Lie bialgebroids and Poisson groupoids: the Mackenzie-Xu duality

Intuition Beginner

A Lie algebra and its dual vector space usually play very different roles: one carries a bracket, the other does not. But in some happy situations the dual also carries a bracket, and the two brackets fit together. When that happens you have a matched pair — two structures looking at the same space from opposite sides, each constraining the other. This double-sided object is the heart of the story.

The geometric version replaces the single vector space with a bundle of them, one sitting over each point of a manifold, with a rule for moving between nearby points. Such a bundle-with-bracket is a Lie algebroid. The new idea here is to ask that its dual bundle is also a Lie algebroid, and that the two are compatible. The result is called a Lie bialgebroid, and it is the infinitesimal fingerprint of a much larger geometric object.

That larger object is a space of arrows — a groupoid — carrying a Poisson structure that respects how arrows compose. Mackenzie and Xu discovered that shrinking such a Poisson space of arrows down to its infinitesimal core produces exactly a Lie bialgebroid, with the bracket on one side and the bracket on the dual side both emerging from a single global rule.

Visual Beginner

Alt text: At the top of the picture sits a Poisson groupoid, drawn as a fan of arrows over a base line, shaded to suggest a Poisson structure that is compatible with composing arrows. A downward arrow labelled "differentiate / shrink to the infinitesimal" leads to the bottom of the picture, where a bundle A and its dual bundle A-star sit side by side over the same base line. Each bundle carries its own bracket, shown as a small looping symbol, and a horizontal link between them is labelled "compatible." The picture conveys that the global Poisson space of arrows leaves behind a two-sided infinitesimal object, the Lie bialgebroid, with matched brackets on a bundle and its dual.

Worked example Beginner

Start with the smallest interesting case: let the base be a single point. Then a Lie algebroid over that point is just a Lie algebra $\mathfrak{g}$, and its dual is the vector space ${\mathfrak{g}}^{\ast }$. Asking the dual to also be a Lie algebra, compatibly, gives a Lie bialgebra — the object Drinfeld built to describe symmetries of integrable systems.

A concrete one: take $\mathfrak{g}$ two-dimensional with bracket making it the "book" or affine Lie algebra, and equip ${\mathfrak{g}}^{\ast }$ with the dual bracket coming from a chosen element of ${\Lambda }^2\mathfrak{g}$. The compatibility is a short algebraic check that the two brackets respect each other. The pair $(\mathfrak{g},{\mathfrak{g}}^{\ast })$ is then a Lie bialgebra, and it is the shadow of a Poisson structure on the corresponding Lie group.

What this shows: the elaborate-sounding "Lie bialgebroid over a manifold" reduces, over one point, to the familiar Lie bialgebra. The manifold version just lets the brackets vary smoothly from point to point and adds an anchor saying how each fibre moves the base around.

Check your understanding Beginner

Formal definition Intermediate+

Throughout, $A\to M$ is a Lie algebroid with anchor $\rho :A\to TM$ and bracket $[\cdot ,\cdot {]}_A$ on $\Gamma (A)$ 03.04.16, and $A^{\ast }\to M$ is its dual vector bundle. The bracket on $A$ extends uniquely to the Schouten (Gerstenhaber) bracket $[\cdot ,\cdot {]}_A$ on the graded algebra $\Gamma ({\Lambda }^{\bullet }A)$ of multisections, making $\Gamma ({\Lambda }^{\bullet }A)$ a Gerstenhaber algebra: a graded-commutative product, a degree $-1$ graded Lie bracket extending $[\cdot ,\cdot {]}_A$ and the anchor action on functions, and the graded Leibniz rule tying the two. Dually, the Lie algebroid cohomology differential $d_A:\Gamma ({\Lambda }^p{A}^{\ast })\to \Gamma ({\Lambda }^{p+1}{A}^{\ast })$ of 03.04.22 encodes the same data. Suppose now $A^{\ast }$ also carries a Lie algebroid structure, with differential $d_{A^{\ast }}$ acting on $\Gamma ({\Lambda }^{\bullet }(A^{\ast }{)}^{\ast })=\Gamma ({\Lambda }^{\bullet }A)$.

Definition (Lie bialgebroid). A Lie bialgebroid is a pair $(A,A^{\ast })$ of Lie algebroids in duality such that the differential $d_{A^{\ast }}$ of $A^{\ast }$ is a derivation of the Schouten bracket $[\cdot ,\cdot {]}_A$ of $A$: $$ d_{A^}[X, Y]A = [d{A^}X,, Y]A + (-1)^{|X|-1}[X,, d{A^}Y]_A $$ for all multisections $X,Y\in \Gamma ({\Lambda }^{\bullet }A)$. This compatibility condition is symmetric in $A$ and $A^$:itholdsforthepair$(A,A^)$ifandonlyif$d_A$isaderivationof$[\cdot,\cdot]_{A^}$.Hencethedualpair$(A^*, A)$ is again a Lie bialgebroid — the Mackenzie-Xu duality at the level of the algebraic data.

An equivalent formulation uses the big bracket (the graded Poisson bracket of Kosmann-Schwarzbach) on $\Gamma ({\Lambda }^{\bullet }(A\oplus A^{\ast }))$. Both Lie algebroid structures are encoded by elements $\mu \in \Gamma ({\Lambda }^2{A}^{\ast }\otimes A)$ and $\gamma \in \Gamma ({\Lambda }^{2}A\otimes A^{\ast })$, and the pair $(A,A^{\ast })$ is a Lie bialgebroid exactly when the combined element $\mu +\gamma$ squares to zero under the big bracket. This packages the two Jacobi identities and the compatibility into the single condition $\{\mu +\gamma ,\mu +\gamma \}=0$.

Definition (Poisson groupoid). A Poisson groupoid is a Lie groupoid $\mathcal{G}\rightrightarrows M$ 03.03.10 equipped with a Poisson tensor $\pi \in \Gamma ({\Lambda }^{2}T\mathcal{G})$ that is multiplicative: the graph of groupoid multiplication $\{(g,h,gh)\}\subset \mathcal{G}\times \mathcal{G}\times \overline{\mathcal{G}}$ is a coisotropic submanifold, where $\overline{\mathcal{G}}$ carries $-\pi$. Equivalently $\pi$ is compatible with composition in the precise sense that the multiplication map is a Poisson map up to the sign convention on the third factor. The base $M$ inherits a Poisson structure, and the source map is Poisson while the target map is anti-Poisson.

Definition (the dual pairing). For a Poisson groupoid $(\mathcal{G},\pi )$ the Lie algebroid $A=A(\mathcal{G})$ 03.04.17 acquires a dual algebroid structure on $A^{\ast }$ from the linearisation of $\pi$ along the units; the resulting bracket on $\Gamma (A^{\ast })$ is the one whose Mackenzie-Xu compatibility with $[\cdot ,\cdot {]}_A$ we record below. When $\pi$ is non-degenerate this recovers the symplectic groupoid of 03.04.19.

Counterexamples to common slips

• A Lie algebroid plus any bracket on $A^{\ast }$ is not a bialgebroid. The compatibility condition $d_{A^{\ast }}[X,Y{]}_A=[d_{A^{\ast }}X,Y{]}_A\pm [X,d_{A^{\ast }}Y{]}_A$ is a genuine constraint coupling the two brackets; most independent choices fail it. The condition is what makes $(A,A^{\ast })$ rigid, not the existence of two brackets.
• "Multiplicative" is stronger than "invariant." A multiplicative Poisson tensor on a groupoid is not merely a left- or right-invariant bivector; the graph-coisotropy condition forces $\pi$ to vanish along the units in a controlled way, $\pi {|}_M=0$, which is what allows the linearisation defining $A^{\ast }$.
• Symplectic groupoid is the non-degenerate, not the general, case. Every symplectic groupoid is a Poisson groupoid, but the Poisson tensor of a general Poisson groupoid is degenerate; the bialgebroid $(A,A^{\ast })$ has $A^{\ast }\ne A$ unless the form is non-degenerate. Reading "Poisson groupoid" as "symplectic groupoid" loses the whole duality.

Key theorem with proof Intermediate+

Theorem (Mackenzie-Xu). Let $(\mathcal{G}\rightrightarrows M,\pi )$ be a Poisson groupoid with Lie algebroid $A=A(\mathcal{G})$. Then the multiplicative Poisson tensor $\pi$ induces a Lie algebroid structure on the dual bundle $A^{\ast }$, and the pair $(A,A^{\ast })$ is a Lie bialgebroid. Conversely, if $\mathcal{G}$ is source-simply-connected and $(A,A^{\ast })$ is the bialgebroid of an integrable Lie bialgebroid, then $\mathcal{G}$ carries a unique multiplicative Poisson tensor with this $(A,A^{\ast })$, so integrable Lie bialgebroids correspond to source-simply-connected Poisson groupoids.

Proof. Write $A=\ker (ds){|}_M$ with anchor $\rho =dt{|}_A$, as for the Lie functor 03.04.17. Because $\pi$ is multiplicative it vanishes on the units, $\pi {|}_M=0$, so its first-order jet along $M$ is well defined and intrinsic. Contracting this linearisation with right-invariant one-forms produces a skew bracket $[\cdot ,\cdot {]}_{A^{\ast }}$ on $\Gamma (A^{\ast })$ together with an anchor ${\rho }_{\ast }={\pi }^{♯}|$-derived map $A^{\ast }\to TM$.

Step 1: $A^$isaLiealgebroid.\ast TheJacobiidentityfor$[\cdot,\cdot]{A^*}$andtheLeibnizrulefor$\rho*$followfromtheintegrability$[\pi,\pi]_{\mathrm{SN}} = 0$ofthePoissontensortogetherwithitsmultiplicativity:theSchouten-Nijenhuisconditionon$\pi$linearises,alongtheunits,totheJacobiidentityfortheinducedbracket.ThisisthegroupoidanalogueofthefactthattheKoszulbracketon$T^*M$satisfiesJacobipreciselywhen$\pi$ is Poisson 03.04.19.

Step 2: the compatibility condition. The multiplicativity of $\pi$ is equivalent, after linearisation, to the statement that $d_{A^{\ast }}$ is a derivation of $[\cdot ,\cdot {]}_A$. Concretely, multiplicativity says the graph of multiplication is coisotropic; differentiating the coisotropy condition along the units, term by term in the convolution, yields exactly the derivation identity $d_{A^{\ast }}[X,Y{]}_A=[d_{A^{\ast }}X,Y{]}_A+(-1{)}^{|X|-1}[X,d_{A^{\ast }}Y{]}_A$. Mackenzie and Xu carry out this linearisation in ^{[Mackenzie-Xu §3]}; the key is that the convolution product on $\mathcal{G}$ differentiates to the Schouten bracket on $\Gamma ({\Lambda }^{\bullet }A)$, and the Poisson tensor differentiates to $d_{A^{\ast }}$. Hence $(A,A^{\ast })$ is a Lie bialgebroid.

Step 3: the converse by integration. Given an integrable Lie bialgebroid $(A,A^{\ast })$, integrate $A$ to its source-simply-connected groupoid $\mathcal{G}$ 03.04.18. The dual structure $d_{A^{\ast }}$, being a derivation of the Schouten bracket, integrates to a multiplicative bivector field on $\mathcal{G}$; the bialgebroid compatibility is precisely the infinitesimal form of multiplicativity and of $[\pi ,\pi ]=0$, so the integrated bivector is a multiplicative Poisson tensor. Uniqueness is the source-simply-connected rigidity of Lie II 03.04.18. Mackenzie-Xu prove the integration in their 2000 sequel ^{[Mackenzie-Xu 2000]}. $■$

Bridge. This theorem builds toward the entire programme of Poisson-Lie symmetry: it is the foundational reason a multiplicative Poisson structure leaves an algebraic shadow, and it appears again in the integration theory where the Drinfeld double and the symplectic double groupoid are constructed. The duality $A\leftrightarrow A^{\ast }$ is dual to the duality of the two algebroid differentials $d_A\leftrightarrow d_{A^{\ast }}$, and the symplectic groupoid of 03.04.19 is exactly the non-degenerate special case in which $A^{\ast }\cong A$ and the bivector is invertible. Putting these together, the single multiplicativity condition on $\pi$ is the central insight that generates both the bracket on $A^{\ast }$ and its compatibility with $A$ — one geometric condition, two algebraic consequences, one duality.

Exercises Intermediate+

Advanced results Master

The Courant algebroid double and Manin triples. For any Lie bialgebroid $(A,A^{\ast })$, Liu, Weinstein, and Xu construct on $E=A\oplus A^{\ast }$ the Courant bracket, a skew (or, in the Dorfman presentation, non-skew) bracket together with the canonical symmetric pairing $⟨X+\xi ,Y+\eta ⟩=\xi (Y)+\eta (X)$ and an anchor ${\rho }_E={\rho }_A+{\rho }_{A^{\ast }}$ ^{[Liu-Weinstein-Xu]}. This $E$ is a Courant algebroid: the bracket fails Jacobi by an exact term, and $A,A^{\ast }$ embed as a pair of transverse Dirac structures — a Manin triple $(E;A,A^{\ast })$. The Lie bialgebroid is recovered as the pair of complementary Dirac structures, and the Courant algebroid is the obstruction-carrying object whose Jacobiator measures the interaction. Over a point this is precisely Drinfeld's double Lie algebra $\mathfrak{d}=\mathfrak{g}\oplus {\mathfrak{g}}^{\ast }$ of a Lie bialgebra, the source of the quantum double.

Integration to Poisson groupoids and the double. Mackenzie and Xu prove that an integrable Lie bialgebroid integrates to a source-simply-connected Poisson groupoid, generalising Drinfeld's integration of a Lie bialgebra to a Poisson-Lie group ^{[Mackenzie-Xu 2000]}. The geometric reason the duality $A\leftrightarrow A^{\ast }$ exists at the global level is the cotangent double: the cotangent bundle $T^{\ast }\mathcal{G}$ of a Poisson groupoid is a double Lie groupoid, and its two algebroid reductions are $A$ and $A^{\ast }$. This double-structure source of the duality is the bridge to the LA-groupoid and double-Lie-algebroid theory developed in the companion unit 03.04.25, where $T^{\ast }A\cong T^{\ast }{A}^{\ast }$ realises the duality as an isomorphism of double vector bundles.

The Poisson-Lie group base case. When $M$ is a point, the whole theory specialises to Poisson-Lie groups: a Lie group $G$ with a multiplicative Poisson tensor, whose tangent bialgebroid is the Lie bialgebra $(\mathfrak{g},{\mathfrak{g}}^{\ast })$ with cobracket $\delta :\mathfrak{g}\to {\Lambda }^2\mathfrak{g}$ ^{[Mackenzie-Xu §4]}. The dual group $G^{\ast }$ integrates ${\mathfrak{g}}^{\ast }$, and the Drinfeld double $D=G\bowtie G^{\ast }$ integrates $\mathfrak{d}$. This is the classical limit of the quantum group / Yang-Baxter story, and the bialgebroid framework is exactly its geometrisation over a manifold rather than a point.

The modular class of a bialgebroid. Each side of a Lie bialgebroid has a modular class in $H^1(A)$ and $H^1(A^{\ast })$ 03.04.22; their interaction defines the modular class of the bialgebroid, an element measuring the failure of the two divergence operators to agree. For $(TM,T^{\ast }M)$ this is the modular class of the Poisson manifold (the obstruction to a volume form invariant under all Hamiltonian flows). The modular class is the primary secondary invariant attached to the duality and is the bialgebroid-level home of the unimodularity question.

Synthesis. The Mackenzie-Xu theory is the foundational reason that a single global object — a multiplicative Poisson tensor on a groupoid — produces a two-sided infinitesimal datum, and it generalises the Lie bialgebra of Drinfeld from a point to an arbitrary base. This is exactly the pattern whereby the cotangent algebroid $T^{\ast }M$ of a Poisson manifold 03.04.19 becomes one half of the prototype $(TM,T^{\ast }M)$, while the symplectic groupoid is the non-degenerate special case in which the duality $A\leftrightarrow A^{\ast }$ is the self-duality of $T^{\ast }M$. Putting these together, the central insight is that the compatibility condition — $d_{A^{\ast }}$ a derivation of $[\cdot ,\cdot {]}_A$ — is symmetric, so the duality is intrinsic and not a choice; and the bridge to the next level is the cotangent double $T^{\ast }\mathcal{G}$, whose double-vector-bundle duality $T^{\ast }A\cong T^{\ast }{A}^{\ast }$ realises $A\leftrightarrow A^{\ast }$ geometrically 03.04.25. The Courant algebroid $A\oplus A^{\ast }$ is dual to this picture: it is the obstruction-carrying Manin double in which $A$ and $A^{\ast }$ sit as transverse Dirac structures, and its Jacobiator is the precise measure of how the two brackets interact.

Full proof set Master

*Proposition (the prototype $(TM,T^{\ast }M)$ is a Lie bialgebroid).* *Let $(M,\pi )$ be a Poisson manifold. Then $(TM,T^{\ast }M)$, with $T^{\ast }M$ the cotangent algebroid and $d_{T^{\ast }M}=[\pi ,\cdot {]}_{\mathrm{SN}}$, is a Lie bialgebroid.*

Proof. The differential on $A=TM$ is the de Rham differential, and the dual differential on multivector fields $\Gamma ({\Lambda }^{\bullet }TM)$ is $d_{A^{\ast }}=[\pi ,\cdot {]}_{\mathrm{SN}}$, the Lichnerowicz operator. The Schouten bracket $[\cdot ,\cdot {]}_A$ on $\Gamma ({\Lambda }^{\bullet }TM)$ is the Schouten-Nijenhuis bracket of multivector fields. That $[\pi ,\cdot {]}_{\mathrm{SN}}$ is a derivation of $[\cdot ,\cdot {]}_{\mathrm{SN}}$ is the graded Jacobi identity for the Schouten-Nijenhuis bracket: $[\pi ,[X,Y]]=[[\pi ,X],Y]\pm [X,[\pi ,Y]]$. This holds for every bivector $\pi$, so the compatibility condition is automatic; the Poisson condition $[\pi ,\pi {]}_{\mathrm{SN}}=0$ is the separate requirement making $d_{A^{\ast }}^2=0$, i.e. making $T^{\ast }M$ a genuine Lie algebroid 03.04.19. Hence $(TM,T^{\ast }M)$ is a Lie bialgebroid. $■$

Proposition (duality of the compatibility condition). If $d_{A^}$isaderivationof$[\cdot,\cdot]A$,then$d_A$isaderivationof$[\cdot,\cdot]{A^}$.

Proof. Encode both algebroid structures in the big-bracket element $\Theta =\mu +\gamma$ on $\Gamma ({\Lambda }^{\bullet }(A\oplus A^{\ast }))$, with $\mu$ the structure of $A$ and $\gamma$ that of $A^{\ast }$. The Jacobi identities are $\{\mu ,\mu \}=0$ and $\{\gamma ,\gamma \}=0$. A computation of Kosmann-Schwarzbach shows that "$d_{A^{\ast }}$ is a derivation of $[\cdot ,\cdot {]}_A$" is equivalent to $\{\mu ,\gamma \}=0$ ^{[Kosmann-Schwarzbach §4]}. Since $\{\mu ,\gamma \}=\{\gamma ,\mu \}$ (graded symmetry of the big bracket in this bidegree), the same equation reads "$d_A$ is a derivation of $[\cdot ,\cdot {]}_{A^{\ast }}$." Hence the compatibility is symmetric and $(A^{\ast },A)$ is a Lie bialgebroid whenever $(A,A^{\ast })$ is. $■$

Proposition (the base inherits a Poisson structure). If $(\mathcal{G}\rightrightarrows M,\pi )$ is a Poisson groupoid, then $M$ carries a unique Poisson structure making the source map $s$ a Poisson map.

Proof. Multiplicativity forces $\pi {|}_M=0$ and makes the unit embedding $M\hookrightarrow \mathcal{G}$ coisotropic. Coisotropic reduction of $(\mathcal{G},\pi )$ along the units yields a Poisson bivector ${\pi }_M$ on $M$; concretely ${\pi }_M=s_{\ast }\pi$ is well defined because $s$ is constant along the symplectic leaves transverse to $M$ and the multiplicativity makes $s_{\ast }\pi$ independent of the chosen fibre. The bracket $\{f,g{\}}_M:=\{s^{\ast }f,s^{\ast }g{\}}_{\mathcal{G}}{|}_M$ is closed in pulled-back functions by multiplicativity, descends to $M$, and satisfies Jacobi because $\pi$ does. Uniqueness is forced by requiring $s$ Poisson. The target map is then anti-Poisson for the same structure, the groupoid analogue of source/target being Poisson/anti-Poisson for a symplectic groupoid 03.04.19. $■$

The full Mackenzie-Xu differentiation (Step 2 of the key theorem) and the integration theorem are stated above; the detailed convolution-linearisation computations are in Mackenzie-Xu 1994 §3 ^{[Mackenzie-Xu §3]} and the integration in Mackenzie-Xu 2000 ^{[Mackenzie-Xu 2000]}.

Connections Master

• Cotangent algebroid of a Poisson manifold 03.04.19. That unit builds $A^{\ast }=T^{\ast }M$ with its Koszul bracket and the symplectic groupoid integrating it; this unit pairs it with $A=TM$ to make the prototype bialgebroid $(TM,T^{\ast }M)$ and reveals the symplectic groupoid as the non-degenerate special case of a Poisson groupoid. The Poisson condition $[\pi ,\pi {]}_{\mathrm{SN}}=0$ proved there is exactly the bialgebroid compatibility here.

• Lie algebroid 03.04.16 and the Lie functor 03.04.17. A Lie bialgebroid is two interlocking copies of the algebroid structure defined there, on $A$ and on $A^{\ast }$; the Mackenzie-Xu theorem differentiates a Poisson groupoid using the same $A=\ker (ds){|}_M$ construction the Lie functor uses, now carrying the extra dual bracket from the Poisson tensor.

• Lie algebroid cohomology and the Chevalley-Eilenberg differential 03.04.22. The compatibility condition is phrased entirely in terms of the two differentials $d_A$ and $d_{A^{\ast }}$ from that unit: $d_{A^{\ast }}$ must be a derivation of the Schouten bracket dual to $d_A$. The modular classes in $H^1(A)$ and $H^1(A^{\ast })$ combine into the modular class of the bialgebroid.

• Poisson bracket and Poisson manifolds 05.02.02. The base of a Poisson groupoid is a Poisson manifold, and the prototype bialgebroid is built directly from the Poisson tensor; the whole theory is the groupoid-level lift of Poisson geometry, with the multiplicative tensor on $\mathcal{G}$ projecting to the Poisson bracket on $M$.

• Double Lie groupoids and double Lie algebroids 03.04.25. The geometric origin of the duality $A\leftrightarrow A^{\ast }$ is the cotangent double $T^{\ast }\mathcal{G}$, a double Lie groupoid whose two reductions are $A$ and $A^{\ast }$; the companion unit develops the double-structure machinery ($T^{\ast }A\cong T^{\ast }{A}^{\ast }$) that makes the Mackenzie-Xu duality a statement about double vector bundles.

Historical & philosophical context Master

The story begins with Drinfeld, who in his 1983 work on the classical Yang-Baxter equation and quantum groups introduced Lie bialgebras $(\mathfrak{g},{\mathfrak{g}}^{\ast })$ and Poisson-Lie groups as the classical limit of quantum groups ^{[Mackenzie-Xu §4]}. Weinstein then proposed Poisson groupoids in his 1988 paper Coisotropic calculus and Poisson groupoids as the common generalisation of Poisson-Lie groups and symplectic groupoids, asking what infinitesimal object they differentiate to ^{[Weinstein 1988]}. The answer was supplied by Kirill Mackenzie and Ping Xu in their 1994 Duke Mathematical Journal paper Lie bialgebroids and Poisson groupoids, which defined the Lie bialgebroid via the derivation-compatibility condition and proved that it is precisely the infinitesimal of a Poisson groupoid, establishing the duality $A\leftrightarrow A^{\ast }$ ^{[Mackenzie-Xu §2-4]}.

The double construction was completed by Liu, Weinstein, and Xu in 1997, who identified $A\oplus A^{\ast }$ as a Courant algebroid and a Lie bialgebroid as a pair of transverse Dirac structures — a Manin triple — feeding directly into Hitchin and Gualtieri's later generalised complex geometry ^{[Liu-Weinstein-Xu]}. Mackenzie and Xu closed the integration question in 2000, showing integrable Lie bialgebroids integrate to Poisson groupoids ^{[Mackenzie-Xu 2000]}. Mackenzie's 2005 monograph General Theory of Lie Groupoids and Lie Algebroids placed the whole theory inside the double-structure framework of Chapter 12, the philosophical payoff being that the seemingly asymmetric "object and its dual" is in fact a symmetric duality governed by a single self-dual condition — a recurring theme, surveyed by Kosmann-Schwarzbach, in which Poisson geometry, Lie theory, and supergeometry meet in one graded bracket ^{[Kosmann-Schwarzbach §3-5]}.

Bibliography Master

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  title   = {Lie bialgebroids and {P}oisson groupoids},
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}

@article{MackenzieXu2000,
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  year    = {2000}
}

@article{LiuWeinsteinXu1997,
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}

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}

@book{Mackenzie2005,
  author    = {Mackenzie, Kirill C. H.},
  title     = {General Theory of {L}ie Groupoids and {L}ie Algebroids},
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  year      = {2005},
  note      = {Chapter 12: Poisson groupoids and Lie bialgebroids}
}

@article{KosmannSchwarzbach2008,
  author  = {Kosmann-Schwarzbach, Yvette},
  title   = {{P}oisson manifolds, {L}ie algebroids, modular classes: a survey},
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  year    = {2008}
}

@article{Drinfeld1983,
  author  = {Drinfeld, Vladimir G.},
  title   = {Hamiltonian structures on {L}ie groups, {L}ie bialgebras and the geometric meaning of the classical {Y}ang--{B}axter equations},
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}