A∞, L∞, and Deformation Complexes

Let $k$ be a field (Field), and let $V_1,\dots ,V_n$ be graded $k$-vector spaces. A $k$-vector space is a unital left $k$-module as in Unital left and right modules over a ring; unqualified module means left module. Tensor powers are formed using the tensor products of Tensor algebra of a vector space. If $f_i:V_i\to W_i$ are homogeneous linear maps of degrees $d_i$, and $x_i\in V_i$ are homogeneous of degrees $p_i$, the graded tensor map is defined on pure tensors by

$(f_1\otimes \cdots \otimes f_n)(x_1\otimes \cdots \otimes x_n)=(-1{)}^{{\sum }_{i>j}{d}_i{p}_j}{f}_1(x_1)\otimes \cdots \otimes f_n(x_n)\mathrm{.}$

It is extended linearly. In particular, for two maps, $(f\otimes g)(x\otimes y)=(-1{)}^{\mathrm{\mid }g\mathrm{\mid }\mathrm{\mid }x\mathrm{\mid }}f(x)\otimes g(y)$.

For homogeneous $x_1,\dots ,x_n$ and a permutation $\sigma \in S_n$, the Koszul sign of reordering them to $x_{\sigma (1)}\otimes \cdots \otimes x_{\sigma (n)}$ is

$ϵ(\sigma ;x_1,\dots ,x_n)=(-1{)}^{{\sum }_{\begin{array}{c}i<j\\ \sigma (i)>\sigma (j)\end{array}}\mathrm{\mid }{x}_{\sigma (i)}\mathrm{\mid }\mathrm{\mid }{x}_{\sigma (j)}\mathrm{\mid }}\mathrm{.}$

Thus swapping two adjacent homogeneous factors $x,y$ contributes $(-1{)}^{\mathrm{\mid }x\mathrm{\mid }\mathrm{\mid }y\mathrm{\mid }}$. Every tensor of maps, permutation, unshuffle, and symmetric power below uses this same convention. A sign is a scalar in $k$ determined only by the parity of the displayed integer exponent.

Let $(A,d,\mu )$ be a cohomological differential graded algebra over a commutative ring, with the degree-one differential and signed Leibniz rule of Cohomological differential graded algebra. For $j\in \mathbb{Z}$, define

$H^j(A)=\ker (d:A^j\to A^{j+1})\mathrm{/}\mathrm{im}(d:A^{j-1}\to A^j)\mathrm{.}$

A triple Massey datum consists of homogeneous closed elements $a\in A^p$, $b\in A^q$, and $c\in A^r$ such that $ab$ and $bc$ are boundaries, together with elements

$u\in A^{p+q-1},v\in A^{q+r-1},du=ab,dv=bc\mathrm{.}$

The associated triple Massey expression is

$\omega (a,b,c;u,v)=(-1{)}^{p+1}av+uc\in A^{p+q+r-1}\mathrm{.}$

This item defines the data and expression only. The next lemma proves that the expression is closed, so its cohomology class is defined.

Let $k$ be a field (Field) and let $A={⨁}_{j\in \mathbb{Z}}{A}^j$ be a $\mathbb{Z}$-graded $k$-vector space; its tensor maps use the convention in Koszul sign rule for graded tensor products and permutations. An uncurved cohomological A∞-algebra in the unsuspended convention is a sequence of homogeneous $k$-linear maps

$m_n:A^{{\otimes }_{k}n}⟶A,\mathrm{\mid }{m}_n\mathrm{\mid }=2-n,n\ge 1,$

such that for every $n\ge 1$ the following equality of maps $A^{{\otimes }_{k}n}\to A$ holds:

${\sum }_{\begin{array}{c}r,t\ge 0,s\ge 1\\ r+s+t=n\end{array}}(-1{)}^{r+st}{m}_{r+1+t}(1^{\otimes r}\otimes m_s\otimes 1^{\otimes t})=0.$

Here $1$ denotes the identity map of $A$, the tensor maps use the Koszul rule, and the displayed sign $(-1{)}^{r+st}$ is part of the fixed convention. The degree condition means that for homogeneous inputs of degrees $j_1,\dots ,j_n$, $m_n$ has output degree $j_1+\cdots +j_n+2-n$.

“Uncurved” means no arity-zero operation $m_0$ is included. “Nonunital” means that no unit is part of this definition. A unit or a strict-unit condition on higher operations is additional data and must be stated separately. No graded commutativity is assumed for $m_2$.

The arity-two identity is the chain-map condition for $m_2$. Its commutative square is shown below; the downward map on $A^{\otimes 2}$ is the tensor differential with the Koszul sign rule. This diagram records the equation only; it is not a proof of any Stasheff identity.

For every triple Massey datum in Triple Massey data in a cohomological differential graded algebra, the expression $\omega =(-1{)}^{p+1}av+uc$ is closed. Thus it determines a class in $H^{p+q+r-1}(A)$.

Let $k$ be a characteristic-zero field (Field), meaning $\mathrm{char}(k)=0$ under The characteristic of a ring: the least $n\ge 1$ with $n\cdot 1_R=0$ when one exists, and $0$ otherwise. By Every field is a commutative ring with $1\mathrm{\ne }0$; it is an integral domain, and it is a commutative division ring, $k$ is a commutative ring. Let $W={⨁}_{j\in \mathbb{Z}}{W}^j$ be a graded $k$-vector space. For $n\ge 1$, define the graded symmetric power

$S^n(W)=W^{{\otimes }_{k}n}\mathrm{/}{J}_n,$

where $J_n$ is spanned by the differences obtained by replacing any adjacent homogeneous factors $x\otimes y$ by $(-1{)}^{\mathrm{\mid }x\mathrm{\mid }\mathrm{\mid }y\mathrm{\mid }}y\otimes x$ inside a pure tensor. Write $w_1\odot \cdots \odot w_n$ for the image of a tensor and set

$S^c(W)={⨁}_{n\ge 1}{S}^n(W)\mathrm{.}$

The reduced shuffle coproduct is the degree-zero linear map $\bar{\mathrm{\Delta }}:S^c(W)\to S^c(W){\otimes }_k{S}^c(W)$ given on a homogeneous pure symmetric tensor by

$\bar{\mathrm{\Delta }}(w_1\odot \cdots \odot w_n)={\sum }_{p=1}^{n-1}{\sum }_{\sigma \in \mathrm{Sh}(p,n-p)}ϵ(\sigma ;w_1,\dots ,w_n)(w_{\sigma (1)}\odot \cdots \odot w_{\sigma (p)})\otimes (w_{\sigma (p+1)}\odot \cdots \odot w_{\sigma (n)}),$

where $\mathrm{Sh}(p,n-p)=\{\sigma \in S_n:\sigma (1)<\cdots <\sigma (p),\sigma (p+1)<\cdots <\sigma (n)\}$, and $ϵ$ is the Koszul sign of Koszul sign rule for graded tensor products and permutations. For $n=1$ the sum is empty and $\bar{\mathrm{\Delta }}(w)=0$. This is a reduced, hence noncounital coalgebra convention; no arity-zero summand is included. The next lemmas verify its coproduct laws, conilpotence, and cofree universal property.

Let $k$ be a characteristic-zero field, let $B$ be a commutative unital $k$-algebra concentrated in degree zero, and let $I$ be an ideal such that . Nilpotent ideals are included. Let $V$ be any graded $k$-vector space; no finite-dimensionality assumption is made. Use the graded tensor convention of Koszul sign rule for graded tensor products and permutations.

The carrier $C$ is either the coaugmented tensor coalgebra $T^c(V)={⨁}_{r\ge 0}{V}^{\otimes r}$ with deconcatenation, or the coaugmented graded symmetric coalgebra $S^c(V)={⨁}_{r\ge 0}{S}^r(V)$ with unshuffle coproduct. For the latter, $S^r$ means graded symmetric coinvariants, and the coproduct sums over all divisions of the inputs into two ordered sublists, with the Koszul sign. The empty word is $1_C$, the counit $ϵ$ is projection to length zero, and the coaugmentation is inclusion of scalars. These formulas specify the carrier without requiring a cofree universal property.

Define its coefficient completion, filtration, and tensor target by

All inverse limits in this definition are in graded modules: explicitly and $C_B={⨁}_j(C_B{)}^j$, and the tensor target is completed in each fixed total degree. All structure maps are extended degreewise through these inverse limits. This is coefficient completion of a direct sum: modulo each $I^n$ every element has finite length support. No unrestricted product over arities is included. Completed tensor powers and polynomial coefficients mean the analogous explicit inverse limits. A homogeneous map is understood in the grading induced by $C$, and continuity always means the displayed coefficient topology.

Let $E^d$ consist of continuous $B$-linear filtration-preserving homogeneous endomorphisms of degree $d$ on $C_B$, and set $E={⨁}_d{E}^d$, with composition $E^d{E}^e\subseteq E^{d+e}$. Completeness of this graded composition algebra always means completeness of each $E^d$ separately, not of its ungraded direct sum. All exponential and logarithm series below have inputs in $E^0$ and remain in $E^0$; their conjugation series on a homogeneous $T\in E^d$ remain in that same fixed $E^d$. Its order filtration is

$F^{p}E=\{T:T(F^q{C}_B)\subseteq F^{p+q}{C}_B\text{ for every }q\ge 0\}(p\ge 0)\mathrm{.}$

Let $\mathfrak{l}\subset E$ be the homogeneous coderivations satisfying $T(1_C)=0$, $ϵT=0$, and

$\mathrm{\Delta }T=(T\otimes 1+1\otimes T)\mathrm{\Delta }\mathrm{.}$

The sign in $1\otimes T$ is the graded tensor sign. In particular these coderivations have zero nullary component. Fix $D\in {\mathfrak{l}}^1$ with $D^2=0$. A coefficient-complete fixed-carrier codifferential deformation is $D+R$ with $R\in F^1{\mathfrak{l}}^1$ and $(D+R{)}^2=0$. Its allowed equivalences are degree-zero continuous $B$-linear counital coaugmented coalgebra automorphisms $G$ such that $G-\mathrm{id}\in F^{1}E$, acting by $D+R\mapsto G(D+R)G^{-1}$.

Here identity modulo $I$ can equivalently be stated as $(G-\mathrm{id})(C_B)\subseteq F^1{C}_B$. To see the filtration implication without assuming that $I$ is finitely generated, observe that $F^q{C}_B$ is the closure of $C\otimes I^q$: modulo $F^n$, $n>q$, every element lifts from $C\otimes I^q$. By $B$-linearity a continuous map with image in $F^1$ carries that dense submodule into $F^{q+1}$; closedness gives the same inclusion on all $F^q$.

This contract uses all continuous coefficient-complete Taylor operations. It does not identify them with a naive algebraic tensor ${\mathrm{Hom}}_k(W,U)\otimes I$ when the coefficient algebra is infinite dimensional. Over a finite-dimensional coefficient algebra, expanding in a finite coefficient basis does give the usual tensor identification, without any finite-dimensionality assumption on $W,U$.

Let $(A,m_n)$ be an uncurved cohomological A∞-algebra with the unsuspended convention of Uncurved cohomological A∞-algebra in the unsuspended convention. Its first three identities are

$m_1{m}_1=0,$

$m_1{m}_2=m_2(m_1\otimes 1+1\otimes m_1),$

and

$m_1{m}_3+m_3(m_1\otimes 1\otimes 1+1\otimes m_1\otimes 1+1\otimes 1\otimes m_1)+m_2(m_2\otimes 1)-m_2(1\otimes m_2)=0.$

In particular, $m_1$ is a cochain differential and a signed derivation of $m_2$. The third identity explicitly gives the associator $m_2(m_2\otimes 1)-m_2(1\otimes m_2)$ as the negative of the displayed $m_1,m_3$ terms.

Let $A$ and $B$ be uncurved cohomological A∞-algebras in the convention of Uncurved cohomological A∞-algebra in the unsuspended convention. An A∞-morphism $f:A\to B$ is a family of homogeneous maps

$f_n:A^{{\otimes }_{k}n}⟶B,\mathrm{\mid }{f}_n\mathrm{\mid }=1-n,n\ge 1,$

such that, for every $n\ge 1$, the following equality of maps $A^{{\otimes }_{k}n}\to B$ holds:

${\sum }_{\begin{array}{c}r,s,t\ge 0,s\ge 1\\ r+s+t=n\end{array}}(-1{)}^{r+st}{f}_{r+1+t}(1^{\otimes r}\otimes m_s^A\otimes 1^{\otimes t})={\sum }_{q=1}^n{\sum }_{\begin{array}{c}{i}_1+\cdots +i_q=n\\ i_j\ge 1\end{array}}(-1{)}^{\eta (i_1,\dots ,i_q)}{m}_q^B(f_{i_1}\otimes \cdots \otimes f_{i_q}),$

where $\eta (i_1,\dots ,i_q)={\sum }_{j=1}^{q-1}(q-j)(i_j-1)$ and all tensor maps use Koszul sign rule for graded tensor products and permutations. The identity at $n=1$ says that $f_1$ is a cochain map from $(A,m_1^A)$ to $(B,m_1^B)$.

An A∞-morphism is an A∞ quasi-isomorphism if $f_1$ induces an isomorphism on cohomology, where $H^j(A,m_1^A)=\ker (m_1^A:A^j\to A^{j+1})\mathrm{/}\mathrm{im}(m_1^A:A^{j-1}\to A^j)$ and likewise for $B$. This definition imposes no unit condition on $f$.

Let $A$ and its cohomology groups be as in Triple Massey data in a cohomological differential graded algebra, and let $\alpha \in H^p(A)$, $\beta \in H^q(A)$ and $\gamma \in H^r(A)$ be cohomology classes. Their triple Massey product in that item's sign convention is the subset

$⟨\alpha ,\beta ,\gamma ⟩=\{[\omega (a,b,c;u,v)]\in H^{p+q+r-1}(A):(a,b,c;u,v)\text{ is a triple Massey datum representing }(\alpha ,\beta ,\gamma )\}\mathrm{.}$

The expression is a cohomology class because it is closed by The triple Massey expression is a cocycle. The set includes every choice of cocycle representatives and null-homotopies for which the adjacent products are boundaries; if there is no such datum, it is empty. No claim that the set is a singleton is part of the definition.

Let $S^c(W)$ and $\bar{\mathrm{\Delta }}$ be as in Reduced graded symmetric coalgebra with its shuffle coproduct. The reduced coproduct is well-defined on the graded-symmetric quotient, is coassociative and graded-cocommutative, and is conilpotent: every element is killed by a sufficiently long iterated reduced coproduct. Its primitive subspace ($\ker \bar{\mathrm{\Delta }}$) is exactly $S^1(W)=W$.

Use Coefficient-complete fixed-carrier codifferential deformation. For every fixed degree $d$, the order filtration on $E^d$ is separated and complete, with ; thus $E$ is complete as a graded algebra in the degreewise sense, with $F^{p}E{F}^{q}E\subseteq F^{p+q}E$. The coderivations form a closed graded Lie subalgebra for

$[P,Q]=PQ-(-1{)}^{\mathrm{\mid }P\mathrm{\mid }\mathrm{\mid }Q\mathrm{\mid }}QP\mathrm{.}$

With $dT=[D,T]$, the positive-order space $\mathfrak{g}=F^1\mathfrak{l}$ is a complete filtered differential graded Lie algebra in this degreewise sense, with $F^p\mathfrak{g}=F^p\mathfrak{l}$ for $p\ge 1$, $d(F^p\mathfrak{g})\subseteq F^p\mathfrak{g}$, and $[F^p\mathfrak{g},F^q\mathfrak{g}]\subseteq F^{p+q}\mathfrak{g}$.

Let $E$ be any unital associative $k$-algebra over a characteristic-zero field, with descending two-sided filtration $F^{0}E=E$, $F^{p}E{F}^{q}E\subseteq F^{p+q}E$, and . Commutativity is not assumed. The formulas

$\exp X={\sum }_{j\ge 0}\frac{X^j}{j!},\log (1+N)={\sum }_{j\ge 1}\frac{(-1{)}^{j-1}{N}^j}{j}$

give mutually inverse bijections $F^{1}E\leftrightarrow 1+F^{1}E$, and $\exp (-X)$ is the inverse of $\exp X$. For $X\in F^{1}E$ and $T\in E$,

$\exp (X)T\exp (-X)={\sum }_{r\ge 0}\frac{({\mathrm{ad}}_X{)}^r(T)}{r!},{\mathrm{ad}}_X(T)=XT-TX\mathrm{.}$

There is also a degreewise-complete graded variant: assume $E={⨁}_d{E}^d$ is a multiplicatively filtered graded algebra with each . Then exp and log give inverse bijections $(F^{1}E{)}^0\leftrightarrow 1+(F^{1}E{)}^0$, and the displayed conjugation identity holds for $X\in (F^{1}E{)}^0$ and any fixed homogeneous $T\in E^d$. The first two series converge in $E^0$ and the conjugation series in $E^d$; no completeness of the ungraded direct sum is required. In particular this variant applies to the degreewise complete endomorphism algebra in Coefficient-complete fixed-carrier codifferential deformation once its completeness is established.

Let $k$ be a field and let $(A,d,\mu )$ be a cohomological differential graded algebra over $k$ in the convention of Cohomological differential graded algebra. Put $m_1=d$, $m_2=\mu$, and $m_n=0$ for all $n\ge 3$. Then these maps make $A$ an uncurved cohomological A∞-algebra in the unsuspended convention of Uncurved cohomological A∞-algebra in the unsuspended convention.

Let $A$ be an uncurved A∞-algebra (Uncurved cohomological A∞-algebra in the unsuspended convention). A minimal A∞-model for $A$ is an uncurved A∞-algebra $H$ with $m_1^H=0$, together with an A∞ quasi-isomorphism $f:H\to A$. Here an A∞ quasi-isomorphism is as defined in A∞-morphism and quasi-isomorphism. In an explicit model whose underlying graded space is identified with $H^{\ast }(A,m_1^A)$, the binary product is ordinarily checked against the product induced by $m_2^A$; this definition does not assert an existence or transfer theorem.

This is a definition of the data required for a minimal model. It does not assert that every A∞-algebra has one or that a transfer theorem has been proved.

Let $W$ be a graded vector space over a characteristic-zero field and let $S^c(W)$ be the reduced graded symmetric coalgebra of Reduced graded symmetric coalgebra with its shuffle coproduct. Let $C$ be any graded $k$-vector space with a degree-zero reduced coproduct ${\bar{\mathrm{\Delta }}}_C:C\to C{\otimes }_{k}C$ that is coassociative and graded-cocommutative (invariant under the graded flip), with no counit. Assume that for every $c\in C$ some iterated reduced coproduct of $c$ is zero; such a coalgebra is called conilpotent here. A coalgebra morphism is a degree-zero linear map commuting with the reduced coproduct. For any degree-zero linear map $\varphi :C\to W$, there is a unique degree-zero coalgebra morphism $G:C\to S^c(W)$ whose length-one projection is $\varphi$. Explicitly, let ${\rho }_n:W^{\otimes n}\to S^n(W)$ be the quotient map, put ${\bar{\mathrm{\Delta }}}_C^{(0)}={\mathrm{id}}_C$, and define iterates recursively by ${\bar{\mathrm{\Delta }}}_C^{(n)}=({\bar{\mathrm{\Delta }}}_C^{(n-1)}\otimes 1){\bar{\mathrm{\Delta }}}_C$; coassociativity makes the parenthesization immaterial. Then

$G(c)={\sum }_{n\ge 1}\frac{1}{n!}{\rho }_n\text{\hspace{0.17em}⁣}({\varphi }^{\otimes n}{\bar{\mathrm{\Delta }}}_C^{(n-1)}(c))\mathrm{.}$

The sum is finite for each $c$ by conilpotence. In particular, the graded symmetric coalgebra is the reduced cofree conilpotent cocommutative coalgebra on $W$.