Let $L=L(V)$ be a free Lie algebra on a vector space $V$ and $A=T(V)$ the tensor algebra. Make $L$ into a module over $A$ consistent with the formula $a\cdot \alpha=[a,\alpha]$ for $a\in V$ and $\alpha\in L$.

What is a canonical resolution of $L$ by free $A$ modules? I'm really most interested in the case where there is a grading and a differential.

Let $L=L(V)$ be a free Lie algebra on a vector space $V$ and $A=T(V)$ the tensor algebra. Make $L$ into a module over $A$ consistent with the formula $a\cdot \alpha=[a,\alpha]$ for $a\in V$ and $\alpha\in L$.

What is a canonical resolution of $L$ by free $A$ modules? I'm really most interested in the case where there is a grading and a differential.

Edited:

After thinking I realize there is the bar construction $B(A,A,L)$. Is there anything smaller in this special case?