What is known about the morphism $H^*_{Lie}(L,L)\to H^*_{Lie}(L,UL)$ induced by $L\hookrightarrow UL$

Question

Let $L$ be a (differential) graded Lie algebra over a field $k$ of characteristic 0, and let $UL$ be the universal enveloping algebra of $L$.

The inclusion $L\hookrightarrow UL$ induces a morphism of the Lie algebra cohomology $H^*_{Lie}(L,L)\to H^*_{Lie}(L,UL)$. Can one deduce any properties of this map (like injectivity, surjectivity, etc...)?

Answer 1

The symmetrization mapping $\sigma$ from the symmetric algebra $S(L)$ to the universal enveloping algebra $U(L)$ is an isomorphism of $L$-modules. Since $L$ is a direct summand of $S(L)$, its isomorphic image $\sigma(L)$ is a direct summand of $U(L)$. Thus the same direct summand property holds for their Lie algebra cohomology. In particular, the induced map on the Lie algebra cohomology is injective.