Hochschild cochains, differential, and Gerstenhaber insertion bracket

Let $A$ be a unital associative real algebra. For each integer $p\ge 1$, the Hochschild $p$-cochains are $C^p(A,A):={\mathrm{Hom}}_{\mathbb{R}}(A^{\otimes p},A)\mathrm{.}$ The Hochschild differential $\delta :C^p\to C^{p+1}$ is $\begin{array}{rl}{\displaystyle (\delta \varphi )(a_1,\dots ,a_{p+1})=}& {\displaystyle a_1\varphi (a_2,\dots ,a_{p+1})}\\ {\displaystyle }& {\displaystyle +\sum _{i=1}^p(-1{)}^i\varphi (a_1,\dots ,a_{i-1},a_i{a}_{i+1},a_{i+2},\dots ,a_{p+1})}\\ {\displaystyle }& {\displaystyle +(-1{)}^{p+1}\varphi (a_1,\dots ,a_p)a_{p+1}\mathrm{.}}\end{array}$ The general cochain-complex convention is degree $+1$ with square-zero differential (Cochain complex in an abelian category); the proposition The shifted Hochschild cochains form a differential graded Lie algebra ↗ proves that this formula has square zero.

A cochain is normalized if it vanishes whenever any input is $1$. If $A=C^{\mathrm{\infty }}(M,\mathbb{R})$ for a smooth manifold $M$, a cochain is multidifferential if in every coordinate chart it is locally a finite sum of terms $c_{{\alpha }_1,\dots ,{\alpha }_p}(x){\prod }_{j=1}^p{\mathrm{\partial }}^{{\alpha }_j}{f}_j(x),$ with smooth coefficients and a finite order bound in each input. The normalized multidifferential cochains are the subcomplex used for differential star-product obstructions; closure under $\delta$ and insertion is proved in the cited proposition. For $r\ge 2$, $H{H}_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f},\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}^r(A,A)$ denotes the kernel of $\delta :C_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f},\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}^r\to C_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f},\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}^{r+1}$ modulo the image from degree $r-1$.

For $\varphi \in C^p(A,A)$ and $\psi \in C^q(A,A)$, define insertion $(\varphi \circ \psi )(a_1,\dots ,a_{p+q-1})={\sum }_{i=0}^{p-1}(-1{)}^{(q-1)i}\varphi (a_1,\dots ,a_i,\psi (a_{i+1},\dots ,a_{i+q}),a_{i+q+1},\dots ,a_{p+q-1}),$ and the Gerstenhaber bracket by $[\varphi ,\psi {]}_G:=\varphi \circ \psi -(-1{)}^{(p-1)(q-1)}\psi \circ \varphi \mathrm{.}$ The shifted degree of a $p$-cochain is $p-1$. These formulas fix all insertion signs used in the local deformation equations below.