Twisted cohomology of Lie algebroids

TL;DR

This paper introduces a new twisted cohomology for Lie algebroids, depending on an odd cocycle, which generalizes several existing cohomology theories and depends only on its cohomology class.

Contribution

It defines the twisted cohomology for Lie algebroids and proves its dependence solely on the cohomology class of the twisting cocycle, unifying various known theories.

Findings

The twisted cohomology depends only on the cohomology class of the cocycle.

Examples show the new cohomology encompasses known theories.

The construction generalizes existing cohomology concepts.

Abstract

In this short note we define a new cohomology for a Lie algebroid $\mathcal{A}$, that we call the \emph{twisted cohomology} of $\mathcal{A}$ by an odd cocycle $\theta$ in the Lie algebroid cohomology of $\mathcal{A}$. We proof that this cohomology only depends on the Lie algebroid cohomology class $[\theta]$ of the odd cocycle $\theta$. We give a few examples showing that this new cohomology encompasses various well-known cohomology theories.

Equations

$$[u,fv]=f[u,v]+(\mathsf{a}(u)\cdot f)v,$$

$$[fu,v]=f[u,v]-(\mathsf{a}(v)\cdot f)u,$$

$$[fu,v]_{x}=\big{[}f(x)u_{x},v_{x}\big{]}_{\mathfrak{g}}=f(x)[u_{x},v_{x}]_{\mathfrak{g}}=\big{(}f[u,v]\big{)}_{x},$$

$$\mathsf{a}(\alpha )\cdot f=\pi (\alpha ,\mathrm{d}f),$$

$$[\alpha,\beta]=\bm{\mathcal{L}}_{{\pi^{\sharp}}(\alpha)}\beta-\bm{\mathcal{L}}_{{\pi^{\sharp}}(\beta)}\alpha-\bm{\mathrm{d}}\big{(}\pi(\alpha,\beta)\big{)},$$

$$[u,v]_{x}=[u_{x},v_{x}]_{\mathfrak{g}}+\big{[}\rho(u_{x})\cdot v\big{]}_{x}-\big{[}\rho(v_{x})\cdot u\big{]}_{x},$$

$$\mathrm{d}\alpha(u_{0},\dots,u_{p})=\sum_{i=0}^{p}(-1)^{i}\mathsf{a}(u_{i})\cdot\alpha(u_{0},\dots,\widehat{u_{i}},\dots,u_{p})\\ +\sum_{0\leq i<j\leq p}(-1)^{i+j}\alpha\big{(}[u_{i},u_{j}],u_{0},\dots,\widehat{u_{i}},\dots,\widehat{u_{j}},\dots,u_{p}\big{)},$$

$$(\omega+\mathrm{d}\alpha)\wedge(\eta+\mathrm{d}\beta)=\omega\wedge\eta+\mathrm{d}\big{(}\alpha\wedge\eta+(-1)^{p}\omega\wedge\beta+\alpha\wedge\mathrm{d}\beta\big{)},$$

$${\nabla }_u(fs)=f{\nabla }_{u}s+(\mathsf{a}(u)\cdot f)s,$$

$${\nabla }_{fu}s=f{\nabla }_{u}s,$$

$${\mathrm{d}}_{\nabla }(\alpha \otimes s)=\mathrm{d}\alpha \otimes s+(-1{)}^p\alpha \wedge \nabla s,$$

$${\mathrm{d}}_{\nabla }(\alpha \wedge \omega )=\mathrm{d}\alpha \wedge \omega +(-1{)}^p\alpha \wedge {\mathrm{d}}_{\nabla }\omega ,$$

$${\mathrm{d}}_{\nabla }(\omega \wedge \alpha )={\mathrm{d}}_{\nabla }\omega \wedge \alpha +(-1{)}^q\omega \wedge \mathrm{d}\alpha ,$$

$$\mathrm{d}_{\nabla}\omega(u_{0},\dots,u_{p})=\sum_{i=0}^{p}(-1)^{i}\nabla_{u_{i}}\omega(u_{0},\dots,\widehat{u_{i}},\dots,u_{p})\\ +\sum_{0\leq i<j\leq p}(-1)^{i+j}\omega\big{(}[u_{i},u_{j}],u_{0},\dots,\widehat{u_{i}},\dots,\widehat{u_{j}},\dots,u_{p}\big{)},$$

$$(\alpha \otimes \Phi )\wedge (\beta \otimes \Psi )=(\alpha \wedge \beta )\otimes (\Phi \circ \Psi ),$$

$$(\alpha \otimes \Phi )\wedge (\beta \otimes s)=(\alpha \wedge \beta )\otimes \Phi (s),$$

$${\mathrm{d}}_{\nabla }^2\alpha =F_{\nabla }\wedge \alpha ,$$

$$({\mathrm{d}}_{\nabla }\omega )(u,v)={\nabla }_u\omega (v)-{\nabla }_v\omega (u)-\omega ([u,v]),$$

$$\big{(}\mathrm{d}_{\nabla}^{2}s\big{)}(u,v)=\nabla_{u}\nabla_{v}s-\nabla_{v}\nabla_{u}s-\nabla_{[u,v]}s=F_{\nabla}(u,v)s=(F_{\nabla}\wedge s)(u,v),$$

$${\mathrm{d}}_{\nabla }^2(\alpha \otimes s)$$

$$\displaystyle=\mathrm{d}_{\nabla}\big{(}\mathrm{d}\alpha\wedge s+(-1)^{p}\alpha\wedge\mathrm{d}_{\nabla}s\big{)}$$

$$={\mathrm{d}}^2\alpha \wedge s+(-1{)}^{p+1}\mathrm{d}\alpha \wedge {\mathrm{d}}_{\nabla }s+(-1{)}^p\mathrm{d}\alpha \wedge {\mathrm{d}}_{\nabla }s+\alpha \wedge {\mathrm{d}}_{\nabla }^{2}s$$

$$=\alpha \wedge (F_{\nabla }\wedge s)$$

$$=F_{\nabla }\wedge (\alpha \otimes s),$$

$$(\alpha+\mathrm{d}\beta)\wedge(\omega+\mathrm{d}_{\nabla}\eta)=\alpha\wedge\omega+\mathrm{d}_{\nabla}\big{(}\beta\wedge\omega+(-1)^{p}\alpha\wedge\eta+\beta\wedge\mathrm{d}_{\nabla}\eta\big{)},$$

$$\mathrm{Hom}(V,{\mathfrak{g}}^{\ast }\otimes V)\cong \mathrm{Hom}(V,\mathrm{Hom}(\mathfrak{g},V))\cong \mathrm{Hom}(\mathfrak{g}\otimes V,V)\cong \mathrm{Hom}(\mathfrak{g},\mathrm{End}(V)),$$

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$${\Omega }^{\text{even}}(\mathcal{A})=\underset{i\ge 0}{⨁}{\Omega }^{2i}(\mathcal{A}),{\Omega }^{\text{odd}}(\mathcal{A})=\underset{i\ge 0}{⨁}{\Omega }^{2i+1}(\mathcal{A}).$$

$${\mathrm{d}}_{\nabla }[\theta ]\omega ={\mathrm{d}}_{\nabla }\omega +\theta \wedge \omega ,$$

$${{\mathrm{d}}_{\nabla }[\theta ]}^2\omega ={\mathrm{d}}_{\nabla }^2\omega +{\mathrm{d}}_{\nabla }(\theta \wedge \omega )+\theta \wedge {\mathrm{d}}_{\nabla }\omega +\theta \wedge \theta \wedge \omega .$$

$${{\mathrm{d}}_{\nabla }[\theta ]}^2\omega =\sum _{i\ge 0}\left({\mathrm{d}}_{\nabla }{\theta }_{2i+1}\right)\wedge \omega ,$$

$${\mathrm{d}}_{\nabla }[\theta ](\alpha \wedge \omega )$$

$$=\mathrm{d}\alpha \wedge \omega +(-1{)}^p\alpha \wedge {\mathrm{d}}_{\nabla }\omega +\theta \wedge \alpha \wedge \omega$$

$$=\mathrm{d}\alpha \wedge \omega +(-1{)}^p\alpha \wedge {\mathrm{d}}_{\nabla }[\theta ]\omega$$

Full text

1. 1 Lie algebroids

2. 1.1 Definition and examples of Lie algebroids

3. 1.2 Cohomology of Lie algebroids

4. 1.3 Representations of Lie algebroids

5. 2 Twisted cohomology of Lie algebroids

6. 2.1 Definition of the twisted cohomology

7. 2.2 Independence from the cocycle cohomology class

8. 2.3 Examples of twisted cohomologies

Introduction

Lie algebroids are objects in differential geometry that generalize both Lie algebras and tangent bundles of smooth manifolds (see the definition 1.2). They appear in many situations, ranging from classical geometric classification problems (see [FS08] for instance) to Poisson geometry (see [Mar08]) and foliations (see [MM03]), but not exclusively. Importantly, a Lie algebroid is the infinitesimal version of a Lie groupoid (see [Pra67]). To any Lie algebroid is associated a cohomology algebra, which is a generalization of both the Chevalley-Eilenberg cohomology of a Lie algebra and the De Rham cohomology of a manifold. After reviewing some definitions in section 1, we define in section 2 a new cohomology, that we will call the twisted cohomology of a Lie algebroid, using an odd differential form $\theta$ that is $\mathrm{d}$-closed, where $\mathrm{d}$ is the differential of the usual Lie algebroid cohomology. We show in the corollary 2.7 that this new cohomology only depends on the (Lie algebroid) cohomology class $[\theta]$ of $\theta$, and recover examples of the literature in section 2.3, in particular the twisted De Rham cohomology of a smooth manifold [MW11].

1 Lie algebroids

In this section, we briefly recall the notion of a Lie algebroid, its cohomology, including the one with coefficients, obtained from a representation, and give some examples. All manifolds that we consider are smooth in the sense of [Lee13, chapter 1], and all vector bundles are smooth, real and of constant rank (see [Hus94, definition 1.1, chapter 3]). The usual interior product, De Rham differential, and Lie derivative, operating on differential forms of a manifold, will be typed in bold type ($\bm{\iota}_{X}$, $\bm{\mathrm{d}}$ et $\bm{\mathcal{L}}_{X}$, where $X\in\mathfrak{X}(M)$ is a vector field on the manifold $M$).

1.1 Definition and examples of Lie algebroids

Definition 1.1**:**

Let $M$ be a manifold. An anchor on a vector bundle $A\to M$ is vector bundle morphism $\mathsf{a}:A\to TM$ covering the identity, where $TM\to M$ denotes the tangent bundle of $M$.

Definition 1.2** ([Mac05, definition 3.3.1]):**

Let $M$ be a manifold. A Lie algebroid $\mathcal{A}$ is a triple $(A\to M,\mathsf{a},[\cdot,\cdot])$, where $A\to M$ is a vector bundle on $M$, $\mathsf{a}$ is an anchor on $A\to M$, and $[\cdot,\cdot]$ is a $\mathbb{R}$-bilinear operation on the $\mathcal{C}^{\infty}(M)$-module $\Gamma(A)$ of sections of $A\to M$ called the bracket, such that $(\Gamma(A),[\cdot,\cdot])$ is a Lie algebra and a right Leibniz rule is satisfied:

[TABLE]

for all $u,v\in\Gamma(A)$ and $f\in\mathcal{C}^{\infty}(M)$.

Remark 1.3**:**

From the definition above, a left Leibniz rule can be obtained using the skew-symmetry of the bracket $[\cdot,\cdot]$, that is,

[TABLE]

for all $u,v\in\Gamma(A)$ and $f\in\mathcal{C}^{\infty}(M)$.

Proposition 1.4** ([KSM90, section 6.1]):**

Let $\mathcal{A}=(A\to M,\mathsf{a},[\cdot,\cdot])$ be a Lie algebroid. The anchor $\mathsf{a}$ induces a morphism of Lie algebras $\Gamma(A)\to\mathfrak{X}(M)$, still denoted by $\mathsf{a}$, where $\mathfrak{X}(M)$ is equipped with the Lie bracket of vector fields on the manifold $M$.

We now give some examples of Lie algebroids.

Example 1.5**:**

Lie algebras are in correspondence with Lie algebroids over a point.

Example 1.6**:**

Let $M$ be a manifold and $A\to M$ be a Lie algebra bundle. Then $A\to M$ can be equipped with a Lie algebroid structure as follows. Let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}})$ be a typical fiber of $A\to M$, that is, a Lie algebra. Then, the Lie algebra bracket $[\cdot,\cdot]_{\mathfrak{g}}$ induces a bracket on the space of sections $\Gamma(A)$, defined by $[u,v]_{x}=[u_{x},v_{x}]_{\mathfrak{g}}$ for all $u$, $v\in\Gamma(A)$ and $x\in M$. This bracket is $\mathcal{C}^{\infty}(M)$-bilinear since

[TABLE]

for all $u$, $v\in\Gamma(A)$, $f\in\mathcal{C}^{\infty}(M)$ and $x\in M$. Therefore, $A\to M$ equipped with the null anchor and the bracket we just defined is a Lie algebroid. Lie algebra bundles are studied in detail in [Mac05, section 3.3] and [Gü10].

Example 1.7**:**

Let $M$ be a manifold. On the tangent bundle $TM\to M$ of $M$, there is a natural Lie algebroid structure given as follows. The anchor is the identity vector bundle morphism $TM\to TM$, and the bracket on $\Gamma(TM)=\mathfrak{X}(M)$ is the Lie bracket of vector fields on $M$ (see [Lee13, chapter 8]). This Lie algebroid will be called the canonical Lie algebroid on $M$, and will be denoted by $\mathcal{T}_{M}$.

Example 1.8**:**

Let $M$ be a manifold and $A\to M$ be an involutive distribution of $TM\to M$ (see [Lee13, chapter 19]). Note that according to the global Frobenius theorem [Lee13, theorem 19.21], this distribution comes from a foliation of $M$. The vector bundle $A\to M$ is equipped with a Lie algebroid structure as follows. The anchor is the inclusion map $A\hookrightarrow TM$, and the bracket is the Lie bracket of vector fields on $M$ restricted to $\Gamma(A)$, which is well-defined thanks to the involutivity of the distribution.

Example 1.9**:**

Let $M$ be a manifold and $\pi\in\Gamma(\Lambda^{2}TM)$ a be bivector field such that $[\pi,\pi]_{\textsf{SN}}=0$ for the Schouten-Nijenhuis bracket [LGPV13, section 3.3]. Then $(M,\pi)$ is a Poisson manifold [LGPV13, section 1.3.2] and the cotangent bundle $T^{*}M\to M$ of $M$ can be given a Lie algebroid structure as follows (see [Mar08, section 6.2]). The anchor $\mathsf{a}$ is defined by

[TABLE]

for all $\alpha\in\Omega^{1}(M)=\Gamma(T^{*}M)$ and $f\in\mathcal{C}^{\infty}(M)$. Writing $\pi^{\sharp}:\Omega^{1}(M)\to\mathfrak{X}(M)$ for the map defined by $\pi^{\sharp}(\alpha)(\beta)=\pi(\alpha,\beta)$, we get $\mathsf{a}=\pi^{\sharp}$. The bracket is defined by

[TABLE]

for all $\alpha$, $\beta\in\Omega^{1}(M)$. Then the Jacobi identity for this bracket is equivalent to the identity $[\pi,\pi]_{\textsf{SN}}=0$, where $[\cdot,\cdot]_{\textsf{SN}}$ denotes the Schouten-Nijenhuis bracket (see [LGPV13, section 3.3]). We will denote by $\mathcal{P}_{M}[\pi]$ this Lie algebroid.

Example 1.10**:**

Let $M$ be a manifold and let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}})$ be a Lie algebra acting (infinitesimally) on $M$ through a Lie algebra homomorphism $\rho:\mathfrak{g}\to\mathfrak{X}(M)$. The trivial vector bundle $M\times\mathfrak{g}$ can be equipped with a Lie algebroid structure (called an action Lie algebroid, see [Mac05, example 3.3.7]). The anchor is defined by $\mathsf{a}((x,\xi))=\rho(\xi)_{x}$, for any $x\in M$ and $\xi\in\mathfrak{g}$. The bracket is defined on sections $u$, $v\in\Gamma(M\times\mathfrak{g})\cong\mathcal{C}^{\infty}(M,\mathfrak{g})$ by

[TABLE]

where the dot $\cdot$ in the right hand side denotes the component-wise action of a vector field.

Some other important examples have been omitted, but we still would like to briefly describe them hereafter. Similarly to Lie algebras that can be obtained by differentiation from a Lie group, starting with a Lie groupoid [Mac05, section 1.1], one can obtain a Lie algebroid by differentiation [Mac05, section 3.5]. Nevertheless, there are obstructions to integrate a given Lie algebroid into a Lie groupoid [CF03]. The last example is actually the Lie algebroid of a Lie groupoid, the so-called action Lie groupoid [Mac05, example 1.1.9], which gives the corresponding action Lie algebroid in the case the infinitesimal action integrates into a global one (see [Mac05, example 3.5.14]). Finally, although we will not need it, we mention the so-called Atiyah Lie algebroid ([Ati57], [Mac05, section 3.2]). Let $G$ be a Lie group and $\pi:P\to M$ be a principal $G$-bundle. Let $\overline{\bm{\mathrm{d}}\pi}$ denote the quotient map $TP/G\to TM$. Taking $\overline{\bm{\mathrm{d}}\pi}$ as anchor and the Lie bracket of ($G$-invariant) vector fields on the manifold $P$ as bracket, the vector bundle $TP/G\to M$ is a Lie algebroid, called the Atiyah Lie algebroid associated to $\pi:P\to M$. It is obtained by differentiation from the Ehresmann gauge groupoid associated to the same principal $G$-bundle (see [Kub89] and the references within).

1.2 Cohomology of Lie algebroids

We now turn to the definition of Lie algebroid cohomology.

Definition 1.11**:**

Let $\mathcal{A}=(A\to M,\mathsf{a},[\cdot,\cdot])$ be a Lie algebroid. We will denote by $\Omega^{\bullet}(\mathcal{A})$ the $\mathbb{Z}$-graded commutative $\mathbb{R}$-algebra $(\Gamma(\Lambda^{\bullet}A^{*}),\wedge)$. Elements of $\Omega^{\bullet}(\mathcal{A})$ will be called differential forms on $\mathcal{A}$.

Definition 1.12** ([ELW99, section 2], [Mar08, section 5.2]):**

Let $\mathcal{A}=(A\to M,\mathsf{a},[\cdot,\cdot])$ be a Lie algebroid. The exterior derivative of $\mathcal{A}$ is the operator $\mathrm{d}:\Omega^{p}(\mathcal{A})\to\Omega^{p+1}(\mathcal{A})$ defined by

[TABLE]

for all $u_{0},\dots,u_{p}\in\Gamma(A)$ and any $\alpha\in\Omega^{p}(\mathcal{A})$; where $\mathsf{a}(u_{i})\cdot\alpha(u_{0},\dots,\widehat{u_{i}},\dots,u_{p})$ denotes the action of the vector field $\mathsf{a}(u_{i})$ on the function $\alpha(u_{0},\dots,\widehat{u_{i}},\dots,u_{p})$. On functions $f\in\mathcal{C}^{\infty}(M)$, $\mathrm{d}$ is defined by $(\mathrm{d}f)(u)=\mathsf{a}(u)\cdot f$, for any $u\in\Gamma(A)$.

The following proposition is a particular case of [Mar08, proposition 5.2.3, point 2].

Proposition 1.13** ([Mar08, proposition 5.2.3]):**

Let $\mathcal{A}=(A\to M,\mathsf{a},[\cdot,\cdot])$ be a Lie algebroid. The exterior derivative $\mathrm{d}$ of $\mathcal{A}$ is a derivation of degree 1 of $\Omega^{\bullet}(\mathcal{A})$ that squares to zero, that is, $\mathrm{d}$ is a differential and $\Omega^{\bullet}(\mathcal{A})$ endowed with this differential becomes a differential graded commutative algebra (see [FHT01, part 1, chapter 3] for the definition).

Definition 1.14**:**

Let $\mathcal{A}=(A\to M,\mathsf{a},[\cdot,\cdot])$ be a Lie algebroid. The cohomology of $\mathcal{A}$ is defined as the homology of the differential graded commutative algebra $(\Omega^{\bullet}(\mathcal{A}),\mathrm{d})$ (see [FHT01, part 1, chapter 3] for the definition).

Lemma 1.15**:**

Let $\mathcal{A}=(A\to M,\mathsf{a},[\cdot,\cdot])$ be a Lie algebroid. On $H^{\bullet}(\mathcal{A})$ there is a cup product that gives $H^{\bullet}(\mathcal{A})$ the structure of a graded commutative algebra.

Proof:.

The cup product comes directly from the wedge product on $\Omega^{\bullet}(\mathcal{A})$, it is defined on cohomology classes by $[\omega]\wedge[\eta]=[\omega\wedge\eta]$, for any $\omega\in\Omega^{p}(\mathcal{A})$ and $\eta\in\Omega^{q}(\mathcal{A})$ that are cocycles, that is, $\mathrm{d}$-closed. The differential form $\omega\wedge\eta$ is $\mathrm{d}$-closed since $\mathrm{d}$ is a derivation according to proposition 1.13. Also, the cup product does not depend on the choice of representation for the cocycles since, according to proposition 1.13 again, we have

[TABLE]

for any $\alpha\in\Omega^{\bullet}(\mathcal{A})$ and $\beta\in\Omega^{\bullet}(\mathcal{A})$. ∎

A detailed study of the cohomology of Lie algebroids, and especially the relationship between the cohomology of a Lie groupoid and the cohomology of its associated Lie algebroid, is available in [Mac05, chapter 7]. We only explore below some simple examples.

Example 1.16**:**

Let $\mathcal{A}$ be the Lie algebroid associated to a Lie algebra $\mathfrak{g}$ (see example 1.5). Then $H^{\bullet}(\mathcal{A})$ corresponds to the Chevalley-Eilenberg cohomology of $\mathfrak{g}$ (see [CE48, section 14]).

Example 1.17**:**

Let $M$ be a manifold and consider the canonical Lie algebroid $\mathcal{T}_{M}$ associated to $M$ (see example 1.7). The cohomology $H^{\bullet}(\mathcal{T}_{M})$ is the De Rham cohomology of $M$ (see [Lee13, chapter 17]).

Example 1.18**:**

Let $M$ be a manifold and $A\to M$ be an involutive distribution, and consider the associated Lie algebroid $\mathcal{A}$ (see example 1.8). Then $H^{\bullet}(\mathcal{A})$ corresponds to the tangential cohomology of the foliation associated to the involutive distribution (see [MS88, chapter 3] and [BDD07, section 1.1.3]).

Example 1.19**:**

Let $(M,\pi)$ be a Poisson manifold, and consider the associated Lie algebroid $\mathcal{P}_{M}[\pi]$. Then $H^{\bullet}(\mathcal{P}_{M}[\pi])$ corresponds to the Lichnerowicz-Poisson cohomology of $(M,\pi)$ (see [Lic77] and [Vai94, definition 5.1]). Note that the differential can be identified with $-[\pi,\cdot]_{\textsf{SN}}$ ([Vai94, proposition 4.3] and [KSM90, proposition 6.4]).

1.3 Representations of Lie algebroids

The subject of this section is representations of Lie algebroids, which allow to consider coefficients in Lie algebroid cohomology.

Definition 1.20**:**

Let $\mathcal{A}=(A\to M,\mathsf{a},[\cdot,\cdot])$ be a Lie algebroid and $V\to M$ be a vector bundle. We will denote by $\Omega^{\bullet}(\mathcal{A},V)$ the $\mathbb{Z}$-graded $\mathbb{R}$-vector space $\Gamma(\Lambda^{\bullet}A^{*}\otimes V)$. Elements of $\Omega^{\bullet}(\mathcal{A},V)$ will be called differential forms on $\mathcal{A}$ with values in $V\to M$. In degree 0, we recover sections of $V\to M$.

The following result is immediate.

Proposition 1.21**:**

Let $\mathcal{A}$ be a Lie algebroid on a manifold $M$, and $V\to M$ be a vector bundle. $\Omega^{\bullet}(\mathcal{A},V)$ is a graded $\Omega^{\bullet}(\mathcal{A})$-module for the multiplication defined by $\alpha\wedge(\beta\otimes s)=(\alpha\wedge\beta)\otimes s$, for any $\alpha$, $\beta\in\Omega^{\bullet}(\mathcal{A})$ and $s\in\Gamma(V)$.

Definition 1.22** ([DZ11, definition 8.4.7], [Fer02, section 0], [ELW99, section 2]):**

Let $\mathcal{A}=(A\to M,\mathsf{a},[\cdot,\cdot])$ be a Lie algebroid, and let $V\to M$ be a vector bundle. A $\mathcal{A}$-connection on $V\to M$ is a $\mathbb{R}$-linear map $\nabla:\Gamma(V)\to\Gamma(A^{*}\otimes V)$, satisfying the relations

[TABLE]

for any function $f\in\mathcal{C}^{\infty}(M)$ and sections $u\in\Gamma(A)$, $s\in\Gamma(V)$, where $\nabla_{u}$ denotes the map $\Gamma(V)\to\Gamma(V)$, $s\mapsto(\nabla s)(u)$, for any $u\in\Gamma(A)$.

Similarly to the covariant exterior derivative of a manifold equipped with a (linear) connection (see [Lee09, theorem 12.57]), there is a similar operator for Lie algebroids equipped with a representation, whose existence is guaranteed by the following proposition.

Proposition 1.23** ([Mar08, section 5.2]):**

Let $\mathcal{A}=(A\to M,\mathsf{a},[\cdot,\cdot])$ be a Lie algebroid, and let $\nabla$ be a $\mathcal{A}$-connection on a vector bundle $V\to M$. There exists a unique operator $\mathrm{d}_{\nabla}:\Omega^{p}(\mathcal{A},V)\to\Omega^{p+1}(\mathcal{A},V)$ called the covariant exterior derivative of $\mathcal{A}$, defined by $\mathrm{d}_{\nabla}s=\nabla s$ for all $s\in\Gamma(V)$, and

[TABLE]

for any $\alpha\in\Omega^{p}(\mathcal{A})$ and $s\in\Gamma(V)$; it is then extended to the whole $\Omega^{\bullet}(\mathcal{A},V)$ by the rules

[TABLE]

for all $\alpha\in\Omega^{p}(\mathcal{A})$ and $\omega\in\Omega^{q}(\mathcal{A},V)$.

Remark 1.24** ([MX94, definition 7.1.1]):**

The operator $\mathrm{d}_{\nabla}$ introduced in the previous proposition admits an intrinsic formula given by

[TABLE]

for all $\omega\in\Omega^{p}(\mathcal{A},V)$ and all $u_{0},\dots,u_{p}\in\Gamma(A)$.

The following result comes from the definition of a Lie algebroid connection.

Lemma 1.25**:**

Let $\mathcal{A}=(A\to M,\mathsf{a},[\cdot,\cdot])$ be a Lie algebroid, and let $\nabla$ be a $\mathcal{A}$-connection on a vector bundle $V\to M$. The map $\Gamma(A)\times\Gamma(A)\times\Gamma(V)\to\Gamma(V)$, $(u,v,s)\mapsto(\nabla_{u}\circ\nabla_{v}-\nabla_{v}\circ\nabla_{u}-\nabla_{[u,v]})(s)$ is $\mathcal{C}^{\infty}(M)$-trilinear. Thus there exists a $2$-differential form on $\mathcal{A}$ with values in $\operatorname{End}V$, $F_{\nabla}\in\Omega^{2}(\mathcal{A},\operatorname{End}V)$, such that $F_{\nabla}(u,v)=\nabla_{u}\circ\nabla_{v}-\nabla_{v}\circ\nabla_{u}-\nabla_{[u,v]}$, for all $u$, $v\in\Gamma(A)$.

Definition 1.26**:**

Let $\mathcal{A}$ be a Lie algebroid on a manifold $M$, and let $\nabla$ be a $\mathcal{A}$-connection on a vector bundle $V\to M$. The $2$-differential form $F_{\nabla}$ introduced in the previous lemma will be called the curvature of the $\mathcal{A}$-connection $\nabla$, and we will say that $\nabla$ is flat if $F_{\nabla}=0$.

The following lemma is immediate.

Lemma 1.27**:**

Let $\mathcal{A}$ be a Lie algebroid on a manifold $M$, and let $V\to M$ be a vector bundle. $\Omega^{\bullet}(\mathcal{A},\operatorname{End}V)$ is a $\mathbb{Z}$-graded $\mathbb{R}$-algebra for the multiplication defined by

[TABLE]

for any $\alpha$, $\beta\in\Omega^{\bullet}(\mathcal{A})$ and $\Phi$, $\Psi\in\Gamma(\operatorname{End}V)$. Moreover, $\Omega^{\bullet}(\mathcal{A},V)$ is a left $\Omega^{\bullet}(\mathcal{A},\operatorname{End}V)$-module for the multiplication defined by

[TABLE]

for any $\alpha$, $\beta\in\Omega^{\bullet}(\mathcal{A})$, $\Phi\in\Gamma(\operatorname{End}V)$ and $s\in\Gamma(V)$.

Proposition 1.28**:**

Let $\mathcal{A}=(A\to M,\mathsf{a},[\cdot,\cdot])$ be a Lie algebroid, and let $\nabla$ be a $\mathcal{A}$-connection on a vector bundle $V\to M$. We have the formula

[TABLE]

for all $\alpha\in\Omega^{\bullet}(\mathcal{A},V)$. Therefore, if $\nabla$ is flat, $\mathrm{d}_{\nabla}$ is a differential on $\Omega^{\bullet}(\mathcal{A},V)$.

Proof:.

To begin with, we have for any $\omega\in\Omega^{1}(\mathcal{A},V)$ that

[TABLE]

for all $u$, $v\in\Gamma(A)$. Then, let $s\in\Gamma(V)$, the above formula for $\omega=\nabla s\in\Omega^{1}(\mathcal{A},V)$ yields

[TABLE]

for all $u$, $v\in\Gamma(A)$. Now we prove the formula for any simple tensor $\alpha\otimes s\in\Omega^{p}(\mathcal{A},V)$. According to (1.1), we have

[TABLE]

where in the last step we used the fact that $F_{\nabla}$ is a $2$-differential form on $\mathcal{A}$. The general formula holds for any element of $\Omega^{\bullet}(\mathcal{A},V)$ using $\mathbb{R}$-linear combinations of simple tensors. ∎

Definition 1.29** ([DZ11, section 8.4], [ELW99, section 1]):**

Let $\mathcal{A}$ be a Lie algebroid on a manifold $M$. A representation of $\mathcal{A}$ (or a $\mathcal{A}$-module) is a vector bundle $V\to M$ equipped with a flat $\mathcal{A}$-connection $\nabla$. Such a representation will be denoted by $(V\to M,\nabla)$.

Definition 1.30**:**

Let $\mathcal{A}$ be a Lie algebroid on a manifold $M$ and $(V\to M,\nabla)$ be a $\mathcal{A}$-module. We define $H^{\bullet}(\mathcal{A};V,\nabla)$ as the homology of the differential graded vector space $(\Omega^{\bullet}(\mathcal{A},V),\mathrm{d}_{\nabla})$. We will call $H^{\bullet}(\mathcal{A};V,\nabla)$ the cohomology of $\mathcal{A}$ with coefficients in $(V\to M,\nabla)$.

Lemma 1.31**:**

Let $\mathcal{A}$ be a Lie algebroid on a manifold $M$ and $(V\to M,\nabla)$ be a $\mathcal{A}$-module. Then $H^{\bullet}(\mathcal{A};V,\nabla)$ is a graded $H^{\bullet}(\mathcal{A})$-module.

Proof:.

The multiplication is directly induced by the one defined in the proposition 1.21: $[\alpha]\wedge[\omega]=[\alpha\wedge\omega]$, for any $\alpha\in\Omega^{p}(\mathcal{A})$ and $\omega\in\Omega^{q}(\mathcal{A},V)$. According to (1.1), it is clear that $\alpha\wedge\omega$ is again $\mathrm{d}_{\nabla}$-closed. Also, the multiplication is well-defined, since, using (1.1) again, we have

[TABLE]

for any $\beta\in\Omega^{\bullet}(\mathcal{A})$ and $\eta\in\Omega^{\bullet}(\mathcal{A},V)$. ∎

Example 1.32**:**

Let $\mathcal{A}=(A\to M,\mathsf{a},[\cdot,\cdot])$ be a Lie algebroid. The trivial representation of $\mathcal{A}$ is given by the trivial vector bundle $V=M\times\mathbb{R}\to M$ of rank 1, together with the connection defined by $\nabla_{u}\lambda=\mathsf{a}(u)\cdot\lambda$, for any $u\in\Gamma(A)$ and $\lambda\in\Gamma(V)\cong\mathcal{C}^{\infty}(M)$. In this case, $H^{\bullet}(\mathcal{A};V,\nabla)\cong H^{\bullet}(\mathcal{A})$.

Example 1.33**:**

Let $\mathcal{A}$ be a Lie algebroid over a point $M$, that is, a Lie algebra $\mathfrak{g}$ (see example 1.5), that we will assume to be of finite dimension. Let $(V\to M,\nabla)$ be a $\mathcal{A}$-module. Since $M$ is a point, $V\to M$ is actually a vector space that we will denote again by $V$. Then, the $\mathcal{A}$-connection $\nabla$ is a linear map $V\to\mathfrak{g}^{*}\otimes V$. Using the linear isomorphisms

[TABLE]

we deduce that, to consider a linear map $V\to\mathfrak{g}^{*}\otimes V$ is equivalent to consider a linear map $\rho:\mathfrak{g}\to V$. Moreover, this map $\rho$ is also a Lie algebra homomorphism thanks to proposition 1.25. Therefore, in this case, a $\mathcal{A}$-module $(V\to M,\nabla)$ corresponds to a $\mathfrak{g}$-module $(V,\rho)$. The covariant exterior derivative is the Chevalley-Eilenberg differential on $\mathfrak{g}$ associated to the $\mathfrak{g}$-module $V$ (see [CE48, section 23]) and $H^{\bullet}(\mathcal{A};V,\nabla)$ is the Chevalley-Eilenberg cohomology $H_{\textsf{CE}}^{\bullet}(\mathfrak{g},V)$ of $\mathfrak{g}$ with coefficients in the $\mathfrak{g}$-module $V$.

Example 1.34**:**

Let $M$ be a manifold and let $\mathcal{T}_{M}$ denote the canonical Lie algebroid associated to $M$ (see example 1.7). A representation of $\mathcal{T}_{M}$ is a vector bundle $V\to M$ together with a flat linear connection $\nabla$ on $V\to M$. The covariant exterior derivative is the differential $\bm{\mathrm{d}}_{\nabla}$ associated to $\nabla$ (see [Lee09, theorem 12.57] and [GHV73, chapitre 7, section 4]), and $H^{\bullet}(\mathcal{T}_{M};V,\nabla)$ is the homology of the chain complex $(\Gamma(\Lambda^{\bullet}T^{*}M\otimes V),\bm{\mathrm{d}}_{\nabla})$.

Example 1.35**:**

Let $(M,\pi)$ be a Poisson manifold and let $\mathcal{P}_{M}[\pi]$ be the associated Lie algebroid (see example 1.9). A $\mathcal{P}_{M}[\pi]$-connection on a vector bundle $V\to M$ corresponds to the notion of a contravariant connection on $(M,\pi)$ (see [Fer00, proposition 2.1.2]). However, it seems that there is no mention of the associated cohomology in the literature.

Example 1.36**:**

Let $\mathcal{A}$ be the action Lie algebroid of example 1.10. We have a representation of $\mathcal{A}$ given by the trivial vector bundle $M\times\mathbb{R}\to\mathbb{R}$ of rank 1, together with the connection $\nabla:\mathcal{C}^{\infty}(M)\to\mathcal{C}^{\infty}(M,\mathfrak{g}^{*})\otimes\mathcal{C}^{\infty}(M)$ defined by $\nabla_{\xi}\lambda=\rho(\xi)\cdot\lambda$, for any $\xi\in\mathfrak{g}$ and $\lambda\in\mathcal{C}^{\infty}(M)$. Note that this connection is flat because $\rho:\mathfrak{g}\to\Gamma({TM})$ is a Lie algebra homomorphism. We have $\Omega^{\bullet}(\mathcal{A},M\times\mathbb{R})\cong\mathcal{C}^{\infty}(M,\Lambda^{\bullet}\mathfrak{g}^{*})$, and we also have an isomorphism of chain complexes

[TABLE]

where $\mathrm{d}_{\textsf{CE}}$ denotes the Chevalley-Eilenberg differential [CE48, section 23], and where $\Phi^{k}$ denotes the natural map $\Lambda^{k}\mathfrak{g}^{*}\otimes\mathcal{C}^{\infty}(M)\to\mathcal{C}^{\infty}(M,\Lambda^{k}\mathfrak{g}^{*})$, $\omega\otimes f\mapsto f\omega$. This map induces an isomorphism of graded modules between the cohomology $H^{\bullet}(\mathcal{A};M\times\mathbb{R},\nabla)$ on one side, and the Chevalley-Eilenberg cohomology $H_{\textsf{CE}}^{\bullet}(\mathfrak{g},\mathcal{C}^{\infty}(M))$ of $\mathfrak{g}$ with coefficients in the $\mathfrak{g}$-module $(\mathcal{C}^{\infty}(M),\rho)$ on the other side.

2 Twisted cohomology of Lie algebroids

In this section, given a Lie algebroid $\mathcal{A}$ and an odd cocycle $\theta$ with respect to the cohomology of $\mathcal{A}$, we define a new cohomology, that we will call the twisted cohomology of $\mathcal{A}$ by $\theta$. We show that it only depends on the cohomology class $[\theta]$ of $\theta$ (in the Lie algebroid cohomology of $\mathcal{A}$). Then we explain how this new cohomology encompasses several cohomology theories that appear in the literature.

2.1 Definition of the twisted cohomology

Definition 2.1**:**

Let $\mathcal{A}$ be a Lie algebroid on a manifold $M$. Define

[TABLE]

The elements of $\Omega^{\text{even}}(\mathcal{A})$ are called even differential forms on $\mathcal{A}$ and elements of $\Omega^{\text{odd}}(\mathcal{A})$ are called odd differential forms on $\mathcal{A}$.

This decomposition $\Omega^{\bullet}(\mathcal{A})=\Omega^{\text{even}}(\mathcal{A})\oplus\Omega^{\text{odd}}(\mathcal{A})$ of differential forms on $\mathcal{A}$ equips $\Omega^{\bullet}(\mathcal{A})$ with a $\mathbb{Z}/2\mathbb{Z}$-graduation, induced by the $\mathbb{Z}$-graduation. We extend this $\mathbb{Z}/2\mathbb{Z}$-graduation to $\Omega^{\bullet}(\mathcal{A},V)$ as soon as we are provided with a representation $(V\to M,\nabla)$ of $\mathcal{A}$, $M$ being the base of $\mathcal{A}$.

Definition 2.2**:**

Let $\mathcal{A}$ be a Lie algebroid on a manifold $M$, $(V\to M,\nabla)$ be a representation of $\mathcal{A}$, and $\theta\in\Omega^{\text{odd}}(\mathcal{A})$. The exterior covariant derivative of $\mathcal{A}$, relatively to $(V\to M,\nabla)$, twisted by the odd differential form $\theta$, is the operator $\mathrm{d}_{\nabla}[\theta]$ defined by

[TABLE]

for any $\omega\in\Omega^{\bullet}(\mathcal{A},V)$.

This operator appeared in [GrMa03, section 3] and [IM01, section 3.1]. Note that, since $\theta$ is odd, the operator ${\mathrm{d}_{\nabla}[\theta]}$ is odd, namely ${\mathrm{d}_{\nabla}[\theta]}$ maps $\Omega^{\text{even}}(\mathcal{A},V)$ into $\Omega^{\text{odd}}(\mathcal{A},V)$ and $\Omega^{\text{odd}}(\mathcal{A},V)$ into $\Omega^{\text{even}}(\mathcal{A},V)$.

Lemma 2.3**:**

Let $\mathcal{A}$ be a Lie algebroid on a manifold $M$, $(V\to M,\nabla)$ be a representation of $\mathcal{A}$, and $\theta\in\Omega^{\text{odd}}(\mathcal{A})$ be a $\mathrm{d}$-closed odd differential form on $\mathcal{A}$, where $\mathrm{d}$ denotes the differential of $\mathcal{A}$ (see definition 1.12). Under these assumptions, the operator ${\mathrm{d}_{\nabla}[\theta]}$ is a differential on the $\mathbb{Z}/2\mathbb{Z}$-graded $\mathbb{R}$-vector space $\Omega^{\bullet}(\mathcal{A},V)$.

Proof:.

For any $\omega\in\Omega^{\bullet}(\mathcal{A},V)$ we have

[TABLE]

But $\mathrm{d}_{\nabla}^{2}\omega=0$ since $\nabla$ is a flat connection (see the definition 1.29). Also $\theta\wedge\theta=0$ since $\theta$ is odd. Indeed, write $\sum_{i\geq 0}\theta_{2i+1}$ for $\theta$, with $\theta_{2i+1}\in\Omega^{2i+1}(\mathcal{A})$. Then $\theta_{2i+1}\wedge\theta_{2i+1}=0$ for any $i\geq 0$ and $\theta_{2i+1}\wedge\theta_{2j+1}=-\theta_{2j+1}\wedge\theta_{2i+1}$ for all $i$, $j\geq 0$ such that $i\neq j$. This implies that $\theta\wedge\theta=0$. Using the same decomposition for $\theta$ again, the above formula yields

[TABLE]

which equals zero since $\mathrm{d}_{\nabla}\theta=\mathrm{d}\theta=0$, $\theta$ being assumed to be $\mathrm{d}$-closed. ∎

Definition 2.4**:**

Let $\mathcal{A}$ be a Lie algebroid on a manifold $M$, $(V\to M,\nabla)$ be a representation of $\mathcal{A}$, and $\theta\in\Omega^{\text{odd}}(\mathcal{A})$ a $\mathrm{d}$-closed odd differential form on $\mathcal{A}$. Let $H^{\bullet}(\mathcal{A};V,\nabla;\theta)$ be the homology of the differential graded vector space $(\Omega^{\bullet}(\mathcal{A},V),{\mathrm{d}_{\nabla}[\theta]})$. We will call $H^{\bullet}(\mathcal{A};V,\nabla;\theta)$ the cohomology of $\mathcal{A}$ relatively to the representation $(V\to M,\nabla)$, twisted by the cocycle $\theta$.

Lemma 2.5**:**

Let $\mathcal{A}$ be a Lie algebroid on a manifold $M$, and $(V\to M,\nabla)$ be a representation of $\mathcal{A}$, and $\theta\in\Omega^{\text{odd}}(\mathcal{A})$ a $\mathrm{d}$-closed odd differential form on $\mathcal{A}$. Then $H^{\bullet}(\mathcal{A};V,\nabla;\theta)$ is a graded $H^{\bullet}(\mathcal{A})$-module.

Proof:.

The multiplication is the same as in lemma 1.31, only the differential changes. Let $\alpha\in\Omega^{p}(\mathcal{A})$ be a $\mathrm{d}$-closed differential form on $\mathcal{A}$ and let $\omega\in\Omega^{q}(\mathcal{A},V)$ be a ${\mathrm{d}_{\nabla}[\theta]}$-closed differential form on $\mathcal{A}$ with values in $V\to M$. Thanks to the identity (1.1), we compute that

[TABLE]

so the multiplication makes sense, that is we obtain a cocycle. But we also need to show that the multiplication is well-defined, that is it doesn’t depend on the choice of representative. Thanks to the identity (1.1) again, we compute for all $\beta\in\Omega^{p-1}(\mathcal{A})$ and $\eta\in\Omega^{q-1}(\mathcal{A},V)$ that

[TABLE]

Therefore we obtain

[TABLE]

which yields the result. ∎

2.2 Independence from the cocycle cohomology class

We can now state the main theorem of this note.

Theorem 2.6**:**

Let $\mathcal{A}$ be a Lie algebroid on a manifold $M$, $(V\to M,\nabla)$ be a representation of $\mathcal{A}$, $\theta\in\Omega^{\text{odd}}(\mathcal{A})$ a $\mathrm{d}$-closed odd differential form on $\mathcal{A}$ and $\Psi\in\Omega^{\text{even}}(\mathcal{A})$. We have an isomorphism of graded $H^{\bullet}(\mathcal{A})$-modules

[TABLE]

Proof:.

The idea of the proof is the following. Consider $\omega\in\operatorname{Ker}\mathrm{d}_{\nabla}[\theta+\mathrm{d}\Psi]$, so $\omega$ satisfies the differential equation ${\mathrm{d}_{\nabla}[\theta]}\omega+\mathrm{d}\Psi\wedge\omega=0$. By analogy with calculus, all solutions can be written as $\omega=\exp(-\Psi)\wedge\eta$, where $\eta$ is a constant for ${\mathrm{d}_{\nabla}[\theta]}$, that is $\eta\in\operatorname{Ker}{\mathrm{d}_{\nabla}[\theta]}$, provided that we can define $\exp(-\Psi)$. Thus, the isomorphism should be $\omega\mapsto\exp(\Psi)\wedge\omega$, for any $\omega\in\Omega^{\bullet}(\mathcal{A},V)$. Now we formulate a rigorous proof. Define $\exp(\Psi)|_{x}=\sum_{k\geq 0}\frac{\Psi^{\wedge k}|_{x}}{k!}$, for any $x\in M$; this series converge because exterior powers are known to vanish when the rank of the vector bundle is exceeded. For all real numbers $s$ and $t$ we have that $\exp(s\Psi)\wedge\exp(t\Psi)=\exp((s+t)\Psi)$ thanks to Newton’s binomial theorem; in particular for $s=1$ and $t=-1$ we get that the map $\omega\mapsto\exp(-\Psi)\wedge\omega$ is the inverse of the map $\omega\mapsto\exp(\Psi)\wedge\omega$. Moreover, $\Psi$ being even, the map $\omega\mapsto\exp(\Psi)\wedge\omega$ is even, that is, maps $\Omega^{\text{even}}(\mathcal{A},V)$ into $\Omega^{\text{even}}(\mathcal{A},V)$ and $\Omega^{\text{odd}}(\mathcal{A},V)$ into $\Omega^{\text{odd}}(\mathcal{A},V)$. By induction we have $\mathrm{d}\Psi^{\wedge k}=k\Psi^{\wedge(k-1)}\wedge\mathrm{d}\Psi$ for any $k\in\mathbb{N}^{*}$, hence $\mathrm{d}\exp(\Psi)=\exp(\Psi)\wedge\mathrm{d}\Psi$. For any $\omega\in\Omega^{\bullet}(\mathcal{A},V)$, we compute that

[TABLE]

which has for consequence that $\varepsilon=\exp(\Psi)\wedge\cdot$ induces a $\mathbb{R}$-linear isomorphism of $\mathbb{Z}/2\mathbb{Z}$-graded vector spaces in cohomology. Furthermore, it is also an isomorphism of graded $H^{\bullet}(\mathcal{A})$-modules, since we have $\varepsilon([\alpha]\wedge[\omega])=[\alpha]\wedge\varepsilon([\omega])$ for any $[\alpha]\in H^{p}(\mathcal{A})$ and $[\omega]\in H^{q}(\mathcal{A};V,\nabla;\theta)$. Indeed, we compute that

[TABLE]

where $\Gamma$ is defined by

[TABLE]

for any $\beta\in\Omega^{p-1}(\mathcal{A})$ and $\eta\in\Omega^{q-1}(\mathcal{A},V)$. ∎

The following corollary is directly deduced from the previous proposition.

Corollary 2.7**:**

Let $\mathcal{A}$ be a Lie algebroid on a manifold $M$, $(V\to M,\nabla)$ be a representation of $\mathcal{A}$, and $\theta\in\Omega^{\text{odd}}(\mathcal{A})$ be a $\mathrm{d}$-closed odd differential form on $\mathcal{A}$. Then $H^{\bullet}(\mathcal{A};V,\nabla;\theta)$ depends only on the (Lie algebroid) cohomology class $[\theta]\in H^{\text{odd}}(\mathcal{A})=\bigoplus_{i\geq 0}H^{2i+1}(\mathcal{A})$ of $\theta$ and will be denoted by $H^{\bullet}(\mathcal{A};V,\nabla;[\theta])$.

2.3 Examples of twisted cohomologies

In this last section, we show that the twisted cohomology of a Lie algebroid $\mathcal{A}$ by an odd cocycle can been found in various places of the literature.

Example 2.8**:**

Let $M$ be a manifold and let $H^{\bullet}_{\textsf{DR}}(M)$ denote the De Rham cohomology of $M$. We are going to apply the corollary 2.7 to the Lie algebroid $\mathcal{T}_{M}$ (see example 1.7). Let $\nabla$ be a flat linear connection on a vector bundle $V\to M$ and $\theta\in\Omega^{\text{odd}}(M)=\Omega^{\text{odd}}(\mathcal{T}_{M})$ be a $\bm{\mathrm{d}}$-closed odd differential form. Then the graded $H^{\bullet}_{\textsf{DR}}(M)$-module $H^{\bullet}(\mathcal{T}_{M};V,\nabla;\theta)$ only depends on the De Rham cohomology class $[\theta]\in H^{\text{odd}}_{\textsf{DR}}(M)=H^{\text{odd}}(\mathcal{T}_{M})$. This fact has been proven in [MW11, section 1] were the authors used twisted De Rham complexes to define a notion of twisted analytical torsion. In the case of the trivial representation (see example 1.32), the differential $\bm{\mathrm{d}}+\theta\wedge\cdot$, where $\theta$ is a $\bm{\mathrm{d}}$-closed differential form on $M$ of odd degree, has been used in various contexts like in twisted K-theory (see [AS06, section 6] and [FHT08, section2.1] for a more topological point of view), the theory of hypergeometric functions [AS97, section 2], or in the study of hyperplane arrangements [Kaw05], and in generalized complex geometry [Gua11, section 2.1], where it can be seen as the natural differential to consider on an exact Courant algebroid with Ševera class a $\bm{\mathrm{d}}$-closed $3$-differential form $\theta$ on the base manifold $M$.

Example 2.9**:**

Let $\mathcal{A}=(A\to M,\mathsf{a},[\cdot,\cdot])$ be a Lie algebroid and let $[\theta]\in H^{1}(\mathcal{A})$. We can modify the trivial representation of $\mathcal{A}$ (see 1.32) by means of the cocycle $\theta$ as follows. We keep the trivial vector bundle $V=M\times\mathbb{R}\to\mathbb{R}$ of rank 1 but we now take for connection $\nabla_{u}\lambda=\mathsf{a}(u)\cdot\lambda+\lambda\theta(u)$. Then the differential ${\mathrm{d}_{\nabla}[\theta]}$ plays a role in the definition of generalized Lie bialgebroids (see [NdCCG04, definition 2.4] and [IM01, section 3.1]) and also in the study of Dirac structures of such generalized Lie bialgebroids [NdCCG04, section 4].

Example 2.10**:**

Let $\mathfrak{g}$ be a Lie algebra, $(V,\rho)$ a $\mathfrak{g}$-module and $\theta$ an element in $\Lambda^{\bullet}\mathfrak{g}^{*}$ of odd degree, $\mathrm{d}_{\textsf{CE}}$-closed. Then the $H_{\textsf{CE}}(\mathfrak{g})$-module $H^{\bullet}(\mathfrak{g};V,\rho;\theta)$ only depends on the Chevalley-Eilenberg cohomology class $[\theta]\in H_{\textsf{CE}}(\mathfrak{g})$. This situation arises when one applies the example 2.8 to a compact simply connected Lie group (see [CE48]).

Example 2.11**:**

Let $(M,\pi)$ be a Poisson manifold and consider the associated Lie algebroid $\mathcal{P}_{M}[\pi]$ (see example 1.9) and let $\mathrm{d}_{\pi}$ denote its differential (see example 1.19). Let $R\in\Gamma(\Lambda^{3}TM)$ be a $\mathrm{d}_{\pi}$-closed $3$-vector field. Then the twisted differential $\mathrm{d}_{\pi}+R\wedge\cdot$ is used in [AMSW14, section 4.3] in relation with a Courant algebroid based on Poisson geometry. The associated twisted cohomology for the trivial representation of $\mathcal{P}_{M}[\pi]$ only depends on the Lichnerowicz-Poisson cohomology class $[R]\in H_{\textsf{LP}}^{3}(M,\pi)$.