A Note on the Schouten--Nijenhuis Bracket of Lie--Rinehart Algebras

Abstract

Let $A$ be a commutative associative unital algebra over a field $K$ of characteristic zero, and let $(G,ρ)$ be a Lie--Rinehart algebra over $A$. We revisit the Gerstenhaber structure carried by the exterior algebra $\bigwedge_A G$ from the point of view of graded derivations. Particular attention is paid to the well-definedness of the Schouten--Nijenhuis extension over the coefficient algebra $A$: the Lie--Rinehart identity is shown to provide exactly the correction terms needed for the bracket to descend to the $A$-balanced exterior algebra. After establishing graded antisymmetry, the Leibniz rule and the graded Jacobi identity without circularity, we study the inner graded derivations $\adSN(P)=[P,\cdot]_{SN}$ and prove \[ [\adSN(P),\adSN(Q)]=\adSN([P,Q]_{SN}). \] Finally, we give a complete proof of the converse construction: Gerstenhaber brackets on $\bigwedge_A G$ are in one-to-one correspondence with Lie--Rinehart structures on $G$.