On the origin of higher braces and higherorder derivations
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Abstract
The classical Koszul braces, sometimes also called the Koszul hierarchy, were introduced in 1985 by Koszul (Astérisque, (Numero Hors Serie):257–271, 1985). Their noncommutative counterparts came as a surprise much later, in 2013, in a preprint by Börjeson (\(A_\infty \)algebras derived from associative algebras with a nonderivation differential, Preprint arXiv:1304.6231, 2013). In Part I we show that both braces are the twistings of the trivial \(L_\infty \) (resp. \(A_\infty \)) algebra by a specific automorphism of the underlying coalgebra. This gives an astonishingly simple proof of their properties. Using the twisting, we construct other surprising examples of \(A_\infty \) and \(L_\infty \)braces. We finish Part 1 by discussing \(C_\infty \)braces related to Lie algebras. In Part 2 we prove that in fact all natural braces are the twistings by unique automorphisms. We also show that there is precisely one hierarchy of braces that leads to a sensible notion of higherorder derivations. Thus, the notion of higherorder derivations is independent of human choices. The results of the second part follow from the acyclicity of a certain space of natural operations.
Keywords
Koszul braces Börjeson braces Higherorder derivationMathematics Subject Classification (2000)
13D99 55S201 Introduction
 (1)
as a tool for checking that the classical braces indeed form \(L_\infty \) resp. \(A_\infty \)algebras,
 (2)
as a machine producing explicit examples of \(L_\infty \), \(A_\infty \) and possibly also other types of strongly homotopy algebras,
 (3)
as a justification that the higherorder derivations are Godgiven, not human, inventions existing since the beginning of time, or
 (4)
as a vanilla version of [4].
Plan of the paper. Section 1 contains several examples of braces, including the classical Koszul \(L_\infty \)hierarchy and Börjeson’s \(A_\infty \)braces. We demonstrate various properties which the braces may posses, in particular those leading to a sensible definition of higherorder derivations.
In Sect. 2 we show how to generate braces by the twisting and interpret all examples in Sect. 1 as emerging this way. This offers a very simple verification that they indeed form \(L_\infty \) resp. \(A_\infty \)structures. We close this section by discussing possible generalizations to Lie and other types of algebras.
In Sect. 3 we analyze natural operations and prove that they form an acyclic space. The main results are Propositions 3.3 and 3.6. The material of this section is a baby version of the analysis of the Hochschild cochains in connection to Deligne’s conjecture as given in [4].
Section 4 formulates the consequences of Sect. 3. Theorems 4.1 and 4.9 state that all natural braces are the twistings by unique automorphisms, Corollaries 4.2 and 4.10 then describe the moduli space of all natural braces. By Theorems 4.3 and 4.11, Börjeson’s resp. Koszul braces are the unique ones leading to a meaningful notion of higherorder derivations of associative resp. commutative associative algebras.
Conventions. If not stated otherwise, all algebraic objects will be considered over a fixed field \({\mathbb {k}}\) of characteristic zero. The symbol \(\otimes \) will denote the tensor product over \({\mathbb {k}}\) and \(\mathrm{Span}(S)\) the \({\mathbb {k}}\)vector space spanned by a set \(S\). We will denote by \(1\!\!1_X\) or simply by \(1\!\!1\) when \(X\) is understood, the identity endomorphism of an object \(X\) (set, vector space, algebra,&c.). We will usually write the product of elements \(a\) and \(b\) of an associative algebra as \(a\cdot b\) or simply as \(ab\).
Part 1. Examples and constructions
1.1 Examples in place of introduction
We start by recalling a construction attributed to Koszul [7] and sometimes referred to as the Koszul hierarchy, see also [1, 2, 3, 5, 17]. It is used to define higher order derivations of commutative associative algebras, see §1.2 below; they play a substantial rôle for instance in the BRST approach to closed string field theory [11, Section 4].
Example 1.1
1.2 Higher order derivations
Let \(A\) be a graded commutative associative algebra with a differential \(\Delta \) as in Example 1.1. One says [2] that \(\Delta \) is an order \(r\) derivation if \(\Phi ^\Delta _{r+1} = 0\). Clearly, being an order \(1\) derivation is the same as being a derivation in the usual sense. It is almost clear that an order \(r\)derivation is determined by its values on the products \(x_1 \ldots x_s\), \(s \le r\), of generators of \(A\); an explicit formula is given in [11, Proposition 3.4].
One may ask whether higherorder derivations are ‘Godgiven,’ i.e. whether the braces that define it are unique. Let us try to find out which properties the braces leading to a sensible notion of higherorder derivations should satisfy. First of all, they must be ‘natural’ in that they use only the data that are available for any graded associative commutative algebra with a differential. The exact meaning of naturality is analyzed in Sect. 3.
Example 1.2
It will follow from Theorem 4.11 that assumptions (3)–(5) already imply that \(l^\Delta _k = \Phi ^\Delta _k\) for each \(k \ge 1\). Let us start our discussion of the noncommutative case by recalling one construction from a recent preprint [6] of Börjeson.
Example 1.3
It is obvious that Börjeson’s braces satisfy assumptions (3)–(5), so they lead to a sensible notion of higherorder derivations of graded associative (noncommutative) algebras. By Theorem 4.3, they are the only \(A_\infty \)braces with these properties.
Example 1.4
Example 1.5
Hereditarity is a very fine property; ‘randomly chosen’ braces will not be hereditary. A systematic method of producing nonhereditary braces, based surprisingly on a rather deep Proposition 4.4, is described in Example 4.7.
2 Constructions of higher braces
2.1 Noncommutative algebras and \(A_\infty \)braces
Recall that an \(A_\infty \)algebra consists of a graded vector space \(V\) together with linear operations \(\mu _k : {V}^{\otimes k} \rightarrow V\), \(k \ge 1\), such that \(\deg (\mu _k) = 2k\), satisfying a system of axioms that say that \(\mu _1\) is a differential, \(\mu _2\) is associative up to the homotopy \(\mu _3\),&c, see e.g. [15, 16].
Definition 2.1
An \(A_\infty \) algebra is a structure \({\mathcal A}= (A,m_1,m_2,m_3,\ldots )\) consisting of a graded vector space \(A\) and degree \(+1\) linear maps \(m_k : {A}^{\otimes k}\rightarrow A\), \(k \ge 1\), satisfying (7).
Let \({\mathbb T}^cA\) be the coalgebra whose underlying space is the tensor algebra \({\mathbb T}A := \bigoplus _{n \ge 1} {A}^{\otimes n}\) and the diagonal (comultiplication) is the deconcatenation. It turns out that \({\mathbb T}^cA\) is a cofree conilpotent coassociative coalgebra cogenerated by \(A\), see e.g. [13, §II.3.7].^{3} Its cofreeness implies that each coderivation \(\vartheta \) of \({\mathbb T}^cA\) is given by its components \(\vartheta _k : {A}^{\otimes k} \rightarrow A\), \(k \ge 1\), defined by \(\vartheta _k := \pi \circ \vartheta \circ \iota _k\), where \(\pi : {\mathbb T}^cA \twoheadrightarrow A\) is the projection and \(\iota _k : {A}^{\otimes k} \hookrightarrow {\mathbb T}^cA\) the inclusion. We write \(\vartheta = (\vartheta _1,\vartheta _2,\vartheta _3,\ldots )\).
Let \(m := (m_1,m_2,m_3,\ldots )\) be a degree \(1\) coderivation of \({\mathbb T}^cA\) determined by the linear maps \(m_k\) as in Definition 2.1. It is wellknown that axiom (7) is equivalent to \(m\) being a differential, i.e. to a single equation \(m^2 = 0\). Therefore equivalently, an \(A_\infty \)algebra is a pair \((A,m)\) consisting of a graded vector space \(A\) and a degree \(+1\) coderivation \(m\) of \({\mathbb T}^cA\) which squares to zero.
In homological algebra one usually considers \(A_\infty \)algebras \((A,m_1,m_2,m_3,\ldots )\) as objects living in the category of differential graded (dg) vector spaces, the linear operation (differential) \(m_1\) being part of its underlying dgvector space, not a structure operation. For this reason we call an \(A_\infty \)algebra with \(m_k = 0\) for \(k\ge 2\) a trivial \(A_\infty \)algebra.
Example 2.2
(Trivial \(A_\infty \)algebra) Let \(\Delta : A \rightarrow A\) be a degree \(+1\) differential on a graded vector space \(A\). It is clear that \({\mathcal A}_\Delta := (A,\Delta ,0,0,\ldots )\) is an \(A_\infty \)algebra. The differential \(\Delta \) extends to a linear coderivation \((\Delta ,0,0,\ldots )\) of \({\mathbb T}^cA\).
Definition
Two \(A_\infty \)algebras \({\mathcal A}' = (A',m')\) and \({\mathcal A}'' = (A',m'')\) are isomorphic if there exists an automorphism \(\phi : {\mathbb T}^cA' \rightarrow {\mathbb T}^cA''\) such that \(\phi m' = m'' \phi \).^{4} They are strictly isomorphic if there exist a linear \(\phi \) as above.
Assume that we are given an \(A_\infty \)algebra \({\mathcal A}= (A,m)\) and an automorphism \(\phi : {\mathbb T}^cA \rightarrow {\mathbb T}^cA\). Then clearly \((A,\phi ^{1} m \phi )\) is an \(A_\infty \)algebra isomorphic to \({\mathcal A}\).
Definition 2.3
In the situation above, we denote \(m^\phi :=\phi ^{1} m \phi \) and call the \(A_\infty \)algebra \({\mathcal A}^\phi := (A,m^\phi )\) the twisting of the \(A_\infty \)algebra \({\mathcal A}= (A,m)\) by the automorphism \(\phi \).
2.2 Explicit formulas
Exercise If \(\Delta \) is a derivation, then \(\Delta ^\phi = \Delta \), i.e. \(\Delta _k = 0\) in (11) for all \(k \ge 2\).
Example 2.4
Example 2.5
Example
2.3 Commutative algebras and \(L_\infty \)braces
Lie counterparts of \(A_\infty \)algebras are \(L_\infty \)algebras. An \(L_\infty \)algebra is a graded vector space \(L\) with linear operations \(\ell _k : {L}^{\otimes k} \rightarrow V\), \(k \ge 1\), \(\deg (\ell _k) = 2k\), that are graded antisymmetric and satisfy axioms that say that \(\ell _1\) is a differential, \(\ell _2\) fulfills the Jacobi identity up to the homotopy \(\ell _3\),&c, see e.g. [8, 9].
Definition 2.6
An \(L_\infty \) algebra is an object \({\mathcal L}= (A,l_1,l_2,l_3,\ldots )\) consisting of a graded vector space \(A\) and degree \(+1\) graded symmetric linear maps \(l_k : {A}^{\otimes k}\rightarrow A\), \(k \ge 1\), satisfying (14) for each \(n \ge 1\).
Let \({\mathbb S}^cA = \bigoplus _{k \ge 1} {\mathbb S}^k A\) be the symmetric coalgebra with the diagonal given by the deconcatenation; it is the cofree conilpotent cocommutative coassociative coalgebra cogenerated by \(A\). Each coderivation \(\omega \) of \({\mathbb S}^cA\) is thus determined by its components \(\omega _k := \pi \circ \omega \circ \iota _k\), \(k \ge 1\), where \(\pi : {\mathbb S}^cA \twoheadrightarrow A\) is the projection and \(\iota _k : {\mathbb S}^k A \hookrightarrow {\mathbb S}^cA\) the inclusion of the \(k\)th symmetric power of \(A\). We write \(\omega = (\omega _1,\omega _2,\omega _3,\ldots )\).
Let \(l := (l_1,l_2,l_3,\ldots )\) be a degree \(1\) coderivation of \({\mathbb S}^cA\) determined by the linear maps \(l_k\) of Definition 2.6. Axiom (14) is equivalent to a single equation \(l^2 = 0\) [8, Theorem 2.3]. So an \(L_\infty \)algebra is a pair \((A,l)\) a graded vector space and a degree \(+1\) coderivation \(l\) of \({\mathbb S}^cA\) which squares to zero.
Example 2.7
(Trivial \(L_\infty \)algebra) The observations of Example 2.2 apply verbatim to the \(L_\infty \)case – if \(\Delta \) is a degree \(+1\) differential on a graded vector space \(A\), then \({\mathcal L}_\Delta := (A,\Delta ,0,0,\ldots )\) is an \(L_\infty \)algebra.
As automorphisms of \({\mathbb T}^cA\) twist \(A_\infty \)algebras, \(L_\infty \)algebras can be twisted by automorphisms \(\phi : {\mathbb S}^cA \rightarrow {\mathbb S}^cA\) determined by their components \(\phi _k : {\mathbb S}^kA \rightarrow A\), \(k \ge 1\). We leave as an exercise to derive formulas for the composition and for the twisting of \({\mathcal L}_\Delta \) analogous to (8) and (9).
2.4 Explicit formulas
Example 2.8
Example
Problem
It is clear that an automorphism \(\phi (t)\) with the generating series (15) leads to recursive braces if and only if \(f_k \in \{1,+1\}\). Which property of the generating function guarantees the hereditarity?
2.5 The Lie case
One may ask how the previous material translates to the Lie algebra case. One could expect to have, for a graded Lie algebra \(L\) with a differential \(\Delta \), natural \(C_\infty \)braces \((L,\Delta ,c_2,c_3,\ldots )\) emerging as the twistings of the trivial \(C_\infty \)algebra by automorphisms of the Lie coalgebra \({\mathbb {L}}^{c}L\) and, among these structures, a particular one that leads to higherorder derivations of Lie algebras.
Recall that a \(C_\infty \)algebra (also called, in [10, §1.4], a balanced \(A_\infty \) algebra) is an \(A_\infty \)algebra as in Definition 2.1 whose structure operations vanish on decomposables of the shuffle product. As in the \(A_\infty \) or \(L_\infty \)cases, \(C_\infty \)algebras can equivalently be described as squarezero coderivations of the cofree conilpotent Lie coalgebra \({\mathbb {L}}^{c}L\) cogenerated by \(L\).
We may try to proceed as in the previous two cases. We have the trivial \(C_\infty \)algebra \({\mathcal C}_\Delta = (L,\Delta ,0,0,\ldots )\), thus any natural automorphism \(\phi : {\mathbb {L}}^{c}L \rightarrow {\mathbb {L}}^{c}L\) determines a coderivation \(\Delta ^\phi := \phi ^{1}\Delta \phi \) that squares to \(0\), i.e. \(C_\infty \)braces on \(L\).
The sting lies in the notion of naturality. In constructing \(A_\infty \)braces we very crucially relied on the fact that the cofree conilpotent coassociative coalgebra cogenerated by \(A\) materialized as the tensor algebra \({\mathbb T}A\) equipped with the deconcatenation diagonal. Therefore natural operations \({\mathbb T}A \rightarrow A\) give rise to natural automorphisms of \({\mathbb T}^cA\) and thus also to natural \(A_\infty \)braces. Similarly, the cofree conilpotent coalgebra cogenerated by \(A\) can be realized as the symmetric algebra \({\mathbb S}V\) with the deconcatenation (unshuffle) diagonal.
Remark 2.9
Let \({\mathbb L}^kL\) be the subspace of \(\mathbb LL\) spanned by elements of the product length \(k\). As in the previous cases, each coalgebra automorphism \(\phi : {\mathbb {L}}^{c}L \rightarrow {\mathbb {L}}^{c}L\) is determined by its components \(\phi _k : {\mathbb L}^kL \rightarrow L\), \(k \ge 1\). One has also the canonical maps \(\lambda _k : {\mathbb L}^kL \rightarrow L\) given by the multiplication in \(L\). Assume we found a natural isomorphism between \(\mathbb LL\) and \({\mathbb {L}}^{c}L\) such that the induced diagonal on \(\mathbb LL\) agrees with (17) on elements of length \(\le \! 3\).
We saw in the \(A_\infty \) resp. \(L_\infty \)cases that the twisting by the automorphism whose components were the canonical maps \(\mu _k : {\mathbb T}^k A \rightarrow A\) resp. \(\mu _k: {\mathbb S}^k A \rightarrow A\), lead to the (unique) braces giving a sensible notion of higherorder derivations. This justifies:
Conjecture
The twisting of \({\mathcal C}_\Delta = (L,\Delta ,0,0,\ldots )\) by \(\phi = (1\!\!1_L,\lambda _2,\lambda _3,\ldots )\) gives rise to \(C_\infty \)braces satisfying the analogs of conditions (3)–(5).
2.6 Other cases
Let us finish this part by formulating the most general context in which our approach may work. We will need the language of operads for which we refer for instance to [12, 13]. Let \({\mathcal P}\) be a quadratic Koszul operad and \(A\) a \({\mathcal P}\)algebra. Denote by \({\mathcal P}^!\) the Koszul (quadratic) dual of \({\mathcal P}\) [13, Def. II.3.37] and by \({\mathbb F}_{{\mathcal P}}A\) (resp. \({\mathbb F}^c_{{\mathcal P}}A\)) the free \({{\mathcal P}}\)algebra (resp. the cofree conilpotent \({{\mathcal P}}\)coalgebra) generated (resp. cogenerated) by \(A\).
A strongly homotopy \({\mathcal P}^!\)algebra, also called a \({\mathcal P}^!_\infty \)algebra, is determined by a squarezero coderivation \(p\) of \({\mathbb F}^c_{{\mathcal P}}A\). If \(\Delta \) is a differential on \(A\), one has as before the trivial \({\mathcal P}_\infty \)algebra \({\mathcal P}_\Delta = (A,\Delta ,0,0,\ldots )\) given by extending \(\Delta \) to \({\mathbb F}^c_{{\mathcal P}}A\).
Under the presence of a natural identification \({\mathbb F}_{{\mathcal P}}A\cong {\mathbb F}^c_{{\mathcal P}}A\), one may speak about natural automorphisms that twist \(P_\Delta \) to \({\mathcal P}^!_\infty \)braces. One has the automorphism \(\phi : {\mathbb F}^c_{{\mathcal P}}A\rightarrow {\mathbb F}^c_{{\mathcal P}}A\) whose components are given by the structure map \({\mathbb F}_{{\mathcal P}}A\rightarrow A\). It is sensible to conjecture that the related \({\mathcal P}^!_\infty \)braces lead to a reasonable notion of higherorder derivations of \({\mathcal P}\)algebras.
In this general setup, \(A_\infty \)braces related to associative algebras correspond to the \({\mathcal P}= {\mathcal A{ ss}}\) case, \(L_\infty \)braces related to commutative associative algebras to \({\mathcal P}= {\mathcal C{ om}}\), and \(C_\infty \)braces related to Lie algebras to \({\mathcal P}= \mathcal L{ ie}\), where \({\mathcal A{ ss}}\), \({\mathcal C{ om}}\) and \(\mathcal L{ ie}\) denote the operad for associative, commutative associative and Lie algebras, respectively.^{6}
Part 2. Naturality and acyclicity
3 Naturality
This section is devoted to natural operations \({A}^{\otimes k} \rightarrow A\) (resp. \({\mathbb S}^k A \rightarrow A\)), where \(A\) is a graded associative (resp. graded commutative associative) algebra with a differential \(\Delta \) of degree \(+1\). Since in the commutative associative case the symmetric group action brings extra complications but nothing conceptually new, we analyze in detail only the associative case.
Associative case. We are going introduce the space \({\mathcal Nat}(k)\) of natural operations \({A}^{\otimes k} \rightarrow A\) and show that \({\mathcal Nat}(k)\), graded by the degrees of maps and equipped with the differential induced by \(\Delta \), is acyclic for each \(k \ge 2\). The content of this section is a kindergarten version of the analysis of Deligne’s conjecture given in [4].
3.1 Natural operations
Intuitively, natural operations \({A}^{\otimes k} \rightarrow A\) are linear maps composed from the data available for an arbitrary graded associative algebra \(A\) with a differential. Equivalently, natural operations are linear combinations of compositions of ‘elementary’ operations, which are the multiplication, the differential, permutations of the inputs and projections to the homogeneous parts. Our categorial definition given below is chosen so that it excludes the projections; the reason is explained in Exercise 4.8. Our theory can, however, easily be extended to include the projections as well, cf. Exercise 3.7. Let us start with:
Example 3.1
Example
Natural operations thus appear as natural transformations \(\beta : \bigotimes ^k \rightarrow \Box \) from the tensor power functor \(\bigotimes ^k \!\! : \mathtt{Algs}^\Delta \rightarrow \mathtt{Vect}\) to the forgetful functor \(\Box \! : \mathtt{Algs}^\Delta \rightarrow \mathtt{Vect}\), where \(\mathtt{Algs}^\Delta \) is the category of graded associative algebras with a differential, with morphisms as in (18), and \(\mathtt{Vect}\) the category of vector spaces. We however prefer a more explicit:
Definition 3.2
We are going to prove a structure theorem for natural operations. Denote by \(\mathcal Fr(x_1,\ldots ,x_k)\) the free graded associative algebra with a differential, generated by degree \(0\) elements \(x_1,\ldots ,x_k\) (an explicit description is given in §3.2). Denote also by \(\mathcal Fr_{1,\ldots ,1}(x_1,\ldots ,x_k)\) the subspace of \(\mathcal Fr(x_1,\ldots ,x_k)\) spanned by the words that contain each generator precisely once.
Proposition 3.3
Proof
Denote, for brevity, \(\mathcal Fr: = \mathcal Fr(x_1,\ldots ,x_k)\). Let \(A = (A,\Delta )\) be an arbitrary algebra with a differential. Given elements \(a_{1},\dots ,a_{k} \in A\), there exists a unique \(\Phi ^A_{a_{1},\dots ,a_{n}} : \mathcal Fr\rightarrow A\) satisfying (18), specified by requiring \(\Phi ^A_{a_{1},\dots ,a_{n}}(x_i) := a_i\) for \(1 \le i \le k\).
Remark
A crucial step of the previous proof was that \(\xi (\beta )\) belonged to \(\mathcal Fr_{{1},\ldots ,{1}}\). It was implied by the multilinearity, which is a particular feature of the monoidal structure given by \(\otimes \). In the cartesian situation, \(\xi (\beta )\) might have been an arbitrary element of the free algebra \(\mathcal Fr(x_{1},\dots ,x_{k})\).
3.2 The algebra \(\mathcal Fr(x_{1},\dots ,x_{k})\).
We can clearly encode elements of \(\mathcal Fr_{{1},\ldots ,{1}}\) by ‘flow diagrams’ that record how the multiplication and the differential are applied. For instance, the diagram encoding (25) is
Its underlying graph is a rooted (=oriented) planar tree with the root pointing upwards. The labels of its leaves (=inputs) mark the position of the generators. The vertices symbolize iterated multiplication while the bullets the application of the differential.

each vertex of \(T\) has at least two inputs,

all internal edges and possibly some external edges are decorated by the bullet \(\bullet \), and

the leaves of \(T\) are labelled by a permutation of \(({1},\ldots ,{k})\).
Corollary 3.4
For each \(k \ge 1\) one has a natural isomorphism \({\mathcal Nat}(k)\cong \mathrm{Span}\big ({\mathcal T}(k)\big )\).
It follows from general theory [13, Proposition II.1.27] that \({\mathcal Nat}(k)\) is the arity \(k\)th piece of the operad \({\mathcal Nat}\) whose algebras are couples \((A,\Delta )\) consisting of an associative algebra and a differential. We will, however, not need this interpretation in the sequel.
Example
Corollary 3.4 offers the following description of \({\mathcal Nat}(2)\):
We leave as an exercise to relate the above description to the operations listed in Example 3.1.
Example 3.5
3.3 Acyclicity
Proposition 3.6
The case of \(k=1\) is a particular one, as the differential \(\delta : {\mathcal Nat}(1)^0 \rightarrow {\mathcal Nat}(1)^1\) is the zero map \(\mathrm{Span}(1\!\!1) \mathop {\rightarrow }\limits ^{0}\mathrm{Span}(\Delta )\). The explanation is that the identity \(1\!\!1: A \rightarrow A\) is the only natural operation that is ‘generically’ a chain map.
Proof of Proposition 3.6
We describe a contracting homotopy. By Corollary 3.4, each natural operation \(\beta \in {\mathcal Nat}(k)\) is represented by a unique linear combination of trees from \({\mathcal T}(k)\). It is therefore enough to specify how the homotopy acts on operations given by a single tree.
Let \(\beta \) be represented by \(T \in {\mathcal T}(k)\). If the root edge of \(T\) is decorated by the bullet, we define \(h(\beta )\) as the operation represented the tree \(T'\) obtained from \(T\) by removing the decoration of the root. We define \(h(\beta ) := 0\) if the root edge of \(T\) is not decorated by the bullet. We leave as an exercise to verify that \(h \delta + \delta h = 1\!\!1\), so that \(h\) is a contracting homotopy. \(\square \)
Notice that the complex \({\mathcal Nat}(k)\) is isomorphic to the direct sum of \(k!\) copies of the contractible subcomplex \(\underline{\mathcal Nat}(k) \subset {\mathcal Nat}(k)\) consisting of operations represented by trees with leaves indexed by the identity permutation \((1,\ldots ,k)\).
Exercise 3.7
3.4 Natural \(A_\infty \)algebras
Let \((A,\Delta )\) be a graded associative algebra with a differential. We call an \(A_\infty \)algebra \({\mathcal A} =(A,m_1,m_2,\ldots )\) natural if \(m_k : {A}^{\otimes k} \rightarrow A\) are natural operations from \({\mathcal Nat}(k)^1\) for each \(k \ge 1\). Formally, a natural \(A_\infty \)algebra should be considered as a family \(\{{\mathcal A}_{(A,\Delta )}\}\) of \(A_\infty \)algebras indexed by algebras with a differential, such that each \(\varphi \) as in (18) induces a strict morphism \({\mathcal A}_{(A,\Delta _A)} \rightarrow {\mathcal A}_{(B,\Delta _B)}\). We however believe that our simplification will not lead to confusion.
An automorphism \(\phi : {\mathbb T}^cA \rightarrow {\mathbb T}^cA\) is natural if all its components \(\phi _k : {A}^{\otimes k} \rightarrow A\) are natural operations from \({\mathcal Nat}(k)^0\). Natural automorphisms with \(\phi _1 = 1\!\!1_A\) form a monoid \(\mathrm{Aut}(A)\). It is clear that the twisting of a natural \(A_\infty \)algebra by a natural automorphism is a natural \(A_\infty \)algebra.
Example 3.8
It follows from Example 3.5 that natural automorphisms \(\phi \in \mathrm{Aut}(A)\) are encoded by sequences \((1\!\!1_A,\omega _2,\omega _3,\ldots )\) of elements \(\omega _k \in {\mathbb {k}}[\Sigma _k]\). One has an important submonoid \({\underline{\mathrm{Aut}}}(A)\subset \mathrm{Aut}(A)\) of sequences such that all \(\omega _k\)’s are the identities \(1\!\!1_{\Sigma _k}\).
Exercise 3.9
Assume that \(\Delta \) is a derivation. The twisting of \({\mathcal A}_\Delta = (A,\Delta ,0,0,\ldots )\) by an arbitrary natural automorphism is then strictly isomorphic to \({\mathcal A}_\Delta \), i.e. \(\Delta ^\phi = \Delta \).
Commutative associative case. We are going to formulate commutative versions of the main statements from the first part of this section. We omit the proofs which are analogous to the noncommutative case.
Let \(A\) be a graded commutative associative algebra with a differential \(\Delta \). Natural operations are, analogously to Definition 3.2, natural transformations \(\beta _{A} : {\mathbb S}^k A \rightarrow A\) from the \(k\)th symmetric power of \(A\) to \(A\). Let us denote by \({\mathcal Nat}(k)\) the abelian group of all these natural operations. To see how \({\mathcal Nat}(k)\) differs from its noncommutative counterpart, we give commutative versions of Examples 3.1 and 3.5.
Example 3.10
Example 3.11
(Commutative version of Example 3.5) The space \({\mathcal Nat}(k)^0\) is onedimensional, spanned by the iterated multiplication \( \mu _k(a_{1},\dots ,a_{k}) = a_1\cdots a_k,~\)so\({\mathcal Nat}(k)^0 \cong {\mathbb {k}}\) for each \(k \ge 1\).
An obvious modification of Proposition 3.3 holds, with \(\mathcal Fr(x_1,\ldots ,x_k)\) this time the free commutative associative algebra. Corollary 3.4 holds as well, with \({\mathcal T}(k)\) replaced by the space of all ‘abstract,’ i.e. nonplanar, trees. As in the noncommutative case, \(\Delta \) induces a differential \(\delta \) so that \(({\mathcal Nat}(k)^*,\delta )\) is acyclic for each \(k \ge 2\).
The notions of a natural \(L_\infty \)algebras and natural automorphisms \(\phi : {\mathbb S}^c(A) \rightarrow {\mathbb S}^c(A)\) translate verbatim. The following example however shows that the space \(\mathrm{Aut}(A)\) of natural automorphisms is much smaller than in the noncommutative case.
Example 3.12
The description of \({\mathcal Nat}(k)^0\) given in Example 3.11 implies that natural automorphisms \(\phi \in \mathrm{Aut}(A)\) are encoded by sequences \((1\!\!1_A,f_2,f_3,\ldots )\) of scalars \(f_k \in {\mathbb {k}}\).
4 Main results
We are going to formulate and prove the main theorems. As in Sect. 3, we treat in detail only the associative noncommutative case.
Associative case. Let \(A\) be a graded associative algebra with a differential \(\Delta \), and \({\mathcal A}_\Delta = (A,\Delta ,0,0,\ldots )\) the trivial \(A_\infty \)algebra of Example 2.2. According to the following result, each natural \(A_\infty \)algebra whose linear operation \(m_1\) equals \(\Delta \), is uniquely given by a twisting of \({\mathcal A}_\Delta \), see §3.4 and Definition 2.3 for the meaning of naturality and twisting. In particular, each such an \(A_\infty \)algebra is (weakly) isomorphic to \({\mathcal A}_\Delta \).
Theorem 4.1
Proof
Theorem 4.1 combined with Example 3.8 gives:
Corollary 4.2
A weak isomorphism of \(A_\infty \)algebras induces a strict isomorphism of their cohomology algebras. We therefore get another
Corollary
The cohomology \(H^*(A,\Delta )\) of any natural \(A_\infty \)algebra \((A,\Delta ,m_2,m_3,...)\) is the trivial associative algebra with the underlying space \(H^*(A,\Delta )\).
The observation made in Example 3.9 combined with Theorem 4.1 lead to another
Corollary
Assume that \(\Delta \) is a derivation of \(A\). Then any natural \(A_\infty \)braces vanish on \(A\).
We finally formulate our characterization of Börjeson braces [6] recalled in Example 1.3.
Theorem 4.3
 (i)\(b^\Delta _2\) measures the deviation of \(\Delta \) from being a derivation, i.e.$$\begin{aligned} b^\Delta _2(a_1,a_2) = \Delta (a_1a_2)  \Delta (a_1)a_2  {(1)^{a_1}}a_1\Delta (a_2), \quad a_1,a_2 \in A, \end{aligned}$$
 (ii)
the coefficient at \(\Delta (a_1a_2a_3)\) in \(b^\Delta _3(a_1,a_2,a_3)\) is either \(+1\) or \(1\) and,
 (iii)the hereditarity is satisfied, that is for all \(k \ge 1\),$$\begin{aligned} b^\Delta _k = 0 \ \text{ implies } \ b^\Delta _{k+1} = 0. \end{aligned}$$
It is obvious that the Börjeson braces are recursive and satisfy (i) and (ii). Their hereditarity established in [6] follows from an inductive formula mentioned in Remark 4.6. It remains to prove that (i)–(iii) characterize Börjeson’s braces up to a strict isomorphism. This will follow from Propositions 4.4 and 4.5 below.
Proposition 4.4
Suppose that \(A = (A,\Delta ,m_2,m_3,m_4,\ldots )\) are natural hereditary \(A_\infty \)braces such that \(m_2 = b^\Delta _2\) and \(m_3 = b^\Delta _3\). Then \(m_k = b^\Delta _k\) for any \(k \ge 2\).
Proof
Inspecting the coefficients at the terms (33) in (32), we see that \(\delta z_{n+1}=0\) in \(A_{n+1}\) only if all \(\xi _\sigma \), \(\sigma \in \Sigma _{n+1}\), are trivial. So \(z_{n+1} = 0\) as required, and the induction goes on.
In the proof of Proposition 4.4, the requirement that \(n \ge 3\) in (30) was crucial. Indeed, the monomial \(\Delta (x_1)x_2x_3\) does occur in \(m_2(x_1,x_2)x_3\), so the argument following formula (33) does not work for \(n=2\). We must therefore prove also the following
Proposition 4.5
Let \({\mathcal A}= (A,\Delta ,m_2,m_3,m_4,\ldots )\) be a natural recursive \(A_\infty \)algebra such that \(m_2=b^\Delta _2\), the coefficient \(C_3\) at \(\Delta (a_1a_2a_3)\) in \(m_3(a_1,a_2,a_3)\) is either \(1\) or \(1\), and \(m_3 = 0\) implies \(m_4=0\). Then in fact \(C_3 = 1\) and \(m_3 = b^\Delta _3\).
Proof
Remark 4.6
Example 4.7
Example 4.8
Commutative associative case. The first part of this section translates to the commutative case in a straightforward manner, so we formulate only the commutative versions of the main theorems. We start with the commutative variant of Theorem 4.1. Recall that \({\mathcal L}_\Delta \) denotes the trivial \(L_\infty \)algebra from Example 2.7.
Theorem 4.9
Theorem 4.9 combined with the description of natural automorphisms given in Example 3.12 leads to:
Corollary 4.10
The following theorem offers a characterization of Koszul \(L_\infty \)braces analogous to that of Börjeson \(A_\infty \)braces given in Theorem 4.3.
Theorem 4.11
 (i)\(\Phi ^\Delta _2\) measures the deviation of \(\Delta \) from being a derivation, i.e.$$\begin{aligned} \Phi ^\Delta _2(a_1,a_2) = \Delta (a_1a_2)  \Delta (a_1)a_2  {(1)^{a_1}}a_1\Delta (a_2), \ a_1,a_2 \in A, \end{aligned}$$
 (ii)
the coefficient at \(\Delta (a_1a_2a_3)\) in \(\Phi ^\Delta _3(a_1,a_2,a_3)\) is either \(+1\) or \(1\) and,
 (iii)the hereditarity is satisfied, that is for all \(k \ge 1\), that is$$\begin{aligned} \Phi ^\Delta _k = 0 \ \text{ implies } \ \Phi ^\Delta _{k+1} = 0. \end{aligned}$$
Exercise
Explain how to construct nonhereditary \(L_\infty \)braces.
Footnotes
 1.
We will give a short and elegant proof of this fact in Example ; \(L_\infty \)algebras are recalled in Definition 2.6.
 2.
A simple proof of this fact is provided by Example 2.4; \(A_\infty \)algebras are recalled in Definition 2.1.
 3.
A general misconception is that \({\mathbb T}^cA\) is cofree in the category of all coassociative coalgebras.
 4.
Sometimes one says that \({\mathcal A}'\) and \({\mathcal A}''\) are weakly isomorphic.
 5.
As explained in Example 3.8, \({\underline{\mathrm{Aut}}}(A)\) is a submonoid of the monoid of all natural automorphisms.
 6.
Recall that \({\mathcal A{ ss}}^! \cong {\mathcal A{ ss}}\), \({\mathcal C{ om}}^! \cong \mathcal L{ ie}\) and \(\mathcal L{ ie}^! \cong {\mathcal C{ om}}\), see e.g. [13, Example II.3.38].
 7.
Although, as indicated in Exercise 3.7, we can extend our theory to include also operations of this type.
 8.
The reason is that \({\mathbb T}(a,b)\) is free also in the category of ungraded associative algebras.
 9.
The notation \(c{\mathcal Nat}(k)\) refers to colored natural operations.
 10.
As customary, we denote both the generators of \(\mathcal Fr(x_{1},\dots ,x_{n+1})\) and their equivalence classes in \(A_{n+1}\) by the same symbols.
Notes
Acknowledgments
I would like to express my thanks to Maria Ronco for Remark 2.9 and Kaj Börjeson for spotting a mistake in Example 1.3. I am also indebted to Olga Kravchenko and the referee for many other useful suggestions.
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