Double Lie groupoids, double Lie algebroids, and VB-groupoids

Intuition Beginner

A groupoid is a system of reversible arrows between points: you can travel from one point to another and back. Now imagine you have not one such system but two living on the same set of objects at once, and the two are laid out at right angles so that you can compose arrows in two independent directions. Picture a grid of city streets: from any corner you can walk one block east, then one block north, or north then east. The grid lets you move in two directions, and the two motions fit together into a square.

A double Lie groupoid is the smooth version of that grid. It is a square of four corner sets, with two horizontal arrow-systems and two vertical arrow-systems, arranged so that horizontal composition and vertical composition agree on the shared squares. Each side is an ordinary smooth groupoid; the new content is the law that lets the two directions interlock.

Why build such a thing? Because the right infinitesimal object behind many dualities in geometry is not a single arrow-system but a square of them. Shrinking a groupoid to its infinitesimal shadow is a one-way street that keeps the local rule. Shrinking a square of groupoids, one direction at a time, lands you at the deepest layer of the theory.

Visual Beginner

Picture a square with a set at each of its four corners. Along the top and bottom edges run horizontal arrows; along the left and right edges run vertical arrows. A single small square in the interior is one element of the top-left corner set, and it has a left edge, a right edge, a top edge, and a bottom edge, each an arrow in one of the four side groupoids.

To compose two interior squares horizontally, glue them along a shared vertical edge; to compose vertically, glue along a shared horizontal edge. The interchange law says that gluing a two-by-two block of squares gives the same result whether you first join the rows or first join the columns. A special, simpler case keeps the vertical direction linear, so each vertical fibre is a flat space of directions rather than a curved system of arrows; that linear-on-one-side square is a VB-groupoid.

Worked example Beginner

Take a single space and form its tangent bundle: over each point sits the flat space of directions you can move in. Now take the pair groupoid of the space, whose arrows are all ordered pairs of points. Apply the tangent-bundle construction to the whole pair groupoid at once. You get a new system of arrows, this time between directions rather than points, and it sits over the original pair groupoid.

What you have built is a square. Along one direction the arrows are the original pairs of points. Along the other direction the arrows are linear: they live in flat tangent spaces. The bottom edge is the pair groupoid; the side edges record how a direction at one point is carried to a direction at another.

This square is the tangent VB-groupoid. It keeps the groupoid law in one direction and a flat vector-space law in the other. The flat direction is what the letters "VB" stand for: a vector bundle living inside a groupoid.

Reading off the lesson: differentiating a whole groupoid in one direction, instead of point by point, produces a square in which one side is an ordinary groupoid and the perpendicular side is linear. The square remembers both the global arrows and their infinitesimal directions in one object.

Check your understanding Beginner

Formal definition Intermediate+

A double vector bundle is a square $$ \begin{array}{ccc} D & \longrightarrow & A \ \downarrow & & \downarrow \ B & \longrightarrow & M \end{array} $$ in which $D\to A$ and $D\to B$ are vector bundles, $A\to M$ and $B\to M$ are vector bundles, and the structure maps of each pair of parallel bundles are vector-bundle morphisms over the structure maps of the other pair. The core is the subbundle $$ C := \ker(D \to A) \cap \ker(D \to B) \longrightarrow M, $$ the intersection of the two projections' kernels along the zero sections; it carries the part of $D$ seen by neither outer bundle. The canonical example is the tangent double $TE$ of a vector bundle $E\to M$, with outer bundles $E$ and $TM$ and core $E$ again ^{[Pradines 1977]}.

A VB-groupoid is a groupoid object in the category of vector bundles, equivalently a vector bundle in the category of Lie groupoids: a square $$ \begin{array}{ccc} \Omega & \rightrightarrows & E \ \downarrow & & \downarrow \ \mathcal G & \rightrightarrows & M \end{array} $$ where $\Omega \rightrightarrows E$ and $\mathcal{G}\rightrightarrows M$ are Lie groupoids, $\Omega \to \mathcal{G}$ and $E\to M$ are vector bundles, and the groupoid source, target, unit, multiplication, and inversion of $\Omega \rightrightarrows E$ are all vector-bundle morphisms over those of $\mathcal{G}\rightrightarrows M$. The core of a VB-groupoid is the vector bundle $C\to M$ obtained by restricting $\ker (\Omega \to \mathcal{G})$ to the units $u(M)$. The model examples are the tangent VB-groupoid $T\mathcal{G}\rightrightarrows TM$ over $\mathcal{G}\rightrightarrows M$, with core the Lie algebroid $A(\mathcal{G})$, and the cotangent VB-groupoid $T^{\ast }\mathcal{G}\rightrightarrows A^{\ast }$, whose core is the cotangent bundle $T^{\ast }M$ ^{[Mackenzie Chs. 9-11]}.

A double Lie groupoid is a groupoid object in the category of Lie groupoids: a square $$ \begin{array}{ccc} S & \rightrightarrows & H \ \downrightarrows & & \downrightarrows \ V & \rightrightarrows & M \end{array} $$ in which $S\rightrightarrows H$ and $S\rightrightarrows V$ are the horizontal and vertical Lie groupoid structures on the top corner $S$, $V\rightrightarrows M$ and $H\rightrightarrows M$ are the side groupoids, and each horizontal structure map is a morphism for the vertical structure and conversely. The data must satisfy the double source condition: the double source map $$ (s^H, s^V) : S \longrightarrow H \times_M V, \qquad x \mapsto (s^H(x), s^V(x)), $$ sending a square to its horizontal and vertical sources, is a surjective submersion. The core of a double Lie groupoid is the Lie groupoid $K\rightrightarrows M$ of squares whose horizontal and vertical sources are both units, with the residual groupoid structure inherited from $S$. A VB-algebroid is the algebroid-side analogue: a vector bundle in the category of Lie algebroids, the linear-on-one-side square obtained by differentiating one direction of a VB-groupoid.

Key theorem with proof Intermediate+

Theorem (The double source map controls differentiation). Let $(S;H,V;M)$ be a double Lie groupoid with double source map $(s^H,s^V):S\to H{\times }_{M}V$ a surjective submersion. Then applying the Lie functor 03.04.17 in the horizontal direction yields a well-defined VB-algebroid $A^H(S)\rightrightarrows A(H)$ — equivalently an LA-groupoid, a Lie groupoid in the category of Lie algebroids — and applying it again in the vertical direction yields a double Lie algebroid. The two single differentiations commute up to canonical isomorphism.

Proof. Write $A^V(S)\to H$ for the Lie algebroid of the vertical groupoid $S\rightrightarrows H$, namely $\ker (d{s}^V)$ restricted to the vertical units, with anchor the vertical target differential and bracket of right-invariant vertical fields, exactly as in 03.04.17. The horizontal groupoid structure on $S\rightrightarrows H$ restricts to the vertical-unit submanifold and differentiates its fibres, so $A^V(S)$ inherits a Lie groupoid structure over $A(H)$ provided the horizontal source restricted to $\ker (d{s}^V)$ remains a submersion onto $A(H)$. The double source condition is precisely what supplies this: because $(s^H,s^V)$ is a surjective submersion, the partial map $s^H$ stays a submersion after fixing or differentiating $s^V$, so $\ker (d{s}^V)\to A(H)$ is a surjective submersion of the required kind. Hence $A^V(S)\rightrightarrows A(H)$ is a Lie groupoid whose structure maps are linear over those of $S\rightrightarrows H$; that linearity is the VB-groupoid / LA-groupoid condition.

Now differentiate this LA-groupoid horizontally. The groupoid $A^V(S)\rightrightarrows A(H)$ is a Lie groupoid in the category of Lie algebroids, so its Lie functor produces a Lie algebroid $A^H{A}^V(S)$ over $A(H)$ that is again linear over the vertical algebroid data; this is a double Lie algebroid. Symmetrically, differentiating $V$ first and then $H$ yields $A^V{A}^H(S)$. Both arise from the iterated kernel-of-differential construction applied to the two compatible foliations by source fibres, and the compatibility of those two foliations — guaranteed once more by the double source map being a submersion onto the fibre product — gives a canonical identification $A^H{A}^V(S)\cong A^V{A}^H(S)$ of the two iterated derivatives, the commutative cube of differentiations. $\square$

Bridge. This commuting square of single differentiations builds toward the cotangent-double picture, where the canonical isomorphism is not abstract but the concrete duality $T^{\ast }A\cong T^{\ast }{A}^{\ast }$; it appears again in the integration of Lie bialgebroids 03.04.24, where the double of a Poisson groupoid is the geometric carrier of the $(A,A^{\ast })$ pairing. The bridge is that the double source condition is the foundational reason the Lie functor can be iterated at all: it is exactly the second-order smoothness that lets one curved direction be differentiated without disturbing the other, and this is dual to the way the core of a VB-groupoid records the algebroid that a single differentiation produces. Putting these together, the central insight is that the square, not the single groupoid, is the natural domain of the duality theory, and the commutative cube generalises the one-step Lie functor of 03.04.17 to a second-order operation.

Exercises Intermediate+

Lean formalization Intermediate+

lean_status: none — Mathlib has neither a LieGroupoid nor a LieAlgebroid type, and no machinery for internal objects in a category of vector bundles or of Lie groupoids, so no double structure on this page can be stated. The sketch below records the missing layers; it does not compile against current Mathlib, which is why no Lean module is declared.

-- Pseudo-Lean: target constructions, none present in Mathlib.
variable {M : Type*} [Manifold M]

structure DoubleVectorBundle (A B : VectorBundle M) where
  total   : Type*
  toA     : total → A        -- vector bundle over A
  toB     : total → B        -- vector bundle over B
  compat  : LinearOverEachOther toA toB
  core    : VectorBundle M := kernelIntersection toA toB

structure VBGroupoid (𝒢 : LieGroupoid M) (E : VectorBundle M) where
  Omega       : LieGroupoid E
  proj        : VectorBundleMorphism Omega.arrows 𝒢.arrows
  linearStruct: AllStructureMapsLinear Omega 𝒢

structure DoubleLieGroupoid (H V : LieGroupoid M) where
  S            : Type*
  horiz        : LieGroupoid_structure_on S over H
  vert         : LieGroupoid_structure_on S over V
  doubleSource : Submersion (fun x => (sH x, sV x)) (FiberProduct H V)

-- the double Lie functor: differentiate twice, get a commuting cube
def LAGroupoidOf  (D : DoubleLieGroupoid H V) : LAGroupoid := lieFunctorVertical D
def DoubleAlgebroidOf (D : DoubleLieGroupoid H V) : DoubleLieAlgebroid :=
  lieFunctorHorizontal (LAGroupoidOf D)
theorem differentiations_commute (D) :
    DoubleAlgebroidOf D ≅ flipDifferentiate D    -- the commutative cube

The first genuine obstacle is doubleSource together with its Submersion hypothesis: Mathlib has no fibre product of Lie groupoids over a common base and no statement that a joint source map is a submersion, and that submersion property is exactly the second-order smoothness the iterated Lie functor consumes.

Advanced results Master

The double Lie functor. The Lie functor of 03.04.17 is a first-order operation: it differentiates a groupoid once. On a double Lie groupoid it can be applied twice, once in each direction, and the two single differentiations assemble into a square of differentiations. Differentiating one direction of a double Lie groupoid $(S;H,V;M)$ produces an LA-groupoid: a Lie groupoid object in the category of Lie algebroids, equivalently a VB-groupoid carrying a fibrewise-linear Lie algebroid structure. Differentiating the remaining groupoid direction of an LA-groupoid produces a double Lie algebroid, a Lie algebroid object in the category of Lie algebroids. The two orders of differentiation agree up to a canonical flip, so the four corners $S$, $A^H(S)$, $A^V(S)$, and $AA(S)$ form a commutative cube with the original double groupoid at one vertex and its double algebroid at the opposite one ^{[Mackenzie Chs. 9-11]}.

The tangent and cotangent doubles. Two doubles dominate the applications. The tangent double of a Lie groupoid $\mathcal{G}$ is $T\mathcal{G}\rightrightarrows TM$, an LA-groupoid whose core is $A(\mathcal{G})$; the cotangent double is $T^{\ast }\mathcal{G}\rightrightarrows A^{\ast }$, an LA-groupoid built from the cotangent lift of the multiplication, whose core is $T^{\ast }M$. The cores $A(\mathcal{G})$ and $T^{\ast }M$ are dual bundles, and that duality is no accident: it is the geometric origin of the pairing between an algebroid and its dual.

The duality of double vector bundles. A double vector bundle $D$ with outer bundles $A$ and $B$ and core $C$ admits two duals, $D\to A$ dualised to $D_A^{\ast }$ and $D\to B$ dualised to $D_B^{\ast }$, and Mackenzie's theorem is that these two duals are canonically isomorphic as double vector bundles, exchanging the roles of the second outer bundle and the core: $D_A^{\ast }$ has outer bundles $A$ and $C^{\ast }$, while $D_B^{\ast }$ has outer bundles $B$ and $C^{\ast }$, and the cotangent double of an algebroid realises the symmetry $T^{\ast }A\cong T^{\ast }{A}^{\ast }$ ^{[Mackenzie-Xu 1994]}. This $\mathcal{R}$-isomorphism is the engine that turns a single Lie algebroid into a paired object $(A,A^{\ast })$.

The bridge to Lie bialgebroids. A Poisson groupoid carries a multiplicative Poisson tensor, and the cotangent double $T^{\ast }\mathcal{G}$ of a Poisson groupoid is itself a double Lie groupoid. Differentiating it produces a double Lie algebroid whose two sides are $A$ and $A^{\ast }$, and the compatibility encoded in the double structure is exactly the Lie-bialgebroid condition 03.04.24. The Mackenzie-Xu theorem, that the algebroid of a Poisson groupoid and its dual form a Lie bialgebroid, is thereby a statement about the cotangent double: the double-structure theory is the apparatus that makes the bialgebroid duality a theorem rather than a coincidence ^{[Mackenzie-Xu 1994]}.

Synthesis. The double-structure theory recasts the Lie functor as a second-order operation, and putting these together gives the central insight that the natural home of duality in groupoid geometry is the square, not the single groupoid. A double Lie groupoid differentiates twice into a double Lie algebroid through a commutative cube, the foundational reason being the double source condition, which supplies exactly the transverse smoothness that lets one curved direction be differentiated without disturbing the other; this is exactly the second-order smoothness that the iterated kernel-of-differential construction consumes. The tangent and cotangent doubles realise the abstract cube concretely, with cores $A(\mathcal{G})$ and $T^{\ast }M$ that are dual bundles, and the duality of double vector bundles $T^{\ast }A\cong T^{\ast }{A}^{\ast }$ generalises the linear duality of a vector bundle to the double setting and is dual to the exchange of outer bundle and core. The cotangent double of a Poisson groupoid carries the $(A,A^{\ast })$ pairing, so the Mackenzie-Xu integration of a Lie bialgebroid 03.04.24 appears again here as the differentiation of a double structure, and the whole apparatus builds toward the curriculum's Poisson-geometry track by making the bialgebroid duality the shadow of a geometric square.

Full proof set Master

Proposition (Core of a VB-groupoid is a Lie algebroid when the base groupoid acts). Let $\Omega \rightrightarrows E$ be a VB-groupoid over the Lie groupoid $\mathcal{G}\rightrightarrows M$, with core $C\to M$. Then $C$ inherits a Lie algebroid structure over $M$, and for the tangent VB-groupoid $T\mathcal{G}\rightrightarrows TM$ this core algebroid is precisely $A(\mathcal{G})$.

Proof. The kernel $\ker (\Omega \to \mathcal{G})$ is a vector bundle over $\mathcal{G}$, and restricting to the unit submanifold $u(M)$ gives $C\to M$. The Lie functor 03.04.17 sends $\Omega \rightrightarrows E$ to its algebroid $A(\Omega )\to E$; because the bundle projection $\Omega \to \mathcal{G}$ is a groupoid morphism of vector bundles, differentiation yields a morphism $A(\Omega )\to A(\mathcal{G})$ whose kernel restricted to the zero section of $E$ is the core $C$. The bracket of right-invariant sections of $\Omega$ tangent to the core directions stays in the core, since the core is cut out by the two linear kernel conditions that the right translations preserve; this gives $C$ an anchor (the restriction of the $A(\Omega )$ anchor) and a bracket satisfying the Leibniz law of 03.04.16. For the tangent VB-groupoid, $\ker (T\mathcal{G}\to \mathcal{G}){|}_{u(M)}$ intersected with $\ker (ds)$ is $\ker (ds){|}_{u(M)}=A(\mathcal{G})$, and the induced bracket is the bracket of right-invariant source-fibre fields, so the core algebroid equals $A(\mathcal{G})$. $\square$

Proposition (Double source submersion implies iterated Lie functor is defined). Let $(S;H,V;M)$ be a double Lie groupoid whose double source map $(s^H,s^V):S\to H{\times }_{M}V$ is a surjective submersion. Then $\ker (d{s}^V)$ restricted to the vertical units is a smooth vector bundle over $H$ that carries a Lie groupoid structure over $A(H)$, so $A^V(S)\rightrightarrows A(H)$ is an LA-groupoid.

Proof. Since $s^V:S\to V$ is a submersion (it is a component of the submersion $(s^H,s^V)$), $\ker (d{s}^V)\subseteq TS$ is a smooth subbundle, and its restriction to the vertical-unit submanifold is the vertical algebroid bundle $A^V(S)$ as in 03.04.17. The horizontal source $s^H$ restricted to this subbundle differentiates to a map $A^V(S)\to A(H)$. To see this map is a surjective submersion, factor it through the differential of $(s^H,s^V)$: surjectivity of $(s^H,s^V)$ onto the fibre product makes the partial differential of $s^H$ along $\ker (d{s}^V)$ surjective onto the corresponding fibre of $A(H)$. The horizontal groupoid multiplication on $S$ is by hypothesis a morphism for the vertical structure, so it restricts to $\ker (d{s}^V)$ and descends to a multiplication on $A^V(S)$ that is linear over the algebroid bracket on $A(H)$. Therefore $A^V(S)\rightrightarrows A(H)$ is a Lie groupoid whose structure maps are fibrewise linear, which is the LA-groupoid condition. $\square$

Proposition (Cotangent double realises algebroid self-duality). For a Lie algebroid $A\to M$, the cotangent double satisfies $T^{\ast }A\cong T^{\ast }{A}^{\ast }$ as double vector bundles, the isomorphism exchanging the outer bundle $A$ with $A^{\ast }$ and fixing the common core $T^{\ast }M$.

Proof sketch. The cotangent bundle $T^{\ast }A$ of the total space of $A$ is a double vector bundle with outer bundles $A$ and $T^{\ast }M$ and core $A^{\ast }$; symmetrically $T^{\ast }{A}^{\ast }$ has outer bundles $A^{\ast }$ and $T^{\ast }M$ and core $A$. The Mackenzie $\mathcal{R}$-map is built from the canonical pairing between $A$ and $A^{\ast }$ together with the tautological one-form, sending a covector on $A$ to a covector on $A^{\ast }$ by Legendre-type transposition along the fibres. It is a diffeomorphism intertwining the two outer projections after the exchange, and it fixes $T^{\ast }M$ because the base directions are common to both. The resulting isomorphism is the geometric carrier of the symmetry that makes the Lie-bialgebroid pairing $(A,A^{\ast })$ symmetric in its two slots ^{[Mackenzie-Xu 1994]}. $\square$

Connections Master

The Lie functor 03.04.17 is the first-order operation this unit iterates: a single groupoid differentiates to a single algebroid, and the double Lie functor here applies that same kernel-of-source-differential recipe twice, once in each direction of a square, with the double source condition the smoothness hypothesis that makes the second application land in a manifold. The commutative cube is the second-order generalisation of the one-step functor.

A Lie groupoid 03.03.10 is the corner object of every double structure: each side of a double Lie groupoid is an ordinary Lie groupoid, and the tangent and cotangent VB-groupoids $T\mathcal{G}\rightrightarrows TM$ and $T^{\ast }\mathcal{G}\rightrightarrows A^{\ast }$ are built directly from one groupoid by lifting its structure maps to the tangent and cotangent bundles. The interchange law is the compatibility of the corner groupoids' compositions.

A Lie algebroid 03.04.16 appears here both as the core of a tangent VB-groupoid — where the core is exactly $A(\mathcal{G})$ — and as the target of differentiation, since a double Lie algebroid is a Lie algebroid object in Lie algebroids. The anchor-bracket-Leibniz data of that unit is what the linearity conditions of a VB-algebroid refine to the fibrewise setting.

The Atiyah algebroid 03.05.22 is the transitive prototype of a VB-algebroid: the gauge groupoid of a principal bundle has a tangent VB-groupoid whose core algebroid is the Atiyah algebroid, so the connection-as-splitting picture of that unit is the linear, one-sided shadow of the double-structure theory, and the adjoint bundle reappears as the core.

Lie bialgebroids and Poisson groupoids 03.04.24 are the application the entire double-structure apparatus serves: the cotangent double of a Poisson groupoid differentiates to a double Lie algebroid whose two sides are $A$ and $A^{\ast }$, so the Mackenzie-Xu duality of that unit is, read through this page, a theorem about a commutative cube of differentiations, and the integration of an integrable Lie bialgebroid to a Poisson groupoid is the statement that the corresponding double Lie groupoid exists.

Historical & philosophical context Master

The double-structure theory grew from Jean Pradines's introduction of double vector bundles and the non-holonomic jet calculus, where the core of a double vector bundle and the iterated tangent construction first appear in print ^{[Pradines 1977]}. Pradines, working in the lineage of Charles Ehresmann's categorical differential geometry, recognised that the second-order objects of geometry are naturally squares rather than single bundles, and that differentiation is an operation that can be iterated once the right smoothness is imposed.

Kirill Mackenzie made the double Lie groupoid and the double Lie algebroid into a systematic theory across a sequence of papers and the second half of his 2005 monograph, where the double source condition, the double Lie functor, and the commutative cube of differentiations are given their standard form ^{[Mackenzie Chs. 9-11]}. With Ping Xu he identified the cotangent double and the duality of double vector bundles as the geometric source of the Lie-bialgebroid pairing, proving that the algebroid of a Poisson groupoid and its dual form a Lie bialgebroid through the symmetry $T^{\ast }A\cong T^{\ast }{A}^{\ast }$ ^{[Mackenzie-Xu 1994]}. Alfonso Gracia-Saz and Rajan Mehta later connected the VB-groupoid picture to representations up to homotopy, recovering the superconnection description of a Lie algebroid representation as the linearisation of a VB-groupoid ^{[Gracia-Saz-Mehta 2017]}. The philosophical thread running through this history is that duality, which for a single vector bundle is the elementary passage to the dual space, becomes a structural theorem only after one ascends to second-order, double-categorical objects.

Bibliography Master

@book{mackenzie2005double,
  author    = {Mackenzie, Kirill C. H.},
  title     = {General Theory of Lie Groupoids and Lie Algebroids},
  series    = {London Mathematical Society Lecture Note Series},
  volume    = {213},
  publisher = {Cambridge University Press},
  year      = {2005}
}

@article{mackenzie-xu1994,
  author    = {Mackenzie, Kirill C. H. and Xu, Ping},
  title     = {Lie bialgebroids and Poisson groupoids},
  journal   = {Duke Mathematical Journal},
  volume    = {73},
  number    = {2},
  pages     = {415--452},
  year      = {1994}
}

@article{mackenzie1992double,
  author    = {Mackenzie, Kirill C. H.},
  title     = {Double Lie algebroids and second-order geometry, I},
  journal   = {Advances in Mathematics},
  volume    = {94},
  number    = {2},
  pages     = {180--239},
  year      = {1992}
}

@article{pradines1977,
  author    = {Pradines, Jean},
  title     = {Fibr\'es vectoriels doubles et calcul des jets non holonomes},
  journal   = {Esquisses Math\'ematiques},
  volume    = {29},
  publisher = {Universit\'e d'Amiens},
  year      = {1977}
}

@article{graciasaz-mehta2017,
  author    = {Gracia-Saz, Alfonso and Mehta, Rajan Amit},
  title     = {VB-groupoids and representation theory of Lie groupoids},
  journal   = {Journal of Symplectic Geometry},
  volume    = {15},
  number    = {3},
  pages     = {741--783},
  year      = {2017}
}

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Double structures — a double Lie groupoid is a groupoid object in Lie groupoids, controlled by the double source condition; its VB-groupoid special case is a vector bundle in groupoids with core a Lie algebroid; iterating the Lie functor in both directions gives a commutative cube ending at a double Lie algebroid, and the tangent and cotangent doubles $T\mathcal{G}\rightrightarrows TM$, $T^{\ast }\mathcal{G}\rightrightarrows A^{\ast }$ with dual cores $A(\mathcal{G})$, $T^{\ast }M$ realise the duality $T^{\ast }A\cong T^{\ast }{A}^{\ast }$ that carries the Mackenzie-Xu Lie-bialgebroid pairing 03.04.24.