{g}',\mathbb{C}) = \Lambda(x\_3,x\_5,\ldots,x\_{2n-1})$. (For other Lie types, the cohomology ring is still an exterior algebra on homogeneous generators of certain odd degrees depending on the root system.)


The inclusion of Lie algebras $\mathfrak{g}' \rightarrow \mathfrak{g}$ gives rise to a corresponding restriction map in cohomology: $H^\bullet(\mathfrak{g},\mathbb{C}) \rightarrow H^\bullet(\mathfrak{g}',\mathbb{C})$.



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> Is the restriction map in cohomology map the "obvious'' map from $\Lambda(x\_3,x\_5,\ldots,x\_{2n+1})$ to $\Lambda(x\_3,x\_5,\ldots,x\_{2n-1})$, that is, the map that takes $x\_i$ to $x\_i$ for $1 \leq i \leq 2n-1$ and that takes $x\_{2n+1}$ to zero? If so, can you provide a reference for this fact? For other Lie types, is the restriction map also the obvious map?
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**Edit:** In response to the comments below, I think I should have phrased the question as follows:



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> **Hopefully clarified version of question:** Is there a choice of generators for the cohomology rings $H^\bullet(\mathfrak{g},\mathbb{C})$ and $H^\bullet(\mathfrak{g}',\mathbb{C})$ such that the restriction map in cohomology has the above described form?
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I acknowledge that things could get messier for restriction maps in type $D\_n$ and for types $E\_6$, $E\_7$, and $E\_8$, because in those cases the degrees of the generators for the cohomolgoy ring aren't as well-behaved. But maybe something can still be said in general about the restriction map (e.g., when is it surjective?).



Restriction map for Lie algebra/Lie group cohomology associated to a complex semisimple Lie algebra and a semisimple Lie-subalgebra
The cohomology rings of Lie groups are not just rings but also Hopf algbras with a coproduct Δ coming from the group structure. Picking primitive (Δ(x) = x⊗1 + 1⊗x) homogeneous generators gets close to giving a canonical set of generators up to scalars (though there is still some ambiguity for Dn which can have 2 generators of the same degree). 


Homomorphisms of groups preserve the coproduct, so they will more or less preserve the primitive generators.


