I am working on the following task:
Let $A$ be an associative algebra and $B$ a basis of $A$. Let $L(B)$ be the free Lie algebra over $A$. Show: There exists a canonical algebra homomorphism $\varphi:U(L(B))→A$. What can we say about the homomorphism when $A$ is commutative?
Initially, we have the inclusion mapping $\phi:B\to A$. In accordance with the definition, a homomorphism $\psi:L(B)\to A$ with $\psi(b)=b$ for every $b\in B$ exists. It follows, that a algebra homomorphism $\varphi:U(L(B))\to A$ exists.
I understand the existence of the homomorphism; however, I am not aware of the additional statement that can be made when $A$ is abelian. Can we make a statement about $[U(L(B)),U(L(B))]?$
Any help is greatly appreciated!