Universal Deformation Formula, Formality and Actions

TL;DR

This paper develops a method to quantize Poisson actions of triangular Lie algebras on manifolds using formality, leading to a deformation quantization framework with quantum groups and generalized quantum actions.

Contribution

It introduces a new quantization approach for Poisson actions via formality, extending the concept of quantum actions through $L_$-morphisms.

Findings

Deformation quantization of manifolds with Poisson actions.

Construction of quantum groups $U_ ext{ extbackslash}hbar( g)$.

Generalization of quantum actions using $L_$-morphisms.

Abstract

In this paper we provide a quantization via formality of Poisson actions of a triangular Lie algebra $(\mathfrak g,r)$ on a smooth manifold $M$. Using the formality of polydifferential operators on Lie algebroids we obtain a deformation quantization of $M$ together with a quantum group $\mathscr{U}_\hbar(\mathfrak{g})$ and a map of associated DGLA's. This motivates a definition of quantum action in terms of $L_\infty$-morphisms which generalizes the one given by Drinfeld.

Equations

$$a{\star }_{J}b=m_{\mathcal{A}}(J▹(a\otimes b))$$

$${\mathcal{U}}_J(\mathfrak{g}):=(\mathcal{U}(\mathfrak{g})[[\hslash ]],{\Delta }_J:=J\Delta J^{-1}).$$

$${}^{\mathfrak{g}}{\mathrm{T}}_{\mathrm{poly}}⟶{\mathrm{T}}_{\mathrm{poly}}(M),$$

$${}^{\mathfrak{g}}{\mathrm{T}}_{\mathrm{poly}}⟶{}^{\mathfrak{g}}{\mathrm{D}}_{\mathrm{poly}}\text{mbox}and{\mathrm{T}}_{\mathrm{poly}}(M)⟶{\mathrm{D}}_{\mathrm{poly}}(M).$$

$${}^{\mathfrak{g}}{\mathrm{D}}_{\mathrm{poly}}⟶{\mathrm{D}}_{\mathrm{poly}}(M).$$

$$\Phi :H_{poly}⟶C(\mathcal{A})$$

$$V^{\bullet }[k{]}^l=V^{l+k}$$

$$S^{c}(\mathfrak{L})=\bigoplus_{k=0}^{\infty}\Lambda^{k}\mathopen{}\mathclose{{}\left(\mathfrak{L}[1]}\right)$$

$$\Delta (1)$$

$$\Delta ({\gamma }_1\wedge \dots \wedge {\gamma }_k)$$

$$\overline{\Delta}(\gamma_{1}\wedge\ldots\wedge\gamma_{k})=\sum_{i=1}^{k-1}\sum_{\sigma\in\mbox{\tiny Sh($i$,$k-i$)}}\epsilon(\sigma)\gamma_{\sigma(1)}\wedge\ldots\wedge\gamma_{\sigma(i)}\bigotimes\gamma_{\sigma(i+1)}\wedge\ldots\wedge\gamma_{\sigma(k)},$$

$${\gamma }_1\wedge \dots \wedge {\gamma }_k=ϵ(\sigma ){\gamma }_{\sigma (1)}\wedge \dots \wedge {\gamma }_{\sigma (k)}.$$

$$(ii+1)({\gamma }_1\otimes \dots \otimes {\gamma }_k)=(-1{)}^{|{\gamma }_i||{\gamma }_{i+1}|}{\gamma }_1\otimes \dots \otimes {\gamma }_{i-1}\otimes {\gamma }_{i+1}\otimes {\gamma }_i\otimes {\gamma }_{i+2}\otimes \dots {\gamma }_k,$$

$$\Delta \circ (\cdot \wedge \cdot )=(\cdot \wedge \cdot )\otimes (\cdot \wedge \cdot )\circ (\mathrm{id}\otimes \tau \otimes \mathrm{id})\circ \Delta \otimes \Delta .$$

$$Q_n:{\Lambda }^n(\mathfrak{L}[1])⟶\mathfrak{L}[2]$$

$$Q(\gamma_{1}\wedge\ldots\wedge\gamma_{n})=\sum_{k=0}^{n}\sum_{\sigma\in\mbox{\tiny Sh($k$,$n-k$)}}\epsilon(\sigma)Q_{k}(\gamma_{\sigma(1)}\wedge\ldots\wedge\gamma_{\sigma(k)})\wedge\gamma_{\sigma(k+1)}\wedge\ldots\wedge\gamma_{\sigma(n)},$$

$$\Delta \circ Q=(Q\otimes \mathrm{id}+\mathrm{id}\otimes Q)\circ \Delta .$$

$$\mathfrak{L}={\mathcal{F}}^0\mathfrak{L}\supset {\mathcal{F}}^1\mathfrak{L}\supset \dots \supset {\mathcal{F}}^k\mathfrak{L}\supset \dots ,$$

$$\bigcap _k{\mathcal{F}}^k\mathfrak{L}=\{0\}.$$

$$F:S^c(\mathfrak{L})⟶S^c(\tilde{\mathfrak{L}})$$

$$F_n:{\Lambda }^n(\mathfrak{L}[1])⟶\tilde{\mathfrak{L}}[1],$$

$$\begin{gathered}F(\gamma_{1}\wedge\ldots\wedge\gamma_{n})=\\ \sum_{p\geq 1}\sum_{\begin{subarray}{c}k_{1},\ldots,k_{p}\geq 1\\ k_{1}+\ldots+k_{p}=n\end{subarray}}\sum_{\sigma\in\mbox{\tiny Sh($k_{1}$,..., $k_{p}$)}}\frac{\epsilon(\sigma)}{p!}F_{k_{1}}(\gamma_{\sigma(1)}\wedge\ldots\gamma_{\sigma(k_{1})})\wedge\ldots\wedge F_{k_{p}}(\gamma_{\sigma(n-k_{p}+1)}\wedge\ldots\wedge\gamma_{\sigma(n)}),\end{gathered}$$

$$\Delta \circ F=F\otimes F\circ \Delta .$$

$$F_1:(\mathfrak{L},Q_1)⟶(\tilde{\mathfrak{L}},{\tilde{Q}}_1).$$

$$F:\mathrm{H}(\mathfrak{L})⟶\mathfrak{L}$$

$$\sum _{n=0}^{\infty }\frac{Q_n({\pi }^n)}{n!}=0.$$

$${\mathcal{C}}^{\infty }(\text{mathbbm}{R}_{\mathrm{formal}}^d):=\underset{k\to \infty }{\underleftarrow{\lim }}{\mathcal{C}}^{\infty }(\text{mathbbm}{R}^d)/{\mathcal{I}}_0^k,$$

$${\mathcal{C}}^{\infty }(\text{mathbbm}{R}_{\mathrm{formal}}^d)\supset {\mathcal{I}}_0\supset {\mathcal{I}}_0^2\supset \dots$$

$${\mathrm{T}}_{\mathrm{poly}}(\text{mathbbm}{R}_{\mathrm{formal}}^d):=\underset{k\ge -1}{⨁}{\mathrm{T}}_{\mathrm{poly}}^k(\text{mathbbm}{R}_{\mathrm{formal}}^d),$$

$$[[f,g]]$$

$$[[X_0,f]]$$

Full text

1. 1 Introduction

2. 2 Preliminaries

3. 2.1 $L_{\infty}$-setting

4. 2.2 Local Formality

5. 3 Formality for Lie algebroids

6. 3.1 Fedosov resolutions

7. 3.2 Twisting procedure

8. 4 Formality and Deformation Symmetries

9. 4.1 Deformation Symmetries

10. 4.2 Twisting Poisson actions

11. 4.3 Twisted structures

12. 4.4 Comparison with Drinfeld’s construction

1 Introduction

The concept of deformation quantization has been introduced by Bayen, Flato, Fronsdal, Lichnerowicz and Sternheimer in their seminal paper [4] based on the theory of associative deformations of algebras [23]. A formal star product on a Poisson manifold $M$ is defined as a formal associative deformation of the algebra of smooth functions $\mathscr{C}^{\infty}(M)$ on $M$ (the name comes from the notation $\star$ for the deformed product) and its existence has been proved as a corollary of the so-called formality theorem in [27] (for more details in deformation quantization we refer to the textbooks [16, 36]). On the other hand, Drinfeld introduced the notion of quantum groups in the setting of formal deformations, see e.g. the textbooks [8, 20] for a detailed discussion. Drinfeld also introduced the idea of using symmetries to get formal deformations. More explicitely, given an action by derivations of a Lie algebra $\mathfrak{g}$ on an associative algebra $(\mathscr{A},m_{\mathscr{A}})$, the definition of the so-called Drinfeld twist [10, 12] $J\in(\mathscr{U}({\mathfrak{g}})\mathbin{\otimes}\mathscr{U}(\mathfrak{g}))[\![\hbar]\!]$ allows us to obtain an associative formal deformation of $\mathscr{A}$ by means of a universal deformation formula

[TABLE]

for $a,b\in\mathscr{A}[\![\hbar]\!]$. Here $\mathbin{\triangleright}$ is the action of $\mathfrak{g}$ extended to the universal enveloping algebra $\mathscr{U}(\mathfrak{g})$ and then to $\mathscr{U}(\mathfrak{g})\mathbin{\otimes}\mathscr{U}(\mathfrak{g})$ acting on $\mathscr{A}\mathbin{\otimes}\mathscr{A}$. The deformed algebra $(\mathscr{A}[\![\hbar]\!],\star_{J})$ is then a module-algebra for the quantum group:

[TABLE]

In other words, Drinfeld obtains a quantized action. We mention here that the relevance of deformations induced via symmetries has been deeply investigated in [25] and in a non-formal setting in [5].

The aim of this paper consists in obtaining a more general notion of deformation through symmetry, by using formality theory. We focus on the quantization of Lie algebra actions in the particular case of triangular Lie algebras. Such actions can be regarded as the infinitesimal version of Poisson Lie group actions (see e.g. [28, 34]) and they are very important in the context of integrable systems. Triangular Lie algebras and their quantizations have been studied by many authors, see e.g. [7, 17, 37]. The idea of applying formality to actions has also been used in [2], where the authors use the Kontsevich formality on a Poisson manifold to construct for each Poisson vector field a derivation of the star product. We recover this result.

The formality theorem states the existence of an $L_{\infty}$-quasi-isomorphism from polyvectorfields to polydifferential operators on a manifold $M$. In [15, 14] Dolgushev proves the theorem for general $M$ using the proof for $M=\mathbbm{R}^{n}$. In order to construct such $L_{\infty}$-quasi-isomorphisms, Dolgushev uses Fedosov’s methods [21] concerning formal geometry, Kontsevich’s quasi-isomorphism [27] and the twisting procedure inspired by Quillen [33]. Following the construction provided by Dolgushev, Calaque proved a formality theorem for Lie algebroids [6]. We consider an infinitesimal action of $\mathfrak{g}$ on $M$, i.e. a Lie algebra homomorphism $\varphi\colon\mathfrak{g}\rightarrow\Gamma^{\infty}(TM)$. This can immediately be extended to a DGLA morphism

[TABLE]

where $\mathop{}\mathopen{\vphantom{\mathrm{T}_{\mathrm{poly}}}}^{\mathfrak{g}}\kern-0.5pt\mathrm{T}_{\mathrm{poly}}=\wedge^{\bullet}\mathfrak{g}$ and $\mathop{}\mathopen{\vphantom{\mathrm{T}_{\mathrm{poly}}}}\kern-0.5pt\mathrm{T}_{\mathrm{poly}}(M)=\Gamma^{\infty}(\wedge^{\bullet}TM)$ with the brackets extended via a Leibniz rule. From the formality theorem we know that we have the following $L_{\infty}$-quasi-isomorphisms

[TABLE]

Using the quasi-invertibility of $L_{\infty}$-quasi-isomorphisms we obtain the existence of an $L_{\infty}$-morphism

[TABLE]

If the Lie algebra $\mathfrak{g}$ is endowed with an $r$-matrix, i.e. an element $r\in\mathfrak{g}\wedge\mathfrak{g}$ satisfying the Maurer–Cartan equation $\llbracket r,r\rrbracket=0$, the action always induces a Poisson structure on $M$ and it is automatically a Poisson action.

Lemma 1.1**.**

1. i.)

Given the formal Maurer–Cartan element $\hbar r\in\mathop{}\mathopen{\vphantom{\mathrm{T}_{\mathrm{poly}}}}^{\mathfrak{g}}\kern-0.5pt\mathrm{T}_{\mathrm{poly}}[\![\hbar]\!]$, we obtain via formality a Maurer–Cartan element $\rho_{\hbar}$. This yields a quantum group $\mathscr{U}_{\rho_{\hbar}}(\mathfrak{g})$ with deformed coproduct $\Delta_{1\otimes 1+\rho_{\hbar}}$. 2. ii.)

Given the Maurer–Cartan element $\hbar\pi=\varphi\wedge\varphi(r)\in\mathop{}\mathopen{\vphantom{\mathrm{T}_{\mathrm{poly}}}}\kern-0.5pt\mathrm{T}_{\mathrm{poly}}(M)[\![\hbar]\!]$, we obtain via formality a Maurer–Cartan element $B_{\hbar}$. This induces a formal deformation $(\mathscr{C}^{\infty}(M)[\![\hbar]\!],\star_{B_{\hbar}})$ of the Poisson algebra $(\mathscr{C}^{\infty}(M),\pi)$.

The DGLA obtained from twisting the DGLA of $\mathfrak{g}$-polydifferential operators on the point $\mathop{}\mathopen{\vphantom{\mathrm{D}_{\mathrm{poly}}}}^{\mathfrak{g}}\kern-0.5pt\mathrm{D}_{\mathrm{poly}}$ as given in [6] turns out to be a special case of a DGLA canonically associated to any Hopf algebra $H$, which we call $H_{poly}$. Given any Maurer-Cartan element $F\in H_{poly}$ there is the associated Drinfeld twist $J=1\otimes 1+F$ (in the formal sense). It turns out that the twisted DGLA $H_{poly}^{F}$ is canonically isomorphic to $(H_{J})_{poly}$ where $H_{J}$ denotes the Hopf algebra twisted by $J$. Thus, using the twisting procedure on the $L_{\infty}$-morphism (1.5) we prove the following theorem.

Theorem 1.2**.**

*Let $\mathfrak{g}$ be a Lie algebra endowed with a classical $r$-matrix and a Lie algebra action $\varphi\colon\mathfrak{g}\rightarrow\Gamma^{\infty}(TM)$ inducing a Poisson structure on $M$ by $\pi:=\varphi\wedge\varphi(r)$. Then, there exists an $L_{\infty}$-morphism $(\mathscr{U}_{F}(\mathfrak{g})[\![\hbar]\!])_{poly}\to C(\mathscr{A}_{\hbar};\mathscr{A}_{\hbar})$ between the DGLA associated to the quantum group $\mathscr{U}_{\rho_{\hbar}}(\mathfrak{g})[\![\hbar]\!]$ and the Hochschild complex of the deformation quantization $\mathscr{A}_{\hbar}$ of $\mathscr{C}^{\infty}(M)$. *

This theorem motivates a definition, which generalizes Drinfeld quantized action.

Definition 1.3** (Deformation Symmetry).**

A deformation symmetry of a Hopf algebra $H$ in a unital associative algebra $\mathscr{A}$ is a map

[TABLE]

*of $L_{\infty}$-algebras. *

Comparing the quantized structures obtained with our approach, it is easy to see that we recover Drinfeld’s universal deformation formulas.

The paper is organized as follows. In Section 2 we recall the language of $L_{\infty}$-algebras and the theorem, due to Kontsevich, stating the existence of an $L_{\infty}$-quasi-isomorphism between polyvector fields and polydifferential operators on the formal completion at $0\in\mathbbm{R}^{d}$. In Section 3 we briefly discuss the proof of formality for Lie algebroids, following [7, 15, 14]. In particular, we recall the twisting procedure in the curved context. Section 4 contains the main results of the paper, i.e. the construction of an $L_{\infty}$-morphism out of a Poisson action and the discussion on twisted structures and deformation symmetry. Finally we compare our approach with Drinfeld’s deformation formulas.

Acknoledgments

The authors are grateful to Ryszard Nest and Stefan Waldmann for the inspiring discussions.

2 Preliminaries

Given a graded vector space $V^{\bullet}$ over $\mathbbm{K}$ we denote the $k$-shifted vector space by $V[k]^{\bullet}$, it is given by

[TABLE]

2.1 $L_{\infty}$-setting

We shall recall the definitions of $L_{\infty}$-algebra and $L_{\infty}$-morphisms for the convenience of the reader (and to fix certain conventions). For the rest of this section we consider a field $\mathbbm{K}$ of characteristic [math]. Although many constructions will also allow for replacement of $\mathbbm{K}$ by a PID containing the rationals.

Definition 2.1** ($L_{\infty}$-algebra).**

*A degree $+1$ coderivation $Q$ on the co-unital conilpotent cocommutative coalgebra $S^{c}(\mathfrak{L})$ cofreely cogenerated by the graded vector space $\mathfrak{L}[1]^{\bullet}$ over $\mathbbm{K}$ is called an $L_{\infty}$-structure on the graded vector space $\mathfrak{L}$ if $Q^{2}=0$. *

In more explicit terms we have

[TABLE]

equipped with the coproduct $\Delta$ given by

[TABLE]

for $k\geq 1$ and any $\gamma_{i}\in\mathfrak{L}[1]$. Here we have

[TABLE]

where $\mathrm{Sh}(i,k-i)$ denotes the $(i,k-i)$ shuffles in the symmetric group $S_{k}$ in $k$ letters and the Koszul sign $\epsilon(\sigma)=\epsilon(\sigma,\gamma_{1},\ldots,\gamma_{k})$ is determined by the rule

[TABLE]

Recall that $S^{c}(\mathfrak{L})$ is given by the co-invariants of the tensor algebra for the action of the symmetric groups generated by

[TABLE]

where we use the vertical bars to denote the shifted degree, i.e. the degree of $\gamma_{i}$ in $\mathfrak{L}[1]$. The co-unit is given by the projection $\mathrm{pr}_{\mathbbm{K}}$ onto the ground field $\mathbbm{K}$.

Remark 2.2**.**

A direct computation shows that, denoting the flip $a\otimes b\mapsto(-1)^{|a||b|}b\otimes a$ by $\tau$, we have

[TABLE]

So we obtain the unital and co-unital bialgebra $(S^{c}(\mathfrak{L}),\,\cdot\,\wedge\,\cdot\,,1\in\mathbbm{K},\Delta,\mathrm{pr}_{\mathbbm{K}})$, i.e. $1\wedge X=X$ for all $X\in S^{c}(\mathfrak{L})$. We sometimes abuse notation by omitting $\wedge$ in favor of simple concatenation or superscripts, e.g. $ab:=a\wedge b$ and $x^{3}:=x\wedge x\wedge x$.

Lemma 2.3** (Characterization of coderivations).**

Every degree $+1$ coderivation $Q$ on $S^{c}(\mathfrak{L})$ is uniquely determined by the components

[TABLE]

by the formula

[TABLE]

*where we use the conventions that $\mathrm{Sh}(n,0)=\mathrm{Sh}(0,n)=\{\operatorname{\mathsf{id}}\}$ and that the empty product equals the unit. *

Proof**:**

It follows by simply writing out both sides of the defining equation

[TABLE]

$\heartsuit$

Note that $Q_{0}(1)$ is of degree $1$ in $\mathfrak{L}[1]$ (thus of degree $2$ in $\mathfrak{L}$). The condition $Q^{2}=0$ can now be expressed in terms of a quadratic equation in the components $Q_{n}$.

Example 2.4** (Curved Lie algebras).**

Our main example of an $L_{\infty}$-algebra is given by (curved) Lie algebras, i.e. the tuple $(\mathfrak{L},R,\mathop{}\!\mathrm{d},[\,\cdot\,,\,\cdot\,])$ where we set $Q_{0}(1)=R$, $Q_{1}=\mathop{}\!\mathrm{d}$, $Q_{2}=[\,\cdot\,,\,\cdot\,]$ and $Q_{i}=0$ for all $i\geq 3$. The condition $Q^{2}=0$ amounts to:

• •

$\mathop{}\!\mathrm{d}R=0$,

• •

$\mathop{}\!\mathrm{d}^{2}(\,\cdot\,)=[R,\,\cdot\,]$,

• •

$\mathop{}\!\mathrm{d}$ is a derivation of $[\,\cdot\,,\,\cdot\,]$,

• •

The graded Jacobi identity for $[\,\cdot\,,\,\cdot\,]$.

Remark 2.5**.**

We should note that the our definition of $L_{\infty}$-algebra is usually called curved $L_{\infty}$-algebra (see e.g.  [30]). Although this definition is also not set in stone, see for instance [26] for yet another notion of curved $L_{\infty}$-algebra. For the purpose of this paper it is, however, more convenient to call the curved version simply $L_{\infty}$-algebra. The only $L_{\infty}$-algebras playing a role in this paper are, however, the flat $L_{\infty}$-algebras, i.e. those having $Q_{0}=0$. The usual definition for an $L_{\infty}$-algebra thus coincides with our definition of flat $L_{\infty}$-algebra.

Remark 2.6**.**

In the following we have to deal with various infinite sums. In order for this to make sense, we always consider only $L_{\infty}$-algebras $\mathfrak{L}$ that are equipped with a decreasing filtration

[TABLE]

respecting the $L_{\infty}$-structure and which is moreover complete, i.e.

[TABLE]

This yields a corresponding complete metric topology and we consider convergence of infinite sums in terms of this topology.

Definition 2.7** ($L_{\infty}$-morphisms).**

Let $\mathfrak{L}$ and $\widetilde{\mathfrak{L}}$ be $L_{\infty}$-algebras. A degree [math] filtration preserving co-unital co-algebra morphism

[TABLE]

*such that $FQ=\widetilde{Q}F$ is called an $L_{\infty}$-morphism. *

Lemma 2.8** (Characterization of co-algebra morphisms).**

*A co-algebra morphism $F$ from

$S^{c}(\mathfrak{L})$ to $S^{c}(\widetilde{\mathfrak{L}})$ is uniquely determined by its components, also called Taylor coefficients,*

[TABLE]

where $n\geq 1$. Namely, we set $F(1)=1$ and use the formula

[TABLE]

*where $Sh(k_{1},...,k_{p})$ denotes the set of $(k_{1},\ldots,k_{p})$-shuffles in $S_{n}$ and $Sh(n)=\{\operatorname{\mathsf{id}}\}$. *

Proof**:**

It simply follows by writing out the defining equation

[TABLE]

$\heartsuit$

Example 2.9**.**

Let $(\mathfrak{L},R,\mathop{}\!\mathrm{d},[\,\cdot\,,\,\cdot\,])$ and $(\mathfrak{L}^{\prime},R^{\prime},\mathop{}\!\mathrm{d}^{\prime},[\,\cdot\,,\,\cdot\,]^{\prime})$ be two curved Lie algebras and consider the morphism $C\colon\mathfrak{L}\longrightarrow\mathfrak{L}^{\prime}$ of curved Lie algebras, i.e. $C(R)=R^{\prime}$, $C\mathop{}\!\mathrm{d}=\mathop{}\!\mathrm{d}^{\prime}C$ and $C$ is a morphism of the underlying Lie algebras. Then the map $F$ given by applying the formula (2.16) to the components $F_{1}=C$ and $F_{i}=0$ for $i\geq 2$ is an $L_{\infty}$-morphism. In general, if $F\colon\mathfrak{L}\longrightarrow\mathfrak{L}^{\prime}$ is an $L_{\infty}$-morphism, then $F_{1}(R)=R^{\prime}$, but we only have $\mathop{}\!\mathrm{d}^{\prime}F_{1}(\gamma)=F_{1}\mathop{}\!\mathrm{d}(\gamma)+F_{2}(R\wedge\gamma)$.

Note that, given an $L_{\infty}$-morphism of flat $L_{\infty}$-algebras $\mathfrak{L}$ and $\widetilde{\mathfrak{L}}$, we obtain the map of complexes

[TABLE]

Definition 2.10** ($L_{\infty}$-quasi-isomorphism).**

*An $L_{\infty}$-morphism $F$ is called $L_{\infty}$-quasi-isomorphism if $F_{1}$ is a quasi-isomorphism of complexes. *

The $L_{\infty}$-quasi-isomorphisms we deal with in this paper happen to be the ones witnessing formality, let us therefore introduce the notion of formal $L_{\infty}$-algebras here.

Definition 2.11** (Formal $L_{\infty}$-algebra).**

An $L_{\infty}$-algebra $\mathfrak{L}$ is called formal if it is flat and admits an $L_{\infty}$-quasi-isomorphism

[TABLE]

*for the $L_{\infty}$-structure canonically induced on the cohomology $\mathrm{H}(\mathfrak{L})$ of $\mathfrak{L}$. *

Finally, a crucial concept for this paper is the one of Maurer–Cartan elements, that we define below.

Definition 2.12** (Maurer-Cartan element).**

Given an $L_{\infty}$-algebra $(\mathfrak{L},Q)$, an element $\pi\in\mathcal{F}^{1}\mathfrak{L}[1]^{0}$ is called a Maurer-Cartan or MC element if it satisfies the following equation

[TABLE]

2.2 Local Formality

Let us denote the formal completion at $0\in\mathbbm{R}^{d}$ by $\mathbbm{R}^{d}_{\mathrm{formal}}$. The smooth functions $\mathscr{C}^{\infty}(\mathbbm{R}^{d}_{\mathrm{formal}})$ on $\mathbbm{R}^{d}_{\mathrm{formal}}$ are given by the algebra

[TABLE]

where $\mathcal{I}_{0}$ denotes the ideal of functions vanishing at $0\in\mathbbm{R}^{d}$. Note that $\mathscr{C}^{\infty}(\mathbbm{R}^{d}_{\mathrm{formal}})$ comes equipped with the complete decreasing filtration

[TABLE]

and its corresponding (metric) topology. The Lie algebra of continuous derivations of $\mathscr{C}^{\infty}(\mathbbm{R}^{d}_{\mathrm{formal}})$ is denoted by $\mathop{}\mathopen{\vphantom{\mathrm{T}_{\mathrm{poly}}^{0}}}\kern-0.5pt\mathrm{T}_{\mathrm{poly}}^{0}(\mathbbm{R}^{d}_{\mathrm{formal}})$. By setting $\mathop{}\mathopen{\vphantom{\mathrm{T}_{\mathrm{poly}}^{-1}}}\kern-0.5pt\mathrm{T}_{\mathrm{poly}}^{-1}:=\mathscr{C}^{\infty}(\mathbbm{R}^{d}_{\mathrm{formal}})$ we obtain the Lie–Rinehart pair $(\mathop{}\mathopen{\vphantom{\mathrm{T}_{\mathrm{poly}}^{-1}}}\kern-0.5pt\mathrm{T}_{\mathrm{poly}}^{-1},\mathop{}\mathopen{\vphantom{\mathrm{T}_{\mathrm{poly}}^{0}}}\kern-0.5pt\mathrm{T}_{\mathrm{poly}}^{0})$ and the graded vector space

[TABLE]

where $\mathop{}\mathopen{\vphantom{\mathrm{T}_{\mathrm{poly}}^{k}}}\kern-0.5pt\mathrm{T}_{\mathrm{poly}}^{k}(\mathbbm{R}^{d}_{\mathrm{formal}}):=\Lambda^{k+1}\mathop{}\mathopen{\vphantom{\mathrm{T}_{\mathrm{poly}}^{0}}}\kern-0.5pt\mathrm{T}_{\mathrm{poly}}^{0}(\mathbbm{R}^{d}_{\mathrm{formal}})$ for $k\geq 0$. Here the tensor product is understood to be over $\mathop{}\mathopen{\vphantom{\mathrm{T}_{\mathrm{poly}}^{-1}}}\kern-0.5pt\mathrm{T}_{\mathrm{poly}}^{-1}(\mathbbm{R}^{d}_{\mathrm{formal}})$ and completed. Notice that there is no confusion about grading here although it may seem unnatural at first glance. It is actually obtained by shifting the natural grading. The natural structure is that of Gerstenhaber algebra, but we are only considering the underlying graded Lie algebra. The Lie bracket $\llbracket\,\cdot\,,\,\cdot\,\rrbracket$ on $\mathop{}\mathopen{\vphantom{\mathrm{T}_{\mathrm{poly}}^{0}}}\kern-0.5pt\mathrm{T}_{\mathrm{poly}}^{0}(\mathbbm{R}^{d}_{\mathrm{formal}})$ extends to a graded Lie algebra structure on $\mathop{}\mathopen{\vphantom{\mathrm{T}_{\mathrm{poly}}}}\kern-0.5pt\mathrm{T}_{\mathrm{poly}}(\mathbbm{R}^{d}_{\mathrm{formal}})$ by the rules

[TABLE]

for all $f,g\in\mathop{}\mathopen{\vphantom{\mathrm{T}_{\mathrm{poly}}^{-1}}}\kern-0.5pt\mathrm{T}_{\mathrm{poly}}^{-1}(\mathbbm{R}^{d}_{\mathrm{formal}})$, $X_{0},\ldots,X_{k}\in\mathop{}\mathopen{\vphantom{\mathrm{T}_{\mathrm{poly}}^{0}}}\kern-0.5pt\mathrm{T}_{\mathrm{poly}}^{0}(\mathbbm{R}^{d}_{\mathrm{formal}})$ and $Y\in\mathop{}\mathopen{\vphantom{\mathrm{T}_{\mathrm{poly}}^{l}}}\kern-0.5pt\mathrm{T}_{\mathrm{poly}}^{l}(\mathbbm{R}^{d}_{\mathrm{formal}})$.

The universal enveloping algebra of the Lie-Rinehart pair $(\mathop{}\mathopen{\vphantom{\mathrm{T}_{\mathrm{poly}}^{-1}}}\kern-0.5pt\mathrm{T}_{\mathrm{poly}}^{-1}(\mathbbm{R}^{d}_{\mathrm{formal}}),\mathop{}\mathopen{\vphantom{\mathrm{T}_{\mathrm{poly}}^{0}}}\kern-0.5pt\mathrm{T}_{\mathrm{poly}}^{0}(\mathbbm{R}^{d}_{\mathrm{formal}}))$ is denoted by $\mathop{}\mathopen{\vphantom{\mathrm{D}_{\mathrm{poly}}^{0}}}\kern-0.5pt\mathrm{D}_{\mathrm{poly}}^{0}(\mathbbm{R}^{d}_{\mathrm{formal}})$. Recall that $\mathop{}\mathopen{\vphantom{\mathrm{D}_{\mathrm{poly}}^{0}}}\kern-0.5pt\mathrm{D}_{\mathrm{poly}}^{0}(\mathbbm{R}^{d}_{\mathrm{formal}})$ is naturally equipped with the structures of a bialgebra (see e.g. [31]). More precisely, $\mathop{}\mathopen{\vphantom{\mathrm{D}_{\mathrm{poly}}^{0}}}\kern-0.5pt\mathrm{D}_{\mathrm{poly}}^{0}(\mathbbm{R}^{d}_{\mathrm{formal}})$ allows an $\mathbbm{R}$-algebra structure $\cdot$ and an $\mathbbm{R}$-coalgebra structure $\Delta$. We extend the algebra structure in the obvious (componentwise) way to

[TABLE]

where $\mathop{}\mathopen{\vphantom{\mathrm{D}_{\mathrm{poly}}^{-1}}}\kern-0.5pt\mathrm{D}_{\mathrm{poly}}^{-1}(\mathbbm{R}^{d}_{\mathrm{formal}}):=\mathop{}\mathopen{\vphantom{\mathrm{T}_{\mathrm{poly}}^{-1}}}\kern-0.5pt\mathrm{T}_{\mathrm{poly}}^{-1}(\mathbbm{R}^{d}_{\mathrm{formal}})$ and \mathop{}\mathopen{\vphantom{\mathrm{D}_{\mathrm{poly}}^{k}}}\kern-0.5pt\mathrm{D}_{\mathrm{poly}}^{k}(\mathbbm{R}^{d}_{\mathrm{formal}}):=\mathopen{}\mathclose{{}\left(\mathop{}\mathopen{\vphantom{\mathrm{D}_{\mathrm{poly}}^{0}}}\kern-0.5pt\mathrm{D}_{\mathrm{poly}}^{0}(\mathbbm{R}^{d}_{\mathrm{formal}})}\right)^{\otimes k+1}. Again the tensor product is understood to be over $\mathop{}\mathopen{\vphantom{\mathrm{D}_{\mathrm{poly}}^{-1}}}\kern-0.5pt\mathrm{D}_{\mathrm{poly}}^{-1}(\mathbbm{R}^{d}_{\mathrm{formal}})$ and completed. This allows us to define two $\mathbbm{R}$-bilinear operations $\bullet$ and $[\,\cdot\,,\,\cdot\,]_{G}$ given by

[TABLE]

and

[TABLE]

where $P_{1}\in\mathop{}\mathopen{\vphantom{\mathrm{D}_{\mathrm{poly}}^{k_{1}}}}\kern-0.5pt\mathrm{D}_{\mathrm{poly}}^{k_{1}}(\mathbbm{R}^{d}_{\mathrm{formal}})$, $P_{2}\in\mathop{}\mathopen{\vphantom{\mathrm{D}_{\mathrm{poly}}^{k_{2}}}}\kern-0.5pt\mathrm{D}_{\mathrm{poly}}^{k_{2}}(\mathbbm{R}^{d}_{\mathrm{formal}})$ and $\Delta^{(k)}$ denotes the $k$-th iteration of $\Delta$ given by $(\Delta\otimes\operatorname{\mathsf{id}}^{\otimes k-1})(\Delta\otimes\operatorname{\mathsf{id}}^{\otimes k-2})\ldots(\Delta\otimes\operatorname{\mathsf{id}})\Delta$. Note that the bracket $[\,\cdot\,,\,\cdot\,]_{G}$ defines a graded Lie algebra structure on $\mathop{}\mathopen{\vphantom{\mathrm{D}_{\mathrm{poly}}}}\kern-0.5pt\mathrm{D}_{\mathrm{poly}}(\mathbbm{R}^{d}_{\mathrm{formal}})$.

Theorem 2.13** (Kontsevich[27]).**

There exists an $L_{\infty}$-quasi-isomorphism between DGLA’s

[TABLE]

where $\partial=[\mu,\,\cdot\,]_{G}$ for $\mu=1\otimes 1\in\mathop{}\mathopen{\vphantom{\mathrm{D}_{\mathrm{poly}}^{1}}}\kern-0.5pt\mathrm{D}_{\mathrm{poly}}^{1}(\mathbbm{R}^{d}_{\mathrm{formal}})$. Moreover

1. i.)

$\mathscr{K}$* is $\mathrm{GL}(d,\mathbbm{R})$ equivariant;* 2. ii.)

$\mathscr{K}_{n}(X_{1},\ldots,X_{n})=0$* for all $X_{i}\in\mathop{}\mathopen{\vphantom{\mathrm{T}_{\mathrm{poly}}^{0}}}\kern-0.5pt\mathrm{T}_{\mathrm{poly}}^{0}(\mathbbm{R}^{d}_{\mathrm{formal}})$ and $n>1$;* 3. iii.)

$\mathscr{K}_{n}(X,Y_{2},\ldots,Y_{n})=0$* for all $Y_{i}\in\mathop{}\mathopen{\vphantom{\mathrm{T}_{\mathrm{poly}}}}\kern-0.5pt\mathrm{T}_{\mathrm{poly}}(\mathbbm{R}^{d}_{\mathrm{formal}})$ and $n\geq 2$ whenever $X\in\mathop{}\mathopen{\vphantom{\mathrm{T}_{\mathrm{poly}}^{0}}}\kern-0.5pt\mathrm{T}_{\mathrm{poly}}^{0}(\mathbbm{R}^{d}_{\mathrm{formal}})$ is induced by the action of $\mathfrak{gl}(d,\mathbbm{R})$.*

3 Formality for Lie algebroids

In this section we recall the formality theorem for Lie algebroids, which is due to Calaque, see [6]. The proof of this theorem follows the lines of Dolgushev’s construction [14, 15] of the $L_{\infty}$-quasi-isomorphism from polyvectorfields to polydifferential operators. The main ingredients are Fedosov’s methods [21] concerning formal geometry, Kontsevich’s quasi-isomorphism [27] and the twisting procedure inspired by Quillen [33] (although we use Dolgushev’s version [15]). Since we only need the result and not in fact the details of the construction we are rather brief here and refer to [6] for details.

3.1 Fedosov resolutions

As a first step, Calaque constructs Fedosov resolutions of polyvector fields and polydifferential operators of Lie algebroids.

Let us recall that a Lie algebroid is a vector bundle $E\to M$ over a manifold $M$, equipped with a Lie bracket on sections $\Gamma^{\infty}(E)$ and an anchor map $\rho\colon E\to TM$, preserving the Lie bracket, such that

[TABLE]

for any $v,w\in\Gamma^{\infty}(E)$ and $f\in\mathscr{C}^{\infty}(M)$. Equivalently, we can consider the algebra of $E$-differential forms $\Gamma^{\infty}(\wedge^{\bullet}E^{*})$ endowed with the differential $\mathop{}\!\mathrm{d}_{E}$ given by $(\mathop{}\!\mathrm{d}_{E}f)(v)=\rho(v)(f)$ for $f\in\mathscr{C}^{\infty}(M)$ and $X\in\Gamma^{\infty}(E)$, by $(\mathop{}\!\mathrm{d}_{E}\alpha)(v,w)=\rho(v)(\alpha(w))-\rho(w)(\alpha(v))-\alpha([v,w]_{E})$ for $X,Y\in\Gamma^{\infty}(E)$ and $\alpha\in\Gamma^{\infty}(E^{*})$ and extended as a derivation for the wedge product.

The definitions of the DGLA’s $\mathop{}\mathopen{\vphantom{\mathrm{T}_{\mathrm{poly}}}}\kern-0.5pt\mathrm{T}_{\mathrm{poly}}(\mathbbm{R}^{d}_{\mathrm{formal}})$ and $\mathop{}\mathopen{\vphantom{\mathrm{D}_{\mathrm{poly}}}}\kern-0.5pt\mathrm{D}_{\mathrm{poly}}(\mathbbm{R}^{d}_{\mathrm{formal}})$ given in Section 2.2 go through mutatis mutandis to define the DGLA’s $\mathop{}\mathopen{\vphantom{\mathrm{T}_{\mathrm{poly}}}}^{E}\kern-0.5pt\mathrm{T}_{\mathrm{poly}}(M)$ and $\mathop{}\mathopen{\vphantom{\mathrm{D}_{\mathrm{poly}}}}^{E}\kern-0.5pt\mathrm{D}_{\mathrm{poly}}(M)$ starting from the Lie-Rinehart pair \mathopen{}\mathclose{{}\left(\mathscr{C}^{\infty}(M),\Gamma^{\infty}(E)}\right). Notice that the resulting spaces $\mathop{}\mathopen{\vphantom{\mathrm{D}_{\mathrm{poly}}^{k}}}^{E}\kern-0.5pt\mathrm{D}_{\mathrm{poly}}^{k}(M)$ can be identified with the spaces of $E$-polydifferential operators of order $k+1$. In order to extend the result of Theorem 2.13 to any Lie algebroid, we need to consider the so-called Fedosov resolutions. The idea (coming from formal geometry) consists in replacing the DGLA’s $\mathop{}\mathopen{\vphantom{\mathrm{T}_{\mathrm{poly}}}}\kern-0.5pt\mathrm{T}_{\mathrm{poly}}(\mathbbm{R}^{d}_{\mathrm{formal}})$ and $\mathop{}\mathopen{\vphantom{\mathrm{D}_{\mathrm{poly}}}}\kern-0.5pt\mathrm{D}_{\mathrm{poly}}(\mathbbm{R}^{d}_{\mathrm{formal}})$ by quasi-isomorphic DGLA’s (using DGLA morphisms in this case). For the rest of this section we consider a Lie algebroid $E$ of rank $d$. We denote by ${}^{E}\mathcal{T}_{\mathrm{poly}}$ the bundle of formal fiberwise $E$-polyvector fields over $M$, this is the bundle associated to the principal bundle of general linear frames in $E$ with fiber $\mathop{}\mathopen{\vphantom{\mathrm{T}_{\mathrm{poly}}}}\kern-0.5pt\mathrm{T}_{\mathrm{poly}}(\mathbbm{R}^{d}_{\mathrm{formal}})$. Similarly, the bundle ${}^{E}\mathcal{D}_{\mathrm{poly}}$ of formal fiberwise $E$-polydifferential operators is the bundle over $M$ associated to the principal bundle of general linear frames in $E$ with fiber $\mathop{}\mathopen{\vphantom{\mathrm{D}_{\mathrm{poly}}}}\kern-0.5pt\mathrm{D}_{\mathrm{poly}}(\mathbbm{R}^{d}_{\mathrm{formal}})$. The Fedosov resolutions are given on the level of vector spaces by the $E$-differential forms with values in ${}^{E}\mathcal{T}_{\mathrm{poly}}$ and ${}^{E}\mathcal{D}_{\mathrm{poly}}$ respectively. We denote these spaces by $\mathop{}\mathopen{\vphantom{\Omega(M;\mathcal{T}_{\mathrm{poly}})}}^{E}\kern-0.5pt\Omega(M;\mathcal{T}_{\mathrm{poly}})$ and $\mathop{}\mathopen{\vphantom{\Omega(M;\mathcal{D}_{\mathrm{poly}})}}^{E}\kern-0.5pt\Omega(M;\mathcal{D}_{\mathrm{poly}})$ respectively. Note that these spaces carry a natural DGLA structure, namely the one induced by the structure on fibers (which is $\mathrm{GL}(d,\mathbbm{R})$-equivariant).

Lemma 3.1**.**

There exist $\mathrm{GL}(d,\mathbbm{R})$-equivariant isomorphisms of algebras

[TABLE]

[TABLE]

*Here the Lie algebra structure on $\mathscr{C}^{\infty}(\mathbbm{R}^{d}_{\mathrm{formal}})\otimes T_{0}\mathbbm{R}^{d}$ is induced from the action of $T_{0}\mathbbm{R}^{d}$ as derivations at [math] on $\mathscr{C}^{\infty}(\mathbbm{R}^{d})$. *

The proof of the above lemma can be found in [9, Prop. 2.1.10] and [32, Theorem 1.1.3]. It implies that

[TABLE]

and similarly

[TABLE]

where $\Lambda$ and $\mathrm{S}$ denote the anti-symmetric and symmetric algebra, respectively.

The next step consists in finding a differential on the Fedosov resolutions which is compatible with the graded Lie algebra structure and which makes them into DGLA’s quasi-isomorphic to $\mathop{}\mathopen{\vphantom{\mathrm{T}_{\mathrm{poly}}}}^{E}\kern-0.5pt\mathrm{T}_{\mathrm{poly}}(M)$ and $\mathop{}\mathopen{\vphantom{\mathrm{D}_{\mathrm{poly}}}}^{E}\kern-0.5pt\mathrm{D}_{\mathrm{poly}}(M)$, respectively. Given some trivializing coordinate neighborhood $V\subset M$ of $E$, a local frame $e_{1},\dots,e_{d}$ and its dual frame $(x_{1},\ldots,x_{d})$, we can define the operators

[TABLE]

by the formula

[TABLE]

In other words $\delta=\llbracket A_{-1},\,\cdot\,\rrbracket$ where $A_{-1}\in\mathop{}\mathopen{\vphantom{\Omega^{1}(V;\mathcal{T}_{\mathrm{poly}}^{0})}}^{E}\kern-0.5pt\Omega^{1}(V;\mathcal{T}_{\mathrm{poly}}^{0})$ is the one-form $A_{-1}=\sum_{i=1}^{d}x_{i}\otimes e_{i}$. One easily checks that $\delta$ does not depend on the choice of coordinates, since $A_{-1}$ is independent of coordinates, and therefore extends to all of $M$. By replacing $\llbracket\,\cdot\,,\,\cdot\,\rrbracket$ by $[\,\cdot\,,\,\cdot\,]_{G}$ in (3.7) we obtain the operators

[TABLE]

Note that, since $\llbracket A_{-1},A_{-1}\rrbracket=[A_{-1},A_{-1}]_{G}=0$, we have $\delta^{2}=0$. Furthermore, since it is given by an inner derivation and $\delta\mu=0$, $\delta$ is compatible with the fiberwise Lie structures and thus yields DGLA structures.

The cohomology of the complexes \mathopen{}\mathclose{{}\left(\mathop{}\mathopen{\vphantom{\Omega^{l}(M;\mathcal{T}_{\mathrm{poly}})}}^{E}\kern-0.5pt\Omega^{l}(M;\mathcal{T}_{\mathrm{poly}}),\delta}\right) and \mathopen{}\mathclose{{}\left(\mathop{}\mathopen{\vphantom{\Omega^{l}(M;\mathcal{D}_{\mathrm{poly}})}}^{E}\kern-0.5pt\Omega^{l}(M;\mathcal{D}_{\mathrm{poly}}),\delta}\right) is given by the following proposition (proved e.g in [6, Prop. 2.1]).

Proposition 3.2**.**

We have that

[TABLE]

while

[TABLE]

Notice that

[TABLE]

as vector spaces. This does not provide us with the quasi-isomorphisms we are looking for, since $\mathrm{H}^{0}(\mathop{}\mathopen{\vphantom{\Omega(M;\mathcal{T}_{\mathrm{poly}})}}^{E}\kern-0.5pt\Omega(M;\mathcal{T}_{\mathrm{poly}}),\delta)$ carries the trivial Lie algebra structure. To correct it, the idea is to construct a perturbation of the differential $\delta$ that does not affect the size of the cohomology, but only the Lie algebra structure on cohomology. Notice that the operator $\delta$ is of degree $-1$ in terms of the filtration and so we may start perturbing at order [math], i.e. adding a connection in the bundle $E$. The fact that the resulting perturbation should square to zero forces us to choose a torsion-free connection $\nabla\!\!\!\!\nabla$. This gives us the operators

[TABLE]

Thus we consider the corresponding operators $-\delta+\nabla\!\!\!\!\nabla$ and $-\delta+\nabla\!\!\!\!\nabla+\partial$. This leads to the problem that there is no reason to assume that we can find $\nabla\!\!\!\!\nabla$ such that $\nabla\!\!\!\!\nabla^{2}=0$ (since not every Lie algebroid is flat). Following the idea of Fedosov, we correct $-\delta+\nabla\!\!\!\!\nabla$ by an inner derivation and make the ansatz

[TABLE]

with $A\in\mathop{}\mathopen{\vphantom{\Omega^{1}(M;\mathcal{T}_{\mathrm{poly}}^{0})}}^{E}\kern-0.5pt\Omega^{1}(M;\mathcal{T}_{\mathrm{poly}}^{0})\hookrightarrow\mathop{}\mathopen{\vphantom{\Omega^{1}(M;\mathcal{D}_{\mathrm{poly}}^{0})}}^{E}\kern-0.5pt\Omega^{1}(M;\mathcal{D}_{\mathrm{poly}}^{0})$ and where $[A,\,\cdot\,]$ means $\llbracket A,\,\cdot\,\rrbracket$ or $[A,\,\cdot\,]_{G}$ depending on the situation. The trick is to find $A$ such that $D^{2}=0$, as proved in [6, Prop. 2.2].

Lemma 3.3**.**

There exists a unique $A$ such that

1. i.)

$\delta A=R+\nabla\!\!\!\!\nabla A+\frac{1}{2}[A,A]$** 2. ii.)

$\delta^{-1}A=0$.

Here $R$ denotes the curvature of $\nabla\!\!\!\!\nabla$ expressed in terms of the bundle ${}^{E}\mathcal{T}_{\mathrm{poly}}(M)$ (or ${}^{E}\mathcal{D}_{\mathrm{poly}}(M)$), i.e. it is given by the equation

[TABLE]

and $\delta^{-1}$ is a particular $\delta$-homotopy from the projection onto degree [math], denoted $\sigma$, to the identity, i.e.

[TABLE]

The condition $\delta^{-1}A=0$ is simply a normalization condition ensuring uniqueness of the solution.

Proposition 3.4**.**

We have

[TABLE]

Furthermore we have

[TABLE]

Proof**:**

[6, Thm. 2.3]** $\heartsuit$

Let us denote the isomorphisms from the above Proposition by $\tau$. Then, using a Poincaré-Birkhoff-Witt-type isomorphism, Calaque constructs an isomorphism (see [6, Sec. 2.3])

[TABLE]

of filtered vector spaces.

Similarly, but in an easier way, we obtain an isomorphism

[TABLE]

of graded vector spaces. Finally, as proved in [6, Prop. 2.4-2.5], we get:

Theorem 3.5** (Fedosov Resolutions).**

The maps

[TABLE]

and

[TABLE]

*both given by $\tau\circ\nu^{-1}$ are DGLA quasi-isomorphisms. *

Let us sketch the remaining steps necessary to obtain the $L_{\infty}$-quasi-isomorphisms from $\mathop{}\mathopen{\vphantom{\mathrm{T}_{\mathrm{poly}}}}^{E}\kern-0.5pt\mathrm{T}_{\mathrm{poly}}(M)$ to $\mathop{}\mathopen{\vphantom{\mathrm{D}_{\mathrm{poly}}}}^{E}\kern-0.5pt\mathrm{D}_{\mathrm{poly}}(M)$. As a second step, one notices that in a trivializing neighborhood $U\subset M$ of $E$ the connection $\nabla\!\!\!\!\nabla$ on both $\mathop{}\mathopen{\vphantom{\Omega(M;\mathcal{T}_{\mathrm{poly}})}}^{E}\kern-0.5pt\Omega(M;\mathcal{T}_{\mathrm{poly}})$ and $\mathop{}\mathopen{\vphantom{\Omega(M;\mathcal{D}_{\mathrm{poly}})}}^{E}\kern-0.5pt\Omega(M;\mathcal{D}_{\mathrm{poly}})$ is given by $\mathop{}\!\mathrm{d}_{E}+[B_{U},\,\cdot\,]$ for some element $B_{U}\in\mathop{}\mathopen{\vphantom{\Omega^{1}(M;\mathcal{T}_{\mathrm{poly}}^{0})}}^{E}\kern-0.5pt\Omega^{1}(M;\mathcal{T}_{\mathrm{poly}}^{0})\hookrightarrow\mathop{}\mathopen{\vphantom{\Omega^{1}(M;\mathcal{D}_{\mathrm{poly}}^{0})}}^{E}\kern-0.5pt\Omega^{1}(M;\mathcal{D}_{\mathrm{poly}}^{0})$. Thus, in this neighborhood, we have $D=\mathop{}\!\mathrm{d}_{E}+[\Gamma,\,\cdot\,]$, where $\Gamma$ is a Maurer-Cartan element. We now observe that the map

[TABLE]

given by applying the map $\mathscr{K}$ from Theorem 2.13 fiberwise commutes with $\mathop{}\!\mathrm{d}_{E}$. The next step consists in twisting this map by $\Gamma$ to obtain $L_{\infty}$-quasi-isomorphisms

[TABLE]

The twisting procedure is essential in our paper and will be discussed in full detail in Section 3.2. By using the properties of Kontsevich’s quasi-isomorphism (2.28) and the fact that $\nabla\!\!\!\!\nabla$ is a $\mathfrak{gl}(d,\mathbbm{R})$ connection we find that these quasi-isomorphisms coincide on intersections and thus we obtain

[TABLE]

Remark 3.6**.**

Although it may seem that we are being sloppy with notation by writing $\mathscr{U}^{\Gamma}$, since it is not a twist a priori, it is still possible to consider it as a twist in the context of curved $L_{\infty}$-algebras. This construction will be discussed in the upcoming paper [18].

Finally we would like to define the $L_{\infty}$-quasi-isomorphism $\lambda_{D}^{-1}\circ\mathscr{U}^{\Gamma}\circ\lambda_{T}\colon\mathop{}\mathopen{\vphantom{\mathrm{T}_{\mathrm{poly}}}}^{E}\kern-0.5pt\mathrm{T}_{\mathrm{poly}}\longrightarrow\mathop{}\mathopen{\vphantom{\mathrm{D}_{\mathrm{poly}}}}^{E}\kern-0.5pt\mathrm{D}_{\mathrm{poly}}.$ One problem remains and it is that, although $\lambda_{D}$ is obviously injective, we cannot be assured that $\mathscr{U}^{\Gamma}\circ\lambda_{T}$ maps $\mathop{}\mathopen{\vphantom{\mathrm{T}_{\mathrm{poly}}}}^{E}\kern-0.5pt\mathrm{T}_{\mathrm{poly}}$ into the image of $\lambda_{D}$. However, Dolgushev [14, Prop. 5] shows that we can always modify $\mathscr{U}^{\Gamma}\circ\lambda_{T}$ using a so-called partial homotopy to obtain a new quasi-isomorphism $\overline{\mathscr{U}}$ which maps into the image of $\lambda_{D}$. Thus we obtain the $L_{\infty}$-quasi-isomorphism

[TABLE]

As a consequence, we obtain the formality theorem for a generic manifold $M$ by considering the case $E=TM$ and formality for Lie algebras by considering the case $E=\mathfrak{g}$ over a point.

Remark 3.7**.**

Note that the constructions of $D$, $\tau$ and so on are not unique, but they depend only on the choice of the torsion-free $E$-connection $\nabla\!\!\!\!\nabla$.

3.2 Twisting procedure

In the following we recall the notions of twisting DGLA’s and $L_{\infty}$-morphisms by Maurer–Cartan elements. The idea of such twisting procedures comes from Quillen’s seminal work [33]. Here we follow Dolgushev’s approach as laid out in [15]. As an example we show how one obtains the local $L_{\infty}$-quasi-isomorphisms $\mathscr{U}^{\Gamma}$ mentioned above.

Lemma 3.8**.**

Suppose $\pi\in\mathcal{F}^{1}\mathfrak{L}[1]^{0}$, then the element

[TABLE]

*is well-defined, invertible and group-like. *

Proof**:**

$\exp(\pi)$* is well-defined, since the partial sums converge by virtue of $\pi$ being in the first filtration (the filtration is respected by $\wedge$). Invertibility follows from the usual direct computations showing that $\exp(-\pi)\exp(\pi)=1=\exp(\pi)\exp(-\pi)$. The fact that $\exp(\pi)$ is group-like can similarly be deduced from a direct computation using the definition of $\Delta$ given in Section 2.1. $\heartsuit$*

Given $\pi\in\mathcal{F}^{1}\mathfrak{L}[1]^{0}$ we define the $\pi$-twist of the $L_{\infty}$-algebra $(\mathfrak{L},Q)$ as the $L_{\infty}$-algebra $\mathfrak{L}^{\pi}$ given by the pair $(\mathfrak{L},Q^{\pi})$ with

[TABLE]

Corollary 3.9**.**

*Suppose $(\mathfrak{L},Q)$ is an $L_{\infty}$-algebra and $\pi\in\mathcal{F}^{1}\mathfrak{L}[1]^{0}$, then the $\pi$-twist $(\mathfrak{L},Q^{\pi})$ is an $L_{\infty}$-algebra. *

Example 3.10**.**

Given a curved Lie algebra $(\mathfrak{L},R,\mathop{}\!\mathrm{d},[\,\cdot\,,\,\cdot\,])$ we find the twisted curved Lie algebra $(\mathfrak{L},R^{\pi},\mathop{}\!\mathrm{d}+[\pi,\,\cdot\,],[\,\cdot\,,\,\cdot\,])$, where

[TABLE]

Note in particular that the $\pi$-twist is flat exactly when $\pi$ satisfies the Maurer-Cartan equation.

Proposition 3.11**.**

*Suppose $\mathfrak{L}$ is an $L_{\infty}$-algebra and $\pi\in\mathcal{F}^{1}\mathfrak{L}[1]^{0}$, then the $\pi$-twist $\mathfrak{L}$ is flat if and only if $\pi$ is a Maurer-Cartan element. *

Proof**:**

We have

[TABLE]

since all terms in $\bigoplus_{k=2}^{\infty}\Lambda^{k}(\mathfrak{L}[1])$ cancel out by virtue of the fact that $Q^{\pi}(1)=Q^{\pi}_{0}(1)\in\mathfrak{L}[1]$. $\heartsuit$

Example 3.12**.**

For a DGLA $(\mathfrak{L},\mathop{}\!\mathrm{d},[\,\cdot\,,\,\cdot\,])$ Eq. (3.28) boils down to the usual Maurer–Cartan equation

[TABLE]

If we have similarly a curved Lie algebra with curvature $-R$ it comes down to the non-homogeneous equation

[TABLE]

Lemma 3.13**.**

*Suppose $\pi\in\mathcal{F}^{1}\mathfrak{L}[1]^{0}$, then $\pi$ is an MC element if and only if $Q(\exp(\pi))=0$. *

Proof**:**

The proof follows from the following equation

[TABLE]

$\heartsuit$

Lemma 3.14**.**

Given an L∞-morphism $F$ from $\mathfrak{L}$ to $\mathfrak{L}^{\prime}$ and an element $\pi\in\mathcal{F}^{1}\mathfrak{L}[1]^{0}$, we define the $F$-associated element $\pi_{F}\in\mathcal{F}^{1}\mathfrak{L}^{\prime}[1]^{0}$ by the formula

[TABLE]

We have

[TABLE]

Proof**:**

It follows from explicit computation using the formula (2.16). $\heartsuit$

Lemmas 3.14 and 3.13 imply the following corollary.

Corollary 3.15**.**

*If $\pi$ is an MC element, then $\pi_{F}$ is also an MC element. *

Let $F\colon(\mathfrak{L},Q)\longrightarrow(\widetilde{\mathfrak{L}},\widetilde{Q})$ be an $L_{\infty}$-morphism and $\pi\in\mathcal{F}^{1}\mathfrak{L}[1]^{0}$.

Definition 3.16** ($\pi$-twist morphism).**

The $\pi$-twist of $F$ is a map

[TABLE]

defined by

[TABLE]

Corollary 3.17**.**

*The $\pi$-twist of an $L_{\infty}$-morphism $F$ is an $L_{\infty}$-morphism. *

Proof**:**

Note that, by Lemma 3.8 and Remark 2.2, the operators of multiplication by $\exp(\pi)$ and $\exp(\pi_{F})$ are co-algebra morphisms. Thus $F^{\pi}$ is a co-algebra morphism. The relation $F^{\pi}Q^{\pi}=\widetilde{Q}^{\pi}F^{\pi}$ follows from the definitions. $\heartsuit$

Remark 3.18**.**

Given two $L_{\infty}$-morphisms $F$ and $G$ from $\mathfrak{L}$ to $\mathfrak{L}^{\prime}$ and $\mathfrak{L}^{\prime}$ to $\mathfrak{L}^{\prime\prime}$, respectively, and the elements $\pi,B\in\mathcal{F}^{1}\mathfrak{L}[1]^{0}$, we have that

[TABLE]

For the proof of the following proposition we refer to [15, Prop. 1].

Proposition 3.19**.**

Let $F\colon\mathfrak{L}\rightarrow\widetilde{\mathfrak{L}}$ be an $L_{\infty}$-quasi-isomorphism such that the induced morphisms

[TABLE]

are also $L_{\infty}$-quasi-isomorphisms for all $k$. Suppose further that $\pi\in\mathcal{F}^{1}\mathfrak{L}[1]^{0}$ is an MC element. Then the $\pi$-twist

[TABLE]

*of $F$ is also a quasi-isomorphism. *

Remark 3.20**.**

The proposition above says that the class of $L_{\infty}$-quasi-isomorphisms is closed under the operation of twisting by a Maurer–Cartan element. This provides the method of showing that an $L_{\infty}$-morphism is an $L_{\infty}$-quasi-isomorphism by showing that it is the twist of a known $L_{\infty}$-quasi-isomorphism.

Example 3.21** (Formality for $\mathbbm{R}^{d}$).**

Here we generalize the result of Theorem 2.13 from $\mathbbm{R}^{d}_{\mathrm{formal}}$ to $\mathbbm{R}^{d}$ by providing an example of the claim in Remark 3.20. From now on we set $E=TM$ and drop the $E$ for notational convenience. Proposition 3.19 allows us to obtain an $L_{\infty}$-quasi-isomorphism witnessing the formality of $\mathop{}\mathopen{\vphantom{\mathrm{D}_{\mathrm{poly}}}}\kern-0.5pt\mathrm{D}_{\mathrm{poly}}(M)$ for any manifold by twisting the formal quasi-isomorphism of Theorem 2.13. We set $M=\mathbbm{R}^{d}$ and recall that we are looking for an $L_{\infty}$-quasi-isomorphism

[TABLE]

since this would complete the diagram

[TABLE]

of $L_{\infty}$-quasi-isomorphisms. Also, recall that $D:=-\delta+\mathop{}\!\mathrm{d}$. We obtain this map $\mathscr{U}^{\delta}$ as follows. First we note that, by applying the map $\mathscr{K}$ from Theorem 2.13 fiberwise, we obtain the $L_{\infty}$-morphism

[TABLE]

By considering the filtrations by exterior degree on both these algebras we construct spectral sequences which show that $\mathscr{U}$ is a quasi-isomorphism. Using this same filtration we may consider the MC element $-A_{-1}\in\mathcal{F}^{1}\Omega(\mathbbm{R}^{d};\mathcal{T}_{\mathrm{poly}})$. Now note that $\Omega(\mathbbm{R}^{d};\mathcal{T}_{\mathrm{poly}})^{-A_{-1}}$ is exactly $(\Omega(\mathbbm{R}^{d};\mathcal{T}_{\mathrm{poly}}),D)$ and $\Omega(\mathbbm{R}^{d};\mathcal{D}_{\mathrm{poly}})^{(-A_{-1})_{\mathscr{U}}}$ is exactly $(\Omega(\mathbbm{R}^{d};\mathcal{D}_{\mathrm{poly}}),D)$, since $(-A_{-1})_{\mathscr{U}}=-A_{-1}$ by point ii*.)* of Theorem 2.13. So we obtain the diagram (3.43) by setting $\mathscr{U}^{\delta}:=\mathscr{U}^{-A_{-1}}$. This concludes the example of the claim in Remark 3.20. In order to obtain the quasi-isomorphism

[TABLE]

we need to invert the final arrow of diagram (3.43). This arrow is actually an identification (by DGLA-morphism) with the kernel of $D$ in exterior degree [math]. Thus it can be inverted without problems if we can guarantee that the map $\mathscr{U}^{\delta}\circ\lambda_{T}$ maps $\mathop{}\mathopen{\vphantom{\mathrm{T}_{\mathrm{poly}}}}\kern-0.5pt\mathrm{T}_{\mathrm{poly}}(\mathbbm{R}^{d})$ into this kernel. We refer to [14, Sect. 4.2] for an explanation of a way to correct $\mathscr{U}^{\delta}$ to have this property.

Example 3.22** (Formality for Lie algebras).**

Let us conclude this section by providing the equivalent of the proof of formality for the case where $M=\{\mbox{pt}\}$ is the connected [math]-dimensional manifold and $E$ is a $d$-dimensional Lie algebra $\mathfrak{g}$. The DGLA of polyvector fields is given by $\mathrm{CE}_{\bullet}(\mathfrak{g})$, the Chevalley-Eilenberg complex with the trivial differential. The complex of $E$-differential forms with values in the fiberwise polyvector fields is thus given by

[TABLE]

where we have denoted $\widehat{\mathcal{S}}(\mathfrak{g}^{*})=\prod_{k\geq 0}S^{k}\mathfrak{g}^{*}$ and the differential $\delta-d_{E}$ coincides with the usual Chevalley-Eilenberg differential. A linear $E$-connection $\nabla\!\!\!\!\nabla$ is simply given by a linear map

[TABLE]

The corresponding map $\nabla\!\!\!\!\nabla\colon\Lambda^{\bullet}\mathfrak{g}^{*}\rightarrow\Lambda^{\bullet+1}\mathfrak{g}^{*}$ is given by extending the formula

[TABLE]

from one-forms as a $\wedge$-derivation. Note that the $\mathfrak{g}$-differential $\mathop{}\!\mathrm{d}_{\mathfrak{g}}$ is simply given by $X\otimes Y\mapsto[X,Y]$. Suppose $\{e_{i}\}_{i=1}^{d}$ is a basis for $\mathfrak{g}$ with dual basis $\{e^{i}\}_{i=1}^{d}$. Then the torsion-freeness of the connection $\nabla\!\!\!\!\nabla$ can be expressed as $\tilde{\Gamma}_{ij}^{k}=\tilde{\Gamma}_{ji}^{k}$ in terms of the Christoffel symbols $\tilde{\Gamma}_{ij}^{k}\in\mathbbm{R}$ defined by

[TABLE]

where we have used the Einstein summation convention. Let us consider also the relative Christoffel symbols $\Gamma_{ij}^{k}$ defined by

[TABLE]

i.e. $\Gamma_{ij}^{k}=\tilde{\Gamma}_{ij}^{k}-\frac{1}{2}c_{ij}^{k}$ where $c_{ij}^{k}$ are the structure constants. In terms of these torsion-freeness is equivalent to the equation

[TABLE]

Note in particular that the connection $\mathop{}\!\mathrm{d}_{\mathfrak{g}}$ is not torsion-free. The most obvious choice of torsion-free connection is given by $\Gamma_{ij}^{k}=\frac{1}{2}c_{ij}^{k}$, but we leave the choice of symmetric part open. Given any connection $\nabla\!\!\!\!\nabla$ it is given on $\mathrm{CE}^{\bullet}(\mathfrak{g};\mathrm{CE}_{\bullet}(\mathfrak{g};\widehat{\mathcal{S}}(\mathfrak{g}^{*})))$ by the formula

[TABLE]

where we have used the hat to signify that we consider $\hat{e}^{j}\in\widehat{S}(\mathfrak{g}^{*})$. Similar statements hold for $\mathop{}\mathopen{\vphantom{\mathrm{D}_{\mathrm{poly}}}}^{\mathfrak{g}}\kern-0.5pt\mathrm{D}_{\mathrm{poly}}$. Now the example proceeds identically to the previous one.

4 Formality and Deformation Symmetries

In this section we prove the main result of this paper, which leads to a new perspective on Drinfeld’s approach to deformation quantization. First we construct certain $L_{\infty}$-algebras related to a Hopf algebra or more generally a unital bialgebra and show how one obtains deformations from Drinfeld twists and maps into a Hochschild cochain complex. Then we briefly recall the basic notions of Poisson action and triangular Lie algebra. We consider the particular case of a Poisson action of a triangular Lie algebra $(\mathfrak{g},r)$ on a manifold $M$ and we show that we can construct a corresponding $L_{\infty}$-morphism between polydifferential operators $\mathop{}\mathopen{\vphantom{\mathrm{D}_{\mathrm{poly}}}}^{\mathfrak{g}}\kern-0.5pt\mathrm{D}_{\mathrm{poly}}$ and $\mathop{}\mathopen{\vphantom{\mathrm{D}_{\mathrm{poly}}}}\kern-0.5pt\mathrm{D}_{\mathrm{poly}}(M)$. This morphism induces a DGLA morphism between a quantum group associated to our Lie algebra and a deformed algebra of smooth functions on $M$.

4.1 Deformation Symmetries

In the following we define the concept of a deformation symmetry. This notion is inspired by Drinfeld’s work on deformation through quantum actions and Drinfeld twists. Let us start by recalling the definition of Drinfeld twist. In this section we shall fix the Hopf algebra $(H,\Delta,\epsilon,S)$ over the PID $\mathsf{R}$ containing $\mathbbm{Q}$.

Definition 4.1** (Drinfeld twist, [10, 12]).**

An element $J\in H\otimes H$ is said to be a twist on $H$ if the following three conditions are satisfied.

1. i.)

$J$* is invertible;* 2. ii.)

$(\Delta\mathbin{\otimes}1)(J)(J\mathbin{\otimes}1)=(1\mathbin{\otimes}\Delta)(J)(1\mathbin{\otimes}J)$* and* 3. iii.)

$(\epsilon\mathbin{\otimes}1)J=(1\mathbin{\otimes}\epsilon)J=1$.

In the following we consider formal deformations. If we consider twists in $H[\![\hbar]\!]$, the condition of invertibility and “co-invertibility” (condition iii*.)* in the above definition) may be replaced by a stronger condition which is easier to check. In fact this condition may be formulated for any Hopf algebra equipped with a complete filtration $H=\mathcal{F}^{0}H\supset\mathcal{F}^{1}H\supset\ldots$.

Definition 4.2** (Formal Drinfeld twist).**

*Let $H$ be equipped with the complete filtration

$H=\mathcal{F}^{0}H\supset\mathcal{F}^{1}H\supset\ldots$. Then an element $J\in H\otimes H$ is said to be a formal twist on $H$ if $J$ satisfies ii.) of Definition 4.1 and $J-1\otimes 1\in\mathcal{F}^{1}(H\otimes H)$. *

Corollary 4.3**.**

*A formal twist on $H$ is a twist on $H$. *

Proof**:**

This follows immediately from the compatibility of the Hopf algebra structure with the complete filtration. $\heartsuit$

It turns out that the definition of formal twist coincides exactly with the definition of Maurer-Cartan element on a certain DGLA that we shall now define. The main observation is that the formulas (2.26) and (2.27) for the Gerstenhaber bracket on $\mathop{}\mathopen{\vphantom{\mathrm{D}_{\mathrm{poly}}}}^{E}\kern-0.5pt\mathrm{D}_{\mathrm{poly}}$ only involve the structure of a unital bialgebra. From now on we denote

[TABLE]

For $P_{1}\in T^{k_{1}+1}H$ and $P_{2}\in T^{k_{2}+1}H$ set

[TABLE]

and

[TABLE]

Proposition 4.4**.**

*The graded vector space $TH[1]$ equipped with the bracket $[\,\cdot\,,\,\cdot\,]_{H}$ is a graded Lie algebra. *

Proof**:**

We can immediately extend $[\,\cdot\,,\,\cdot\,]_{H}$ to non-homogeneous elements, since $\bullet$ can be extended by bilinearity. Thus the bilinearity and anti-symmetry of $[\,\cdot\,,\,\cdot\,]_{H}$ follow immediately from the bilinearity of $\bullet$, which follows in turn from the linearity of the coproduct and the bilinearity of the product. Finally denote the associator of $\bullet$ by $\alpha$, i.e.

[TABLE]

Then the average of $\alpha$ over the symmetric group $S_{3}$ is [math], i.e

[TABLE]

*Here $S_{3}$ acts on $(TH[1])^{\otimes 3}$ through the usual signed permutation of tensor legs. The last equation is obviously equivalent to the Jacobi identity for $[\,\cdot\,,\,\cdot\,]_{H}$. *$\heartsuit$

Remark 4.5**.**

The structure $\bullet$ on $TH[1]$ is actually the pre-Lie structure coming from a brace algebra structure. As such the identity (4.5) can actually be proved by showing the pre-Lie identity

[TABLE]

The braces underlying the brace algebra structure are given by

[TABLE]

where $P\in T^{k+1}H$ and $Q_{j}\in T^{k_{j}+1}H$ for all $j$. Note that restricting this brace algebra structure to $H=T^{1}H=(TH[1])^{0}$ we find the brace algebra given in [1, Section 6].

Let us denote the twist of $TH[1]$ by the “Maurer-Cartan" element $1\otimes 1$ as

[TABLE]

Lemma 4.6**.**

*An element $J\in T^{2}H$ is a formal twist on $H$ if and only if $J-1\otimes 1$ is a Maurer-Cartan element in $H_{poly}$. *

Proof**:**

Suppose first that $J\in T^{2}H$ is a formal twist on $H$. Then, by definition, $F:=J-1\otimes 1$ is an element of $\mathcal{F}^{1}H_{poly}^{1}$ and

[TABLE]

by condition ii.) of Definition 4.1. Conversely, suppose $F$ is a Maurer-Cartan element in $H_{poly}$, then, for $J=1\otimes 1+F$,

[TABLE]

*by the Maurer-Cartan equation. So $J$ satisfies ii.) of 4.1, while clearly $J-1\otimes 1=F\in\mathcal{F}^{1}(H\otimes H)$. *$\heartsuit$

Drinfeld discovered [12, 11] that one can twist the Hopf algebra structure on $H$ by any (formal) twist $J$. More explicitely, one obtains the twisted Hopf algebra $H_{J}$ by changing only the coproduct $\Delta$ to $\Delta_{J}$ given by

[TABLE]

Let us fix the (formal) twist $J$ on $H$. It is convenient to introduce the following notation

[TABLE]

and we set $J_{0}=1$. Note that $J_{k}\in H^{\otimes k+1}$ is invertible with $J_{k}^{-1}$ given by reversing the order of terms above and replacing $J$ by $J^{-1}$.

Lemma 4.7**.**

The iterates of $\Delta_{J}$ are given by the formula

[TABLE]

Proof**:**

The proof is given by straightforward computation. $\heartsuit$

We also need the following (slightly technical) lemma.

Lemma 4.8**.**

The $J_{k}$’s satisfy the relation

[TABLE]

*for all $0\leq i\leq k\in\mathbbm{Z}$ and $l\in\mathbbm{Z}_{\geq 0}$. *

Proof**:**

For $k=i=0$, the formula (4.12) reads

[TABLE]

which is obviously satisfied. For $k=1$ and $i=0$ we get

[TABLE]

which follows immediately from the definition of $J_{l+1}$. For $k=1$, $i=1$ and $l=0$ we find

[TABLE]

We will fully establish the $k=1$ case by induction now. So suppose the formula 4.12 holds for $k=1$, $i=1$, $l-1\in\mathbbm{Z}_{\geq 0}$. Then

[TABLE]

Thus we have established the $k=0$ and $k=1$ cases completely. To establish the $k\geq 2$ cases, we proceed by induction. Suppose that the formula (4.12) is satisfied for all triples $(\kappa,i,l)\in(\mathbbm{Z}_{\geq 0})^{3}$ where $i\leq\kappa$ and $\kappa<k$. Then we have

[TABLE]

Thus, Eq. (4.12) is satisfied for all triples $(k,i,l)$. $\heartsuit$

Proposition 4.9**.**

*The map $\mathscr{J}\colon(H_{J})_{poly}\longrightarrow(H_{poly})^{J-1\otimes 1}$ given by $\mathscr{J}(P)=J_{k}\cdot P$ for $P\in(H_{J})_{poly}^{k}$ is a DGLA isomorphism. *

Proof**:**

First we note that $\mathscr{J}^{-1}(P)=J_{k}^{-1}P$ for $P\in H_{poly}^{k}$ is obviously an inverse of $\mathscr{J}$. Furthermore we observe that $\mathscr{J}(1\otimes 1)=J$, which shows that we only need to check that $\mathscr{J}$ preserves the brackets. This follows by direct computation from Lemmas 4.7 and 4.8 $\heartsuit$

Definition 4.10** (Deformation Symmetry).**

Let $\mathscr{A}$ be a unital associative algebra over $\mathsf{R}$ equipped with a complete filtration $\mathscr{A}=\mathcal{F}^{0}\mathscr{A}\supset\mathcal{F}^{1}\mathscr{A}\supset\ldots$ and consider the Hochschild DGLA structure on the complex $C^{\bullet}(\mathscr{A})=C^{\bullet}(\mathscr{A};\mathscr{A})[1]$. A deformation symmetry of $H$ in $\mathscr{A}$ is a map

[TABLE]

*of $L_{\infty}$-algebras. *

The previous proposition implies the following claim.

Corollary 4.11**.**

*Any formal twist on $H$ produces deformations of all algebras $\mathscr{A}$ equipped with a deformation symmetry of $H$. *

The notion of deformation symmetry is a generalization of the standard notion of universal deformation via Drinfeld twist, see e.g. [5, 19, 25]. Universal deformation formula relies on the notion of Hopf algebra action that we recall in the following definition.

Definition 4.12** (Hopf algebra action).**

Let $\mathscr{A}$ be as in Definition 4.10. Then an action of the Hopf algebra $H$ on $\mathscr{A}$ is defined as a map

[TABLE]

such that

[TABLE]

*Here $\mu$ denotes the multiplication of $\mathscr{A}$, $\mu_{\mathsf{R}}$ denotes the multiplication of $\mathsf{R}$, $\mu_{H}$ denotes the multiplication of $H$, $\eta_{\mathscr{A}}\colon\mathsf{R}\rightarrow A$ denotes the unit of $A$, $\eta_{H}\colon\mathsf{R}\rightarrow H$ denotes the unit of $H$ and $\tau_{H\otimes A}$ denotes the flip $h\otimes a\mapsto a\otimes h$. *

Notice that $\phi$ can be regarded as a map $H\rightarrow\operatorname{\mathsf{End}}_{\mathsf{R}}(A)=C^{1}(A;A)$ satisfying certain conditions.

Proposition 4.13**.**

Given a Hopf algebra action $\phi$ of $H$ on $A$, the map $\Phi$ defined by

[TABLE]

*is a deformation symmetry. Here $\mu_{\mathscr{A}}^{(k)}$ denotes the $k$-th iteration $\mu_{\mathscr{A}}\circ(\operatorname{\mathsf{id}}\otimes\mu_{\mathscr{A}})\circ\ldots\circ(\operatorname{\mathsf{id}}^{\otimes k-1}\otimes\mu_{\mathscr{A}})$ of $\mu_{\mathscr{A}}$. *

Proof**:**

We prove this proposition by showing that $\Phi$ is a map of DGLA’s. Note that for $P=P_{0}\otimes\ldots\otimes P_{k}$, $Q=Q_{0}\otimes\ldots\otimes Q_{l}$ and $a=a_{0}\otimes\ldots\otimes a_{k+l}$ we have

[TABLE]

where we have used the brace notation, see e.g. **[24]**. Note that $[A,B]_{G}=A\{B\}-(-1)^{|A||B|}B\{A\}$. In short the above computation boils down to

[TABLE]

*Thus $\Phi$ respects the brackets and observing that $\Phi(1\otimes 1)=\mu_{\mathscr{A}}$ the proposition is proved. *$\heartsuit$

Remark 4.14**.**

Recall that $TH[1]$ is endowed with a brace algebra structure, as described in Remark 4.5. The above proposition actually follows from the fact that a Hopf algebra action $\phi$ of $H$ on $\mathscr{A}$ actually induces a brace algebra morphism

[TABLE]

where the braces on $C^{\bullet}(\mathscr{A})$ are defined as usual by

[TABLE]

for $P\in C^{k+1}(\mathscr{A};\mathscr{A})$, $Q_{j}\in C^{k_{j}+1}(\mathscr{A};\mathscr{A})$ for all $j$ and $a_{i}\in\mathscr{A}$ for all $i$. So we have

[TABLE]

for all $P,Q_{1},\ldots,Q_{r}\in TH[1]$.

4.2 Twisting Poisson actions

A Lie bialgebra is a pair $(\mathfrak{g},\gamma)$ where $\mathfrak{g}$ is a Lie algebra and $\gamma$ is a $1$-cocycle, $\gamma\colon\mathfrak{g}\to\mathfrak{g}\wedge\mathfrak{g}$. In this paper we consider a particular class of Lie bialgebras. Recall that an element $r\in\mathfrak{g}\wedge\mathfrak{g}$ is called $r$-matrix if it satisfies the Maurer–Cartan equation $\llbracket r,r\rrbracket=0$. It can be proved that $r$-matrices always induce a Lie bialgebra structure on $\mathfrak{g}$, by setting $\gamma=\llbracket r,\,\cdot\,\rrbracket$. We refer to the pair $(\mathfrak{g},r)$ as triangular Lie algebra. For further details on Lie bialgebras we refer to [28].

Let us consider a Lie algebra action $\varphi:\mathfrak{g}\to\Gamma^{\infty}(TM)$.

Definition 4.15** (Poisson action).**

The action $\varphi$ is Poisson if it satisfies

[TABLE]

*where $\mathop{}\!\mathrm{d}_{\pi}=\llbracket\pi,\,\cdot\,\rrbracket$. *

Proposition 4.16**.**

Let $\mathfrak{g}$ be a finite dimensional triangular Lie algebra and $\varphi\colon\mathfrak{g}\to\Gamma^{\infty}(TM)$ a Lie algebra action.

1. i.)

The bitensor $\pi$ defined as the image of $r$ via $\varphi$ is a Poisson tensor. 2. ii.)

$\varphi$* is a Poisson action wrt $\pi=\varphi\wedge\varphi(r)$.*

Proof**:**

Let us consider the $r$-matrix $r=\frac{1}{2}\sum_{i,j}r^{ij}e_{i}\wedge e_{j}$ and define

[TABLE]

From $\llbracket r,r\rrbracket=0$, using the fact that $\varphi$ is a Lie algebra morphism, it follows that $\llbracket\pi,\pi\rrbracket=0$. The second claim is a straightforward computation. $\heartsuit$

It is easy to see that the notion of Poisson action can be extended to a morphism of DGLA’s

[TABLE]

We now show how one may use the formality of $\mathop{}\mathopen{\vphantom{\mathrm{D}_{\mathrm{poly}}}}^{\mathfrak{g}}\kern-0.5pt\mathrm{D}_{\mathrm{poly}}$ and $\mathop{}\mathopen{\vphantom{\mathrm{D}_{\mathrm{poly}}}}\kern-0.5pt\mathrm{D}_{\mathrm{poly}}(M)$ to obtain a deformation symmetry of $\mathscr{U}(\mathfrak{g})$ in $\mathscr{C}^{\infty}(M)$ given an infinitesimal action $\varphi$ of $\mathfrak{g}$ on $M$. Thus, given an r-matrix of $\mathfrak{g}$, we obtain deformations of all relevant structures and we comment on these. In Section 3 we have obtained the formal DGLA’s $\mathop{}\mathopen{\vphantom{\mathrm{D}_{\mathrm{poly}}}}^{\mathfrak{g}}\kern-0.5pt\mathrm{D}_{\mathrm{poly}}$ and $\mathop{}\mathopen{\vphantom{\mathrm{D}_{\mathrm{poly}}}}\kern-0.5pt\mathrm{D}_{\mathrm{poly}}(M)$. The classical $r$-matrix $r\in\mathop{}\mathopen{\vphantom{\mathrm{T}_{\mathrm{poly}}^{1}}}^{\mathfrak{g}}\kern-0.5pt\mathrm{T}_{\mathrm{poly}}^{1}$ yields a Maurer-Cartan element. Although it satisfies the Maurer-Cartan equation $\llbracket r,r\rrbracket=0$ it is in fact not an MC element according to our Definition 2.12 as we have neglected to consider any filtration. The filtration is needed since we go through formality, which means we encounter infinite sums. More precisely, we obtain a filtration by considering the formal power series ring $\mathbbm{R}[\![\hbar]\!]$. Given a DGLA $\mathfrak{L}$ we denote the DGLA obtained by extending scalars to the formal power series ring by $\mathfrak{L}[\![\hbar]\!]$. We consider these DGLA’s as filtered by the degree in $\hbar$. Note that, given $L_{\infty}$-morphisms of DGLA’s $\mathfrak{L}\rightarrow\mathfrak{L}^{\prime}$ we obtain also $L_{\infty}$-morphisms of the extended DGLA’s $\mathfrak{L}[\![\hbar]\!]\rightarrow\mathfrak{L}^{\prime}[\![\hbar]\!]$. In this way $L_{\infty}$-quasi-isomorphism go to $L_{\infty}$-quasi-isomorphisms.

From Section 3 it follows that we have a horse-shoe diagram in the category of $L_{\infty}$-algebras:

[TABLE]

Here the vertical maps $F_{\mathfrak{g}}$ and $F_{M}$ are $L_{\infty}$-quasi-isomorphisms constructed as discussed in Section 3 and the horizontal arrow is induced by $\varphi$ as in (4.28). Thus we obtain, given an r-matrix $r\in\mathfrak{g}\wedge\mathfrak{g}$, the MC elements:

[TABLE]

Remark 4.17**.**

Notice that we obtain the deformed versions of $\mathscr{C}^{\infty}(M)$ and $\mathscr{U}(\mathfrak{g})$ by twisting by the Maurer-Cartan elements listed above. We would like then to obtain a deformation symmetry completing the square in the horse-shoe diagram above, such that it commutes. Commutativity ensures that the two formal deformations of $\mathscr{C}^{\infty}(M)$ induced by $\hbar r$ by transporting it along the bottom or the top to $\mathop{}\mathopen{\vphantom{\mathrm{D}_{\mathrm{poly}}}}\kern-0.5pt\mathrm{D}_{\mathrm{poly}}(M)[\![\hbar]\!]$ coincide. An obvious candidate for such a map would be

[TABLE]

i.e. the map induced by the map of Lie-Rinehart pairs $(\mathfrak{g},\mathbbm{R})\rightarrow(\Gamma^{\infty}(TM),\mathscr{C}^{\infty}(M))$. In other terms the deformation symmetry induced by the obvious Hopf algebra action of $\mathscr{U}(\mathfrak{g})$ on $\mathscr{C}^{\infty}(M)$ through proposition 4.13. However this map may not make the diagram commute. In fact it is the opinion of the authors that such commutation would involve some condition of compatibility of the connections used in defining these maps, see Remark 3.7.

We complete the horse-shoe diagram (4.29) to a commuting square by observing that $L_{\infty}$-quasi-isomorphisms are invertible. The following lemma is essentially contained in [29, Chapt. 10.4]. We remind the reader that the field underlying $L_{\infty}$-algebras is of characteristic [math].

Lemma 4.18**.**

Suppose $\mathfrak{L}$ and $\mathfrak{L}^{\prime}$ are formal $L_{\infty}$-algebras and $f\colon H(\mathfrak{L})\rightarrow H(\mathfrak{L^{\prime}})$ is an $L_{\infty}$-morphism, then there exists a lift $\tilde{f}\colon\mathfrak{L}\rightarrow\mathfrak{L}^{\prime}$ of $f$. In other words there exists a commuting diagram

[TABLE]

*in the category of $L_{\infty}$-algebras such that the vertical arrows are quasi-isomorphisms. *

Proof**:**

Note that we start by hypothesis with the horse-shoe

[TABLE]

in the category of $L_{\infty}$-algebras such that the vertical arrows are quasi-isomorphisms. As shown in [29, Sect. 10.4.4] we can always find a quasi-inverse $l_{i}\colon\mathfrak{L}\rightarrow H(\mathfrak{L})$ of $i_{l}$ such that the induced maps in cohomology are inverse to each other.  We define $\tilde{f}:=i_{r}\circ f\circ l_{i}$. Note that this already proves that $\tilde{f}\circ i_{l}=i_{r}\circ f$ in cohomology (and thus “up to homotopy”). To get the stronger statement in the lemma we will need to consider the construction of the map $l_{i}$. This construction involves the notion of a so-called $\infty$-isomorphism. An $\infty$-isomorphism is a morphism of $L_{\infty}$-algebras $F$ such that $F_{1}$ is an isomorphism. The main observations for the construction of $l_{i}$ are two-fold. First, the homotopy transfer theorem [29, Sect. 10.3] yields an $L_{\infty}$-structure on $H(\mathfrak{L})$ which is unique up to $\infty$-isomorphism. It is obtained by picking a retraction of $\mathfrak{L}$ onto $H(\mathfrak{L})$, we may pick the retraction given by $i_{l}$. Secondly in section 10.4.2 of [29] it is shown that any $L_{\infty}$-algebra $\mathfrak{L}$ is $\infty$-isomorphic to the sum $H(\mathfrak{L})\oplus K$ where $K$ is an acyclic chain complex (with trivial $Q_{0}$, $Q_{2}$, $Q_{3}$ and so on). Now the construction of $l_{i}$ follows by the fact that $\infty$-isomorphisms are invertible (shown in section 10.4.1 of [29]). Our lemma follows from the fact that, for a formal $L_{\infty}$-algebra the $L_{\infty}$-structure induced on $H(\mathfrak{L})$ by homotopy transfer equals the canonically induced structure up to $\infty$-isomorphism. Thus we may simply consider the splittings $\mathfrak{L}\simeq H(\mathfrak{L})\oplus K_{\mathfrak{L}}$ and $\mathfrak{L}^{\prime}\simeq H(\mathfrak{L}^{\prime})\oplus K_{\mathfrak{L}^{\prime}}$, where $\simeq$ means $\infty$-isomorphism, given by the retractions induced by $i_{l}$ and $i_{r}$. Then the map $\tilde{f}$ simply maps $H(\mathfrak{L})$ to $H(\mathfrak{L}^{\prime})$ by $f$. $\heartsuit$

Corollary 4.19**.**

The diagrams

[TABLE]

and

[TABLE]

*commute. *

Proof**:**

Applying the above lemma to our situation we immediately find that the diagram (4.34) commutes. Also, by applying the results of Section 3.2 we obtain diagram (4.35), which commutes thanks to Remark 3.18. $\heartsuit$

4.3 Twisted structures

In the following we show that the twisted complexes obtained above are coming from a formal deformation quantization of $\mathscr{C}^{\infty}(M)$ (in the case of $\mathop{}\mathopen{\vphantom{\mathrm{D}_{\mathrm{poly}}^{B_{\hbar}}}}\kern-0.5pt\mathrm{D}_{\mathrm{poly}}^{B_{\hbar}}(M)$) and a deformation of $\mathscr{U}(\mathfrak{g})$ into a quantum group (in the case of $\mathop{}\mathopen{\vphantom{\mathrm{D}_{\mathrm{poly}}^{\rho_{\hbar}}}}^{\mathfrak{g}}\kern-0.5pt\mathrm{D}_{\mathrm{poly}}^{\rho_{\hbar}}$).

Proposition 4.20**.**

There is a formal deformation quantization $\mathscr{A}_{\hbar}$ of $(\mathscr{C}^{\infty}(M),\pi)$ such that

[TABLE]

*i.e. $\mathop{}\mathopen{\vphantom{\mathrm{D}_{\mathrm{poly}}^{B_{\hbar}}}}\kern-0.5pt\mathrm{D}_{\mathrm{poly}}^{B_{\hbar}}(M)$ is a subcomplex of the Hochschild cochain complex of $\mathscr{A}_{\hbar}$. *

Proof**:**

Note that the Maurer–Cartan equation (2.20) matches exactly the associativity condition of $m+B_{\hbar}$, where $m$ denotes the pointwise multiplication in $\mathscr{C}^{\infty}(M)[\![\hbar]\!]$. Since

[TABLE]

we see that $m+B_{\hbar}$ defines a deformation quantization $\mathscr{A}_{\hbar}$. The differential on the Hochschild complex $C^{\bullet}(\mathscr{A},\mathscr{A})$ is given by taking the Gerstenhaber bracket with the multiplication for any associative algebra $\mathscr{A}$. Thus the twisted differential on $\mathop{}\mathopen{\vphantom{\mathrm{D}_{\mathrm{poly}}^{B_{\hbar}}}}\kern-0.5pt\mathrm{D}_{\mathrm{poly}}^{B_{\hbar}}(M)$ coincides with the differential of $C^{\bullet}(\mathscr{A}_{\hbar},\mathscr{A}_{\hbar})$. Finally we recall that

[TABLE]

Since $(F_{M})_{1}$ is a quasi-isomorphism we find that the alternating part of $B_{\hbar}$ is $\hbar\pi$ modulo $\hbar^{2}$ and

[TABLE]

for all $f,g\in\mathscr{C}^{\infty}(M)$. Here $[\,\cdot\,,\,\cdot\,]_{\star}$ denotes the commutator bracket of $\mathscr{A}_{\hbar}$. $\heartsuit$

Thus we obtain in particular the deformation quantization $\mathscr{A}_{\hbar}$. On the other hand we also obtain the MC element $\rho_{\hbar}$ in $\mathop{}\mathopen{\vphantom{\mathrm{D}_{\mathrm{poly}}}}^{\mathfrak{g}}\kern-0.5pt\mathrm{D}_{\mathrm{poly}}$. This yields the coproduct $\Delta_{J_{\hbar}}$ (where $J_{\hbar}=1\otimes 1+\rho_{\hbar})$ thus establishing a quantum group, obtained by quantization of the Lie bialgebra $(\mathfrak{g},\llbracket r,\,\cdot\,\rrbracket)$. Finally we obtain the following theorem as a corollary of Prop. 4.9 and Corollary 4.19.

Theorem 4.21**.**

Suppose $(\mathfrak{g},r)$ is a triangular Lie algebra and $\varphi\colon\mathfrak{g}\rightarrow\Gamma^{\infty}(TM)$ is an action on the manifold $M$. Then there exist a formal deformation quantization $\mathscr{A}_{\hbar}$ of $(M,\pi)$ (where $\varphi\wedge\varphi(r)=\pi$) and a quantization $\mathscr{U}_{\hbar}(\mathfrak{g})$ of $(\mathfrak{g},r)$ which allow a deformation symmetry

[TABLE]

4.4 Comparison with Drinfeld’s construction

First, let us briefly recall the original construction of Drinfeld (see [3, 13, 25]). Consider a formal twist $J$ on $\mathscr{U}(\mathfrak{g})[\![\hbar]\!]$ and a generic $\mathscr{U}(\mathfrak{g})$-module algebra $\mathscr{A}$. Drinfeld proved that we can then always define an associative star product on $\mathscr{A}$. In particular, consider $\mathscr{A}=\mathscr{C}^{\infty}(M)$ with pointwise multiplication $m$. Given a Lie algebra action $\varphi:\mathfrak{g}\to\Gamma^{\infty}(TM)$ we obtain a Hopf algebra action

[TABLE]

which makes $\mathscr{C}^{\infty}(M)$ into a left $\mathscr{U}(\mathfrak{g})$-module algebra. More precisely, $X\mathbin{\triangleright}f=\mathscr{L}_{\varphi(X)}f$ where $\mathscr{L}$ denotes the Lie derivative. The action $\mathbin{\triangleright}$ extends to formal power series

[TABLE]

Thus the product defined by

[TABLE]

for $f,g\in\mathscr{C}^{\infty}(M)[\![\hbar]\!]$ is a star product. The classical limit of (4.43) is given by

[TABLE]

where $r:=\frac{J-\tau(J)}{\hbar}|_{\hbar=0}$ is the $r$-matrix associated to the twist $J$, here $\tau$ denotes the flip $X\otimes Y\mapsto Y\otimes X$. It is important to underline that the deformed algebra $(\mathscr{C}^{\infty}(M)[\![\hbar]\!],\star_{J})$ is then a module-algebra for the quantum group:

[TABLE]

In other words, $\mathbin{\triangleright}$ is a Hopf agebra action of the twisted Hopf algebra $\mathscr{U}_{J}(\mathfrak{g})$ on $(\mathscr{C}^{\infty}(M)[\![\hbar]\!],\star_{J})$.

Thus, the construction takes a formal twist $J\in\mathscr{U}(\mathfrak{g})[\![\hbar]\!]$ and an infinitesimal action of $\mathfrak{g}$ on $M$ as input and produces a deformation quantization $\mathscr{A}_{\hbar}$ together with an action of the quantum group $\mathscr{U}_{J}(\mathfrak{g})$ on it. In our approach one starts with an $r$-matrix $r\in\mathfrak{g}\wedge\mathfrak{g}$ and an infinitesimal action of $\mathfrak{g}$ on $M$ and obtains a formal twist $J$ and a deformation symmetry $\tilde{\varphi}$. These then also yield a deformation quantization given by

[TABLE]

and another deformation symmetry $\tilde{\phi}^{\rho_{\hbar}}\circ\mathscr{J}$ of the quantum group $\mathscr{U}_{J}(\mathfrak{g})$ in the deformed algebra $\mathscr{A}_{\hbar}$. The main difference of the two approaches is that the formal twist is taken as given in Drinfeld’s approach while we obtain it through quantization of an $r$-matrix. The trade-off is however that we do not obviously obtain an action of a quantum group anymore, instead we obtain the deformation symmetry. A direct comparison of the two approaches will be nontrivial and it implies a study of the compatibility condition between connections mentioned in Remark 4.17. In particular it is of interest whether the process of quantization so obtained can be made functorial for equivariant maps between the manifolds. We will come back to this in a future project.