关于 Lie--Rinehart 代数的 Schouten--Nijenhuis 括号的一个注记
A Note on the Schouten--Nijenhuis Bracket of Lie--Rinehart Algebras
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中文总结 AI 辅助
本文从分级微分角度重新考察 Lie--Rinehart 代数上的 Schouten--Nijenhuis 括号，证明其良定义性、相关恒等式及 Gerstenhaber 括号与 Lie--Rinehart 结构的一一对应。
中文摘要 AI 辅助
设 $A$ 是特征为零的域 $K$ 上的交换结合单位代数，并设 $(G,\rho)$ 是 $A$ 上的 Lie--Rinehart 代数。我们从分级微分的观点重新审视外代数 $\bigwedge_A G$ 所承载的 Gerstenhaber 结构。特别关注 Schouten--Nijenhuis 扩张在系数代数 $A$ 上的良定义性：证明 Lie--Rinehart 恒等式恰好提供了使括号下降到 $A$-平衡外代数所需的修正项。在无循环地建立分级反对称性、Leibniz 法则和分级 Jacobi 恒等式之后，我们研究内分级微分 $\adSN(P)=[P,\cdot]_{SN}$ 并证明 \\[ [\adSN(P),\adSN(Q)]=\adSN([P,Q]_{SN})。\\] 最后，我们给出逆构造的完整证明：$\bigwedge_A G$ 上的 Gerstenhaber 括号与 $G$ 上的 Lie--Rinehart 结构一一对应。
英文摘要
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 $G$.
