1 Introduction

Definition 2.3 .

For $p\in Q$ and $\eta\in\operatorname{dom}(p)$ we let

p^{\langle\eta\rangle}=p\restriction\{\rho\in\operatorname{dom}(p)\,:\,\eta% \subseteq\rho\}.

Definition 2.1 (coalgebra, coassociativity)

Let $C$ be (generally infinite dimensional) graded vector space. When a coproduct $\triangle:C\longrightarrow C\otimes C$ is defined on $C$ and it is coassociative , i.e.

(\triangle\otimes{\bf 1})\triangle=({\bf 1}\otimes\triangle)\triangle

then $C$ is called a coalgebra .

Definition 2.2 (coderivation)

A linear operator ${\mathfrak{m}}:C\rightarrow C$ raising the degree of $C$ by one is called coderivation when

\triangle{\mathfrak{m}}=({\mathfrak{m}}\otimes{\bf 1})\triangle+({\bf 1}% \otimes{\mathfrak{m}})\triangle

is satisfied. Here, for $x,y\in C$ , the sign is defined $({\bf 1}\otimes{\mathfrak{m}})(x\otimes y)=(-1)^{x}(x\otimes{\mathfrak{m}}(y))$ through this operation where the $x$ on $(-1)$ denotes the degree of $x$ .

Definition 2.4 ( $A_{\infty}$ -algebra)

Let ${\cal H}$ be a graded vector space and $C({\cal H})=\oplus_{k\geq 1}{{\cal H}^{\otimes k}}$ be its tensor coalgebra. An $A_{\infty}$ -algebra is a coalgebra $C({\cal H})$ with a coderivation ${\mathfrak{m}}$ which satisfies

({\mathfrak{m}})^{2}=0\ .

Definition 3.4 .

Let R be a commutative ring. An isotropy ring $\cal I=I^{{\mathrm{R},}O^{\prime}}_{\mathrm{D}}$ is generated by ${\mathrm{m}or}(h\cal O^{\prime})$ as a free $R-\textup{mod}$ . Define the multiplication on the generators by

fg=\left\{\begin{array}[]{rl}f\mathbin{\circ}g,&\textup{if\;}\mathrm{codom}({g% })=\mathrm{dom}({f})\\ 0,&\textup{otherwise}\end{array}\right.

and extend the definition to the rest of the elements of $\cal I$ by linearity.

Definition 2.3 .

Definition 2.1 (coalgebra, coassociativity)

Definition 2.2 (coderivation)

Definition 2.4 ( A ∞ subscript 𝐴 A_{\infty} italic_A start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT -algebra)

Definition 3.4 .

Definition 2.4 ( $A_{\infty}$ -algebra)