aa r X i v : . [ m a t h - ph ] A p r MATCHED PAIRS OF DISCRETE DYNAMICAL SYSTEMS
OĞUL ESEN AND SERKAN SÜTLÜAbstract.
Matched pairs of Lie groupoids and Lie algebroids are studied. Discrete Euler-Lagrangeequations are written for the matched pairs of Lie groupoids. As such, a geometric framework toanalyse a discrete system by decomposing it into two mutually interacting subsystems is established.Two examples are provided to illustrate this strategy; the discrete dynamics on the trivial groupoid,and the discrete dynamics on the special linear group.
Key words:
Discrete dynamics, Lie groupoids, matched pairs.
MSC2010: Introduction
Consider two dynamical/mechanical systems, and their equations of motion presented eitherin Lagrangian or in Hamiltonian framework. Let also the systems be mutually interacting, so thatthey cannot preserve their individual motions. As a result, the equation of motion of the coupledsystem demands more effort than merely putting together the equations of motions of the individualsystems. Such systems have been first studied within the semidirect product theory [6, 33, 36],where only one of the systems is allowed to act on the other. Many physical systems fit into thisgeometry; such as the heavy top [44], and the Maxwell-Vlasov equations [15].The present paper is a part of the project which investigates a purely geometric framework(via a purely algebraic strategy) for decoupling the (discrete) dynamical equations of a system.More precisely, we strive to realize the dynamical equations of a system by means of the dynamicalequations of two simpler systems, together with the additional terms that emerge from the mutualinteractions of these, [12, 13]. This theory of “matched pair dynamics” may also be regarded as ageneralization of the semi-direct product theory, which, in particular, corresponds to the case thatone of the mutual representations being trivial.The fundamental geometric object in the matched pair dynamics is that of a Lie group,whereas the algebraic strategy we follow is nothing but the matched pair theory of [48, 27]. A pairof Lie groups that act on each other, subject to compatibility conditions ensuring a group structureon their cartesian product, is called a “matched pair of Lie groups”. We shall then call the totalspace (the cartesian product) “the matched pair Lie group” in order to emphasize the “matching”,while it is also referred as a bicrossedproduct group in [29, 28], the twilled extension in [20], thedouble Lie group in [23], or the Zappa-Szép product in [4]; see also [45, 46, 47, 51] . Conversely,from the decomposition point of view, if a Lie group is isomorphic (as topological sets) to thecartesian product of two of its subgroups (with trivial intersection), then it is a matched pair Liegroup. In this case, the mutual actions of the subgroups are derived from the group multiplication.
Motivated by the fact that the tangent space of a Lie group has the structure of a Lie group(called the “tangent group”), and that the tangent group of a matched pair Lie group is a matchedpair tangent group, we successfully studied in [13] the equations of motion of systems whoseconfiguration spaces being matched pair tangent groups within the Lagrangian framework. Wethus obtained the matched Euler-Lagrange equations, and the matched Euler-Poincaré equations.It was expected then, to carry out the similar discussions along the cotangent bundles andthe “cotangent groups” of Lie groups. This allowed us to apply in [12] the algebraic machinery (ofmatched pairs) to the Hamiltonian formalism of the equations of motion. The matched Hamilton’sequations, and the matched Lie-Poisson equations were obtained this way. The “matched pairHamiltonian dynamics” found an application even in the field of fluid dynamics, [11].On the other hand, the transition from the Kleinian conception of geometry to the Ehres-mannian point of view led to the evolution of Lie groups into Lie groupoids, [41, 43], which maybe considered as the central objects in “discrete dynamics”, [37, 32]. In this geometry, a discretesystem is generated by a Lagrangian on a Lie groupoid, and the dynamical equations are obtainedby the directional derivatives of the Lagrangian with left and right invariant vector fields at a finitesequence of “composable” elements.The theory of discrete dynamics on the Lie groupoid framework has been studied extensivelyin the literature. We refer the reader to [16, 31] for the discrete dynamics involving constraints,and to [49] for the field theoretic approach. In particular, there is an even richer theory of discretedynamics on Lie groups. We cite [3] for the discrete time Lagrangian mechanics on Lie groups,[34] for the discrete Lie-Poisson and the discrete Euler-Poincaré equations, and [35, 18] for thereduction of discrete systems under symmetry. For the discrete Hamiltonian dynamics in the realmof variational integrators we refer to [21], and [8, 7] for the higher order discrete dynamics. Thelocal description of discrete mechanics can be found in [30], whereas the inverse problem of thecalculus of variations in [1]. An implicit formulation of the discrete dynamics has been introducedin [17].In much the same way the Lie groups are matched, the Lie groupoids (as well as their Liealgebroids) can be matched. As a result, the very same algebraic strategy of “matched pairs” maybe used once more; this time to study the discrete dynamical systems. What we achieve then, in thepresent paper, is to represent the equations of motion of such a system in terms of the equations ofmotion of two simpler systems, decorated by the additional terms reflecting the mutual interactionsof the subsystems.The paper is organized in four sections. In the following two sections, Section 2 and Section3, we recall what we shall use on Lie groupoids, Lie algebroids, and their matched pair theory.It is Section 4 where the novelty of the paper lies. The discerete Euler-Lagrange equations arerecalled in Subsection 4.1, and then revisited both in Subsection 4.2 for matched pairs of Liegroupoids, and in Subsection 4.3 for the Lie groups; along the way towards the discrete dynamicson the matched pairs of Lie groups. Section 5 is reserved for concrete examples. More precisely,in Subsection 5.1 we present the (discrete) Euler-Lagrange equations explicitly for the trivialgroupoid, decomposing it into the action groupoid and the coarse groupoid. As such, we realisethe (discrete) Euler-Lagrange equations of the trivial groupoid in terms of the (discrete) Euler-Lagrange equations of the action groupoid and the coarse groupoid, glued together via the termsthat emerge from the mutual actions of those. Finally, in Subsection 5.2 we illustrate the (discrete)
ATCHED PAIRS OF DISCRETE DYNAMICAL SYSTEMS 3
Euler-Lagrange equations on the Lie group SL ( , C ) , the matched pair decomposition of whichincarnates as the Iwasawa decomposition.
2. Lie groupoids and Lie algebroids
In order to fix the notation, as well as the convenience of the reader, we devote the presentsection to a brief summary of the basics of Lie groupoids and Lie algebroids. The reader mayconsult to [24, 26, 39] for further details.2.1.
Lie groupoids and their actions.
Definition and basic examples.
Let G and B be two manifolds, and let there be two surjective submersions G α / / β / / B , called the “source map” and the “target map”, respectively. We assume also that, there exists asmooth map ε : B −→ G , b e b , called the “object inclusion”. The product space G ∗ G : = (cid:8) ( g , g ′ ) ∈ G × G | β ( g ) = α ( g ′ ) (cid:9) is called the “space of composable elements”, and is equipped with the partial multiplication G ∗ G −→ G , ( g , g ′ ) 7→ gg ′ . The five-tuple (G , B , α, β, ǫ ) with a partial multiplication is called a “Lie groupoid” if(i) α ( gg ′ ) = α ( g ) , and β ( gg ′ ) = β ( g ′ ) ,(ii) g ( g ′ g ′′ ) = ( gg ′ ) g ′′ ,(iii) α ( e b ) = β ( e b ) = b ,(iv) g g β ( g ) = g = g α ( g ) g ,(v) there is g − ∈ G such that α ( g − ) = β ( g ) and β ( g − ) = α ( g ) , and that g − g = g β ( g ) , gg − = g α ( g ) , for any ( g , g ′ ) , ( g ′ , g ′′ ) ∈ G ∗ G , any b ∈ B , and any g ∈ G .The elements of B are called “objects”, whereas the elements of G are referred as “arrows”,or “morphisms” . A groupoid may also be considered as a category such that all arrows areinvertible. We shall denote a Lie groupoid by G ⇒ B , or simply by G when there is no confusionon the base. Let us, now, recall the examples of Lie groupoids that we shall need in the sequel. Example 2.1.
Any Lie group G gives rise to a Lie groupoid over the identity element { e } ; thesource map and the target map being the constant maps α = β : G → { e } , and the object inclusionmap being the obvious inclusion e e = e . Moreover, the partial multiplication of this groupoid is thegroup multiplication. We denote this Lie groupoid by G ⇒ { e } , or simply by G . OĞUL ESEN AND SERKAN SÜTLÜ
Example 2.2.
Let G be a Lie group, and M a manifold with a smooth G -action M × G → M fromthe right. Then M × G ⇒ M has the structure of a Lie groupoid over M equipped with the sourcemap, the target map, and the object inclusion given by α : M × G −→ M , α ( m , g ) : = m ,β : M × G −→ M , β ( m , g ) : = m g ,ε : M −→ M × G , ε ( m ) : = ( m , e ) . The partial multiplication is given by(2.1) ( m , g ) · ( m ′ , g ′ ) : = ( m , gg ′ ) , if m g = m ′ . The groupoid M × G is called as the “action groupoid”. Example 2.3.
Let M be a manifold. Then the cartesian product M × M of M with itself is agroupoid over M via the source, target, and the object inclusion maps given by α : M × M −→ M , α ( m , m ′ ) : = m ,β : M × M −→ M , β ( m , m ′ ) : = m ′ ,ε : M −→ M × M , ε ( m ) : = ( m , m ) , whereas the partial multiplication is ( m , m ′ ) · ( n , n ′ ) : = ( m , n ′ ) , if m ′ = n . This groupoid M × M ⇒ M is called the “coarse groupoid”, the “pair groupoid”, or the “banalgroupoid”. Example 2.4.
Given a manifold M , and a Lie group G , the triple product M × G × M is a Liegroupoid over M by the source, target, and the object inclusion maps α : M × G × M −→ M , α ( m , g , m ′ ) : = m ,β : M × G × M −→ M , β ( m , g , m ′ ) : = m ′ ,ε : M −→ M × G × M , ε ( m ) = e m : = ( m , e , m ) , and the partial multiplication ( m , g , m ′ ) · ( n , g ′ , n ′ ) : = ( m , gg ′ , n ′ ) , if m ′ = n . The groupoid M × G × M ⇒ M is called the “trivial groupoid”.2.1.2. Left and right invariant vector fields.
Given a Lie groupoid G ⇒ B , a vector field Z ∈ Γ ( T G) is called “left invariant” if Z ( gg ′ ) = T g ′ ℓ g Z ( g ′ ) , for any ( g , g ′ ) ∈ G ∗ G , where ℓ g : G → G is the left translation induced from the partialmultiplication, and T g ′ ℓ g : T g ′ G → T gg ′ G is the tangent lift of this mapping at g ′ ∈ G . Similarly, a“right invariant” vector field Z ∈ Γ ( T G) is one that satisfies Z ( gg ′ ) = T g r g ′ Z ( g ) for any ( g , g ′ ) ∈ G ∗ G , where r g ′ : G → G is the right translation, and T g r g ′ : T g G → T gg ′ G is itsthe tangent lift at g ∈ G . ATCHED PAIRS OF DISCRETE DYNAMICAL SYSTEMS 5
Morphisms of Lie groupoids.
Let G an H be two Lie groupoids over the bases B and C , respectively. A morphism of Liegroupoids is a pair of smooth maps Φ : G −→ H and Φ : B −→ C compatible with the groupoidmultiplications, source, target, and the object inclusion maps. More precisely, a Lie groupoidmorphism is a pair ( Φ , Φ ) that satisfies(i) ( Φ ( g ) , Φ ( g ′ )) ∈ H ∗ H ,(ii) Φ ( gg ′ ) = Φ ( g ) Φ ( g ′ ) ,(iii) α ( Φ ( g )) = Φ ( α ( g )) ,(iv) β ( Φ ( g )) = Φ ( β ( g )) ,(v) Φ ( e b ) = (cid:157) Φ ( b ) ,for any ( g , g ′ ) ∈ G ∗ G , and any b ∈ B . The requirements may be summarized by the commutativityof the diagram(2.2) G α (cid:15) (cid:15) β (cid:15) (cid:15) Φ / / H α (cid:15) (cid:15) β (cid:15) (cid:15) B ε Φ / / C ε f f for each source, target and inclusion map. In order to avoid the notation inflation, we shall notdistinguish the source maps of the Lie groupoids G and H with different notations. Instead, weshall make it clear from the context.2.1.4. Lie groupoid actions.
Let G be a Lie groupoid over the base B , and let f : P → B be a smooth map from a manifold P to the base manifold B . Given the product space P ∗ G : = (cid:8) ( p , g ) ∈ P × G | f ( p ) = α ( g ) (cid:9) , a smooth map ⊳ : P ∗ G −→ P , ( p , g ) 7→ p ⊳ g is called the (right) action of G on f if(i) f ( p ⊳ g ) = β ( g ) ,(ii) ( p ⊳ g ) ⊳ g ′ = p ⊳ ( gg ′ ) ,(iii) p ⊳ g f ( p ) = p , OĞUL ESEN AND SERKAN SÜTLÜ for any ( p , g ) ∈ P ∗ G , any ( g , g ′ ) ∈ G ∗ G , and any p ∈ P . The definition may be summarized bythe commutativity of the diagram(2.3) P : p f (cid:15) (cid:15) ⊳ g / / ⊳ gg ′ ) ) p ⊳ g f (cid:15) (cid:15) ⊳ g ′ / / ( p ⊳ g ) ⊳ g ′ f (cid:15) (cid:15) B : α ( g ) g / / gg ′ β ( g ) = α ( g ′ ) g ′ / / β ( g ′ ) The letters P and B refers to the manifolds in which the objects in the corresponding rows belong.The left action of a Lie groupoid on a smooth map is defined similarly, [24, 25]. Let H bea Lie groupoid over the base B , and let f : P B be a smooth function from a manifold P to B .Then, given the product space H ∗ P : = (cid:8) ( h , p ) ∈ H × P | β ( h ) = f ( p ) (cid:9) , the smooth mapping ⊲ : H ∗ P −→ P , ( h , p ) 7→ h ⊲ p is called a (left) action of H on f if(i) f ( h ⊲ p ) = α ( h ) ,(ii) h ′ ⊲ ( h ⊲ p ) = ( h ′ h ) ⊲ p ,(iii) g f ( p ) ⊲ p = p ,for any ( h , p ) ∈ H ∗ P , any ( h ′ , h ) ∈ H ∗ H , and any p ∈ P . In other words, the following diagramis commutative:(2.4) P : h ′ ⊲ ( h ⊲ p ) f (cid:15) (cid:15) h ⊲ p f (cid:15) (cid:15) h ′ ⊲ o o p f (cid:15) (cid:15) h ⊲ o o h ′ h ⊲ t t B : α ( h ′ ) h ′ / / h ′ h β ( h ′ ) = α ( h ) h / / β ( h ) We are now ready to conclude with the left and the right actions of a Lie groupoid on anotherLie groupoid. Let G be a Lie groupoid over the base B , and let H be another Lie groupoid over C . The left action of H on G is defined to be the (left) action of H on the source map α : G → B ,while the right action of G on H is similarly defined to be the (right) action of G on the targetmap β : H → C . For further details on the representations of Lie groupoids we refer the readerto [5, 25].2.2. Lie algebroids and their actions.
ATCHED PAIRS OF DISCRETE DYNAMICAL SYSTEMS 7
Definition of a Lie algebroid.
A Lie algebroid over a manifold M may be thought of a generalization of the tangent bundle T M of M , [24, 40, 42]. More technically, given a manifold M , a “Lie algebroid” A over thebase M is a (real) vector bundle τ : A → M , together with a map a : A →
T M of vectorbundles, called the “anchor map”, and a Lie bracket [• , •] (bilinear, anti-symmetric, satisfying theJacobi identity) on the space Γ (A) of sections, so that the induced C ∞ ( M ) -module homomorphism a : Γ (A) → Γ ( T M ) satisfies [ X , f Y ] = f [ X , Y ] + L a ( X ) ( f ) Y for any X , Y ∈ Γ (A) , and any f ∈ C ∞ ( M ) , where L a ( X ) ( f ) stands for the directional derivative of f ∈ C ∞ ( M ) in the direction of a ( X ) ∈ T M . It, then, follows that a ([ X , Y ]) = [ a ( X ) , a ( Y )] for any X , Y ∈ Γ (A) . Accordingly, a Lie algebroid is denoted by a quintuple (A , τ, M , a , [• , •]) , oroccasionally by a triple (A , τ, M ) when there is no confusion on the bracket, and the anchor map.Let us now take a quick tour on a bunch of critical examples. Example 2.5.
Any Lie algebra g is a Lie algebroid τ : g → {∗} ; taking the base manifold B = {∗} to be a one-point set, and the anchor map a : g → T B to be the zero map.
Example 2.6.
Given any manifold M , the tangent bundle T M is a Lie algebroid over the base M ,where the anchor a : T M → T M is the identity map.
Example 2.7.
Let M be manifold admitting an infinitesimal left action of a Lie algebra g . Assuch, there exists a linear map g → Γ ( T M ) , ξ X ξ , preserving the Lie brackets. Then the trivialbundle M × g → M , via the projection onto the first component, can be made into a Lie algebroidvia the (fiber preserving) anchor map a : M × g → T M , a ( m , ξ ) : = X ξ ( m ) , and the bracket given by [ u , v ]( m ) = [ u ( m ) , v ( m )] g + (L X u ( m ) v )( m ) − (L X v ( m ) u )( m ) regarding the sections of the trivial bundle M × g as (smooth) maps u , v : M → g , referring theLie derivative of a Lie algebra valued function w : M → g along the vector field X ∈ Γ ( T M ) by L X ( w ) , and denoting the Lie bracket on g as [• , •] g . This Lie algebroid is called the “action Liealgebroid” (or the “transformation Lie algebroid”), [14, 19, 22, 40].2.2.2. Lie algebroid of a Lie groupoid.
We now recall from [10, 26, 42] that how a Lie algebroid associates to a given Lie groupoidin a canonical way. Let G be a Lie groupoid over B , α : G → B be the source map, and T g α : T g G → T α ( g ) B be its tangent lift at g ∈ G . Along the lines of [32], the “Lie algebroidassociated to the Lie groupoid” G is defined to be the vector bundle (AG , τ, B ) whose fibers aregiven by A b G : = ker T ε ( b ) α, In other words, AG corresponds to the vertical bundle on G with respect to the fibration α : G → B .We shall denote a typical section of the fibration τ : AG → B by X ∈ Γ (AG) . Moreover, theanchor map a : AG →
T B is given by(2.5) a ( X ( b )) = T e b β ◦ X ( b ) OĞUL ESEN AND SERKAN SÜTLÜ where T e b β : T e b G → T b B is the tangent lift of the target map β : G → B at e b = ε ( b ) ∈ G . Finally,the definition of the Lie bracket [• , •] AG on the sections of AG → B follows from the directanalogy with that of the Lie groups. More precisely, the bracket on the (associated) Lie algebroidis defined by means of the Jacobi-Lie bracket of the left (or the right) invariant vector fields on thegroupoid; [9, 24, 32].For a Lie groupoid G ⇒ B , there exists an isomorphism of C ∞ (G) -modules between thesections Γ (AG) of the Lie algebroid of the Lie groupoid and the left-invariant (resp. the rightinvariant) vector fields on G , [26]. To any X ∈ Γ (AG) , there corresponds a left invariant vectorfield ←− X ∈ Γ ( T G) via(2.6) ←− X ( g ) : = T g β ( g ) ℓ g X ( β ( g )) . As such, a left invariant vector field satisfies ←− X ( g g ) = T g ℓ g ←− X ( g ) , for any (composable) g , g ∈ G . Conversely, if ←− X ∈ Γ ( T G) is a left invariant vector field, then X ( b ) : = ←− X ( e b ) defines a section X ∈ Γ (AG) . This identification of the sections of the Lie algebroid AG with theleft invariant vector fields of G allows us to define (induce) the Lie bracket on the sections. Given X , Y ∈ Γ (AG) , we define their Lie bracket as(2.7) [ X , Y ] AG ( b ) : = [←− X , ←− Y ]( e b ) where the bracket on the right hand side is the Jacobi-Lie bracket of left invariant vector fields onthe Lie algebroid G .As for the right invariant vector fields, given any X ∈ Γ (AG) , a right invariant vector field −→ X ∈ Γ ( T G) is defined through(2.8) −→ X ( g ) : = − T g α ( g ) r g ◦ T g α ( g ) inv ( X ( α ( g )) , where inv : G → G stands for the inversion, and T g α ( g ) r g : T g α ( g ) G → T g G is the tangent lift of theright translation by g ∈ G at ε ( α ( g )) = g α ( g ) ∈ G .The right invariant vector field −→ X ∈ Γ ( T G) corresponding to X ∈ Γ (AG) is equivalentlygiven by(2.9) −→ X ( g ) = − ddt (cid:12)(cid:12)(cid:12)(cid:12) t = x − ( t ) g , where x ( t ) is any curve through G with constant source, that is α ( x ( t )) = α ( g ) ∈ B , and is tangentto X ( α ( g )) ∈ A α ( g ) G ⊆ T g α ( g ) G , that is Û x ( ) = X ( α ( g )) .It follows at once from the definition that −→ X ( g g ) = T g r g −→ X ( g ) for any (composable) g , g ∈ G . In this case, the relation between the Lie brackets is given by −−−−→[ X , Y ] AG = −[−→ X , −→ Y ] , ATCHED PAIRS OF DISCRETE DYNAMICAL SYSTEMS 9 where the bracket on the left side is the one on the Lie algebroid level, whereas the bracket on theright hand side is the Jacobi-Lie bracket of vector fields on the manifold G .Let us next present examples on the construction of the left and the right invariant vectorfields on the Lie groupoids in Example 2.1 - 2.4 which will be needed in the sequel. Example 2.8.
Let G be a Lie group, and G ⇒ { e } be the Lie groupoid in Example 2.1. Since theinclusion map is given by ε ( e ) = e e = e , the tangent space at e e ∈ G is T e G = g , and the source map α : G → { e } is constant, the kernel of the tangent lift of the source map (the total space of theassociated Lie algebroid A G ) may be identified with the Lie algebra g of the group G . The rightand the left invariant vector fields associated to ξ ∈ g are then given by −→ ξ : G −→ T G , g T e r g ( ξ ) , (2.10) ←− ξ : G −→ T G , g T e ℓ g ( ξ ) . (2.11) Example 2.9.
The Lie algebroid of the coarse groupoid M × M ⇒ M of Example 2.3 is the Liealgebroid ( T M , τ M , M ) of Example 2.6. Indeed, consider a curve ( m , n t ) ∈ M × M with constantsource, so that n = n ∈ M and Û n = X ∈ T n M . Then its derivative (at t =
0) yields a vector ( θ m , X ) ∈ T m M × T n M . Given any section ( θ m , X ) ∈ A m ( M × M ) , the corresponding right and leftinvariant vector fields on M × M are computed to be −−−−−→( θ m , X ) : M × M −→ T ( M × M ) = T M × T M , ( m , n ) 7→ (− X , θ m ) , (2.12) ←−−−−−( θ m , X ) : M × M −→ T ( M × M ) = T M × T M , ( m , n ) 7→ ( θ m , X ) . (2.13) Example 2.10.
The Lie algebroid of the action groupoid M × G ⇒ M of Example 2.2 is thetransformation Lie algebroid ( M × g , pr , M ) of Example 2.7. Indeed, the derivative of a curve ( m , g t ) ∈ M × G of constant source, such that g = e and that Û g = ξ ∈ g , yields a vector ( θ m , ξ ) ∈ T m M × T e G . As such, the total space A( M × G ) may be identified with the cartesianproduct M × g . For any ( θ m , ξ ) ∈ A m ( M × G ) , the left invariant and the right invariant vector fieldsare given by −−−−−→( θ m , ξ ) ( m , g ) = (cid:16) − ξ † ( m ) , −→ ξ ( g ) (cid:17) , ←−−−−−( θ m , ξ ) ( m , g ) = (cid:16) θ m , ←− ξ ( g ) (cid:17) , where the infinitesimal generator of the right action is computed to be(2.14) ξ † ( m ) : = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = me t for any curve e t ∈ G with e = e ∈ G , and Û e = ξ ∈ g . Example 2.11.
The Lie algebroid of the trivial groupoid of Example 2.4 is the Lie algebroid ( M × g ) ⊕ T M . Indeed, given a curve ( m t , e t , n t ) ∈ M × G × M with e = e ∈ G , m = n = m ∈ M , Û e = ξ ∈ g , Û m = X ∈ T m M , and Û n = Y ∈ T m M , T ( m , e , m ) α ( X , ξ, Y ) = X . As such, A m ( M × G × M ) = (cid:8) ( θ m , ξ, Y ) | ξ ∈ g , Y ∈ T m M (cid:9) which yields A( M × G × M ) (cid:27) ( M × g ) ⊕ T M . In order to compute the left (resp. right) invariantvector fields, let ( m , g , n ) ∈ M × G × M , and let ( θ n , ξ, X ) ∈ A n ( M × G × M ) . Accordingly, let ( n , e t , n t ) ∈ M × G × M be a curve such that e = n , n = n , Û e = ξ , and Û n = Y ∈ T n M . We thensee that(2.15) ←−−−−−−−( θ n , ξ, Y )( m , g , n ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( m , g , n )( n , e t , n t ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( m , g e t , n t ) = ( θ m , ←− ξ ( g ) , Y ) . Similarly, given ( θ m , ξ, X ) , if ( m , e t , m t ) ∈ M × G × M be a curve such that e = e , m = m , Û e = ξ ,and Û m = X ∈ T m M , then −−−−−−−→( θ m , ξ, X )( m , g , n ) = − ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( m , e t , m t ) − ( m , g , n ) = − ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( m t , e − t , m )( m , g , n ) = − ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( m t , e − t g , n ) = (− X , −→ ξ ( g ) , θ n ) . Morphisms of Lie algebroids.
Given two Lie algebroids (A , τ, M ) and (A ′ , τ ′ , M ′ ) , a morphism from (A , τ, M ) to (A ′ , τ ′ , M ) is a vector bundle morphism preserving the achors as well as the brackets. That is, a pair ( φ : A → A ′ ; φ : M → M ′ ) such that A φ / / τ (cid:15) (cid:15) A ′ τ ′ (cid:15) (cid:15) M φ / / M ′ , that a ′ ◦ φ = T φ ◦ a , where a : A →
T M and a ′ : A ′ → T M ′ are the respective anchor maps, and finally that φ ([ X , Y ]) = [ φ ( X ) , φ ( Y )] . It is possible to derive a Lie algebroid morphism starting from a Lie groupoid morphism asfollows. Given two Lie groupoids G ⇒ B and H ⇒ C . Let Φ : G → H ; Φ : M → N be a morphism of Lie groupoids. Then, for A m G ∋ X = ddt (cid:12)(cid:12) t = x t , where α ( x t ) = m , A m Φ : A m G → A Φ ( m ) H ; X ddt (cid:12)(cid:12)(cid:12)(cid:12) t = Φ ( x t ) defines a morphism AG → AH of Lie algebroids associated with the Lie groupoids G and H ,respectively. We refer the reader to [24, Sect. 3.5] for further details.
3. Matched pairs of Lie groupoids and matched pairs of Lie algebroids
Matched pairs of Lie groupoids.
Definition of a matched Lie groupoid.
In this subsection we recall, mainly from [40, 25], the groupoid level of the matched pair theoryof [28, 29]. Let G ⇒ B and H ⇒ B be two Lie groupoids over the same base B , and let H act on G from the left by(3.1) ⊲ : H ∗ G −→ G , ( h , g ′ ) 7→ h ⊲ g ′ , ATCHED PAIRS OF DISCRETE DYNAMICAL SYSTEMS 11 where we recall that the set
H ∗ G of composable elements consists of the pairs ( h , g ) ∈ H × G such that β ( h ) = α ( g ) . Being a left action, (3.1) satisfies(i) α ( h ) = α ( h ⊲ g ′ ) ,(ii) ( h ′ h ) ⊲ g ′ = h ′ ⊲ ( h ⊲ g ′ ) (iii) g α ( h ) ⊲ g ′ = g ′ for any h ∈ H for any ( h , g ′ ) ∈ H ∗ G , any ( h ′ , h ) ∈ H ∗ H , and any h ∈ H . Let also G act on H from the rightby(3.2) ⊳ : H ∗ G −→ H , ( h , g ′ ) 7→ h ⊳ g ′ . Then, being a right action, (3.2) satisfies(iv) β ( g ′ ) = β ( h ⊳ g ′ ) ,(v) h ⊳ g ′ g = ( h ⊳ g ′ ) ⊳ g ,(vi) h ⊳ (cid:157) β ( g ′ ) = h ,for any ( h , g ′ ) ∈ H ∗ G , any ( g ′ , g ) ∈ G ∗ G , and any g ′ ∈ G . Now, the pair (G , H ) is called a“matched pair of Lie groupoids” if, in addition, the compatibilities(vii) β ( h ⊲ g ′ ) = α ( h ⊳ g ′ ) ,(viii) h ⊲ ( g ′ g ) = ( h ⊲ g ′ )(( h ⊳ g ′ ) ⊲ g ) ,(ix) ( h ′ h ) ⊳ g ′ = ( h ′ ⊳ ( h ⊲ g ′ ))( h ⊳ g ′ ) ,are also satisfied for any ( h , g ′ ) ∈ H ∗ G , any ( g ′ , g ) ∈ G ∗ G , and any ( h ′ , h ) ∈ H ∗ H .Then, the product space G ⊲⊳ H : = G ∗ H = (cid:8) ( g , h ) ∈ G × H | β ( g ) = α ( h ) (cid:9) becomes a Lie groupoid by the partial multiplication, on (G ⊲⊳ H ) ∗ (G ⊲⊳ H ) : = (cid:8) (( g , h ) , ( g ′ , h ′ )) ∈ (G ⊲⊳ H ) × (G ⊲⊳ H ) | β ( h ) = α ( g ′ ) (cid:9) , given by(3.3) (G ⊲⊳ H ) ∗ (G ⊲⊳ H ) −→ (G ⊲⊳ H ) , (( g , h ) , ( g ′ , h ′ )) 7→ ( g ( h ⊲ g ′ ) , ( h ⊳ g ′ ) h ′ ) . The source and target maps of the “matched pair Lie groupoid” G ⊲⊳ H are given by α : G ⊲⊳ H −→ B , ( g , h ) 7→ α ( g ) ,β : G ⊲⊳ H −→ B , ( g , h ) 7→ β ( h ) , respectively. The object inclusion map of the matched pair Lie groupoid is defined in terms ofthose on G and H as ε : B −→ G ⊲⊳ H , b
7→ ( e b , e b ) . The relation between the matched pair Lie groupoid G ⊲⊳ H and the individual Lie groupoids G and H is given in [25, Thm. 2.10] that we record below. Proposition 3.1.
A pair (G , H ) of groupoids is a matched pair of groupoids if and only if themanifold
G ∗ H has the structure of a Lie groupoid, such that (i) the maps
G → G ∗ H given by g
7→ ( g , g β ( g )) , and H → G ∗ H given by h
7→ ( g α ( h ) , h ) are morphisms of Lie groupoids, and (ii) the multiplication (( g , g β ( g )) , ( g α ( h ) , h )) 7→ ( g , h ) ∈ G ∗ H is a diffeomorphism. Picturing an element ( g , h ) ∈ G ⊲⊳ H by(3.4) β ( h ) α ( g ) g / / β ( g ) = α ( h ) , h O O the partial multiplication (3.3) on the matched pair groupoid may be illustrated as(3.5) β ( h ′ ) β ( h ) = α ( g ′ ) g ′ / / t ( g ′ ) = β ( h ⊳ g ′ ) h ′ O O α ( g ) g / / β ( g ) = α ( h ) = α ( h ⊲ g ′ ) h O O h ⊲ g ′ / / β ( h ⊲ g ′ ) = α ( h ⊳ g ′ ) . h ⊳ g ′ O O The following remark concerns the actions on the identity elements.
Remark 3.2.
Let (G , H ) be a matched pair of Lie groupoids. Given g ∈ G with α ( g ) = b and β ( g ) = b , and h ∈ H with α ( h ) = b and β ( h ) = b , we see at once that(3.6) h ⊲ e b = e b ∈ G , e b ⊳ g = e b ∈ H , and that,(3.7) e b ⊲ g = g , h ⊳ e b = h . Matched pair decomposition of the trivial groupoid.
Given the action groupoid G = M × G of Example 2.2, and the coarse groupoid H = M × M ofExample 2.3, let us consider the set H ∗ G = ( M × M ) ∗ ( M × G ) = (cid:8) (( m ′ , m ; m , g )) ∈ ( M × M ) × ( M × G ) (cid:9) of composable elements. The left action(3.8) ⊲ : ( M × M ) ∗ ( M × G ) −→ ( M × G ) , ( m ′ , m ) ⊲ ( m , g ) : = ( m ′ , g ) of the action groupoid G = M × G on the coarse groupoid H = M × M , and the right action(3.9) ⊳ : ( M × M ) ∗ ( M × G ) −→ ( M × M ) , ( m ′ , m ) ⊳ ( m , g ) : = ( m ′ g , m g ) ATCHED PAIRS OF DISCRETE DYNAMICAL SYSTEMS 13 of the action groupoid G = M × G on the coarse groupoid H = M × M satisfies the conditions(i)-(ix) of the previous subsection. Thus, the set(3.10) G ∗ H = ( M × G ) ∗ ( M × M ) = (cid:8) ( m , g ; m g , m ′ ) ∈ ( M × G ) × ( M × M ) (cid:9) of composable elements form a Lie groupoid G ⊲⊳ H = ( M × G ) ⊲⊳ ( M × M ) over the basemanifold M . The source, target and object inclusion maps are computed to be α : ( M × G ) ⊲⊳ ( M × M ) −→ M , ( m , g ; m g , m ′ ) 7→ m , (3.11) β : ( M × G ) ⊲⊳ ( M × M ) −→ M , ( m , g ; m g , m ′ ) 7→ m ′ , (3.12) ε : M −→ ( M × G ) ⊲⊳ ( M × M ) , m
7→ ( m , e ; m , m ) . (3.13)In order to proceed to the partial multiplication, we consider the product space (( M × G ) ⊲⊳ ( M × M )) ∗ (( M × G ) ⊲⊳ ( M × M )) : = (cid:8) ( m , g ; m g , m ′ ) , ( m ′ , h ; m ′ h , n ) : m ′ , m , n ∈ M and g , h ∈ G (cid:9) . The partial multiplication, given by (3.3), then appears as ( m , g ; m g , m ′ ) ∗ ( m ′ , h ; m ′ h , n ) = (( m , g ) (( m g , m ′ ) ⊲ ( m ′ , h )) ; (( m g , m ′ ) ⊳ ( m ′ , h )) ( m ′ h , n )) = (( m , g )( m g , h ) ; ( m g h , m ′ h )( m ′ h , n )) = ( m , g h ; m g h , n ) . Accordingly, the inversion is computed to be ( m , g ; m g , n ) − = ( n , g − ; n g − , m ) . The matched pair Lie groupoid ( M × G ) ⊲⊳ ( M × M ) is identified with the trivial Lie groupoid M × G × M via(3.14) Φ : M × G × M −→ ( M × G ) ⊲⊳ ( M × M ) , ( m , g , n ) 7→ ( m , g ; m g , n ) , see for instance, [40].Let us finally note that the map (3.14) that gives the matched pair decomposition of the trivialgroupoid is differentiated to a Lie algebroid morphism A m Φ : A m ( M × G × M ) −→ A m (( M × G ) ⊲⊳ ( M × M )) , ( θ m , ξ, Y ) 7→ ( θ m , ξ ; ξ † ( m ) , Y ) . (3.15)Indeed, for a curve ( m , e t , m t ) ∈ M × G × M with e = e , m = m , Û e = ξ ∈ g , and Û m = Y ∈ T m M ,recalling (2.14) we compute (A m Φ )( θ m , ξ, Y ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = Φ ( m , e t , m t ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( m , e t ; me t , m t ) = ( θ m , ξ ; ξ † ( m ) , Y ) . Matched pairs of Lie algebroids.
Let us begin with a brief discussion on the representation of a Lie algebroid on a vector bundlefrom [14, 40, 22].Let (A , τ, M , a , [• , •]) be a Lie algebroid, and let ( E , π, M ) be a vector bundle over the samebase manifold M . A left representation of (A , τ, M ) to ( E , π, M ) is a bilinear map ρ : Γ (A) × Γ ( E ) −→ Γ ( E ) , ( X , s ) 7→ ρ X ( s ) = ρ ( X , s ) such that (i) ρ f X ( Y ) = f ρ X ( Y ) ,(ii) ρ X ( f Y ) = f ρ X ( Y ) + ( a ( X ) f ) Y ,(iii) ρ [ X , e X ] ( Y ) = ρ X ( ρ e X ( Y )) − ρ e X ( ρ X ( Y )) ,for any X , e X ∈ Γ (A) , any Y ∈ Γ ( E ) , and any f ∈ C ∞ ( M ) . A right representation of a Lie algebroidon a vector bundle is defined similarly.Two Lie algebroids with the same base manifold form a matched pair of Lie algebroids ifthe direct sum of the total spaces of the Lie algebroids has a Lie algebroid structure on the samebase such that the individual Lie algebroids are Lie subalgebroids of the direct sum, [40]. Moreprecisely, let (A , τ, M , a , [• , •]) and (B , κ, M , b , [• , •]) be two Lie algebroids over the same base M with mutual representations ρ : Γ (B) × Γ (A) → Γ (A) , ρ ′ : Γ (A) × Γ (B) → Γ (B) , satisfying(i) ρ Y [ X , e X ] = [ ρ Y ( X ) , e X ] + [ X , ρ Y ( e X )] − ρ ρ ′ X ( Y ) ( e X ) + ρ ρ ′ e X ( Y ) ( X ) ,(ii) ρ ′ X [ Y , e Y ] = [ ρ X ( Y ) , e Y ] + [ Y , ρ X ( e Y )] − ρ ′ ρ Y ( X ) ( e X ) + ρ ′ ρ e Y ( X ) ( Y ) ,(iii) [ b ( Y ) , a ( X )] = a ( ρ Y ( X )) − b ( ρ ′ X ( Y )) ,for any X , e X ∈ Γ (A) , and any Y , e Y ∈ Γ (B) . Then, the direct sum vector bundle A ⊲⊳ B : = A ⊕ B has the structure of a Lie algebroid by the bracket given by(3.16) [ Y , X ] = ρ ( Y , X ) − ρ ′ ( X , Y ) for any X ∈ Γ (A) , and any Y ∈ Γ (B) . The pair (A , B) of Lie algebroids is called a “matched pairof Lie algebroids”, whereas the vector bundle A ⊲⊳ B is called a “matched pair Lie algebroid”.3.3. Lie algebroid actions induced from Lie groupoid actions.
We devote the present subsection on the infinitesimal versions of the mutual actions of a matchedpair (G , H ) of Lie groupoids, over a base manifold B .To this end, let h ∈ H , and let x t ∈ G be a curve so that β H ( h ) = α G ( x t ) = b ∈ B . Sincethe curve has a constant source, its derivative at t = X ∈ A b G . Now, using the left action, we define the curve x t : = h ⊲ x t ∈ G , whosesource is constant as well; α ( h ⊲ x ( t )) = α ( h ) = c ∈ B . As such, the time derivative (at t = h ⊲ X : = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( h ⊲ x t ) ∈ A c G . On the other hand, the right action allows us to define the curve h ⊳ x t ∈ H which passes through h ∈ H at t = h ⊳ x t ∈ H is not necessarily constant,and that its time derivative(3.18) X † ( h ) = h ⊳ X : = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( h ⊳ x t ) ∈ T h H ATCHED PAIRS OF DISCRETE DYNAMICAL SYSTEMS 15 may not be in A c H .Let, next, y t ∈ H be a curve with constant source, say α H ( y t ) = c = α G ( g ) ∈ B for some g ∈ G , and thus generating a Lie algebroid element Y ∈ A c H . Then the left infinitesimal actionof AH on G is defined as(3.19) Y † ( g ) : = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = y − t ⊲ g ∈ T g G . Similarly, the right action of G on H can be lifted to the right action of G on AH as follows.Starting with the curve y t ∈ H , we define the curve ( y − t ⊳ g ) − ∈ H . The source of the lattercurve is constant, hence, at t = A c H , which is given by(3.20) Y ⊳ g : = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( y − t ⊳ g ) − ∈ A c H . Lie algebroid of a matched pair Lie groupoid.
In Subsection 2.2.2, we have seen how a Lie algebroid is associated to a Lie groupoid. In thepresent subsection, we shall revisit this construction for a matched Lie groupoid, say G ⊲⊳ H toarrive at the Lie algebroid A(G ⊲⊳ H ) . We shall conclude with the explicit isomorphism between
A(G ⊲⊳ H ) and the matched pair Lie algebroid AG ⊲⊳ AH .To begin with, let G ⊲⊳ H be a matched pair of Lie groupoids, X ∈ A b G , and Y ∈ A b H ;that is, let there be curves x t ∈ G and y t ∈ H so that α ( x t ) = b = α ( y t ) , and that x = ε G ( b ) ∈ G with y = ε H ( b ) ∈ H . In view of Proposition 3.1, the (groupoid) embeddings G −→ G ⊲⊳ H , x t
7→ ( x t , ( ε H ◦ β )( x t )) (3.21) H −→ G ⊲⊳ H , y t
7→ ( ε G ( b ) , y t ) . (3.22)induce morphisms A b G −→ A b (G ⊲⊳ H ) , X
7→ ( X , T ( ε H ◦ β )( X )) (3.23) A b H −→ A b (G ⊲⊳ H ) , Y
7→ ( θ ε G ( b ) , Y ) (3.24)of Lie algebroids. Furthermore, we have the following proposition, see [40, Prop. 5.1]. Proposition 3.3.
Given two Lie groupoids G and H over the same base B , the map (3.25) A b G ⊕ A b H → A b (G ⊲⊳ H ) , ( X , Y ) 7→ ( X , T ( ε H ◦ β )( X ) + Y ) . is an isomorphism.Proof. Let ( x t , z t ) ∈ G ⊲⊳ H be a curve such that α ( x t ) = b ∈ B , and that β ( x t ) = α ( z t ) . Let also Û x = X ∈ A b G , and Û z = Z ∈ T ε H ( b ) H . Multiplying both sides of (( ε G ◦ β )( z t ) , z − t )( x − t , ε H ( b ))( x t , z t ) = (( ε G ◦ β )( z t ) , ( ε H ◦ β )( z t )) . by (( ε G ◦ α )( z t ) , z t ) ∈ G ⊲⊳ H , from the left, we obtain ( x − t , ε H ( b ))( x t , z t ) = (( ε G ◦ α )( z t ) , z t ) . Differentiating the latter equality, while keeping the product rule in mind, we arrive at ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( x − t , ε H ( b )) ! + ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( x t , z t ) ! = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = (( ε G ◦ α )( z t ) , z t ) ! , that is,(3.26) ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( x t , z t ) ! = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = (( ε G ◦ α )( z t ) , z t ) ! − ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( x − t , ε H ( b )) ! . On the other hand, differentiating ( x − t , ε H ( b ))( x t , ( ε H ◦ β )( x t )) = (( ε G ◦ β )( x t ) , ( ε H ◦ β )( x t )) , we obtain ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( x − t , ε H ( b )) ! + ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( x t , ( ε H ◦ β )( x t )) ! = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = (( ε G ◦ β )( x t ) , ( ε H ◦ β )( x t )) ! , that is,(3.27) ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( x − t , ε H ( b )) ! = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = (( ε G ◦ β )( x t ) , ( ε H ◦ β )( x t )) ! − ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( x t , ( ε H ◦ β )( x t )) ! . Now, (3.26) and (3.27) together imply ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( x t , z t ) ! = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( x t , ( ε H ◦ β )( x t )) ! + ddt (cid:12)(cid:12)(cid:12)(cid:12) t = (( ε G ◦ α )( z t ) , z t ) ! − ddt (cid:12)(cid:12)(cid:12)(cid:12) t = (( ε G ◦ β )( x t ) , ( ε H ◦ β )( x t )) ! = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( x t , ( ε H ◦ β )( x t )) ! + ddt (cid:12)(cid:12)(cid:12)(cid:12) t = (( ε G ◦ β )( x t ) , z t ) ! − ddt (cid:12)(cid:12)(cid:12)(cid:12) t = (( ε G ◦ β )( x t ) , ( ε H ◦ β )( x t )) ! , where, on the second equality we used the fact that β ( x t ) = α ( z t ) . Hence, we see at once that ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( x t , z t ) ! − ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( x t , ( ε H ◦ β )( x t )) ! = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = (( ε G ◦ β )( x t ) , z t ) ! − ddt (cid:12)(cid:12)(cid:12)(cid:12) t = (( ε G ◦ β )( x t ) , ( ε H ◦ β )( x t )) ! = ( θ ε G ( b ) , Z − T ( ε H ◦ β )( X )) ∈ A b (G ⊲⊳ H ) . As a result, Z − T ( ε H ◦ β )( X ) ∈ T ε H ( b ) H is the derivative, at t =
0, of a curve y t ∈ H with α ( y t ) = b ;that is, Z − T ( ε H ◦ β )( X ) ∈ A b H . In other words, the mapping A(G ⊲⊳ H ) −→ AG ⊕ AH , ( X , Z ) 7→ ( X , Z − T ( ε H ◦ β )( X )) is well-defined, and makes the inverse of (3.25). (cid:3) The next example illustrates the isomorphism(3.25) for the matched pair ( M × G , M × M ) of the action Lie groupoid of Example 2.2 and the coarse groupoid M × M of Example 2.3. Example 3.4.
Let ( θ m , ξ ; ξ † ( m ) , Y ) ∈ A m (( M × G ) ⊲⊳ ( M × M )) be a generic element obtained bythe differentiation at t = ( m , g t ; m g t , n t ) ∈ ( M × G ) ⊲⊳ ( M × M ) of constant source,with g = e ∈ G , n = m , Û g = ξ ∈ g and Û n = Y ∈ T m M . Similarly, the inclusion M × G ∋ ( m , g t ) 7→ ( m , g t ; ( ε M × M ◦ β )( m , g t )) = ( m , g t ; m g t , m g t )) ∈ ( M × G ) ⊲⊳ ( M × M ) ATCHED PAIRS OF DISCRETE DYNAMICAL SYSTEMS 17 yields A m ( M × G ) ∋ ( θ m , ξ ) 7→ ( θ m , ξ ; ξ † ( m ) , ξ † ( m )) ∈ A m (( M × G ) ⊲⊳ ( M × M )) , while M × M ∋ ( m , n t ) 7→ ( m , e ; m , n t )) ∈ ( M × G ) ⊲⊳ ( M × M ) leads to A m ( M × M ) ∋ ( θ m , X ) 7→ ( θ m , θ ; θ m , X ) ∈ A m (( M × G ) ⊲⊳ ( M × M )) . As a result, we see that (3.25) takes the form A m ( M × G ) ⊕ A m ( M × M ) ∋ ( θ m , ξ ) ⊕ ( θ m , X ) 7→( θ m , ξ ; ξ † ( m ) , X + ξ † ( m )) ∈ A m (( M × G ) ⊲⊳ ( M × M )) . (3.28)3.5. Left and right invariant vector fields on matched pairs of Lie groupoids.
In this subsection we shall determine the nature of the left invariant (resp. the right invariant)vector fields on a matched pair Lie groupoid.
Proposition 3.5.
Let G ⊲⊳ H be a matched pair Lie groupoid over a base manifold B . Then, theleft invariant vector field corresponding to U ∈ A b (G ⊲⊳ H ) is given by (3.29) ←− U ( g , h ) = (←−−−− h ⊲ X ( g ) , X † ( h ) + ←− Y ( h )) , where g ∈ G , h ∈ H , so that β ( h ) = b , and X ∈ A b G , Y ∈ A b H .Proof. In view of the isomorphism 3.3, any U ∈ A b (G ⊲⊳ H ) may be written as U = ( X , T ( ε ◦ β )( X )) + ( θ ε G ( b ) , Y ) for some X ∈ A b G , and Y ∈ A b H . As such, ←− U ( g , h ) = ←−−−−−−−−−−−−−−( X , T ( ε ◦ β )( X ))( g , h ) + ←−−−−−−−−( θ ε G ( b ) , Y )( g , h ) . More precisely, assuming α ( x t ) = β ( h ) , we have ←−−−−−−−−−−−−−−( X , T ( ε ◦ β )( X ))( g , h ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( g , h )( x t , ( ε H ◦ β )( x t )) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( g ( h ⊲ x t ) , ( h ⊳ x t )( ε H ◦ β )( x t )) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( g ( h ⊲ x t ) , ( h ⊳ x t )) = (←−−−− h ⊲ X ( g ) , h ⊳ X ) = (←−−−− h ⊲ X ( g ) , X † ( h )) . Similarly, assuming β ( y t ) = α ( g ) and β ( h ) = b = α ( y t ) , we have ←−−−−−−−−( θ ε G ( b ) , Y )( g , h ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( g , h )( ε G ( b ) , y t ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( g ( h ⊲ ε G ( b )) , ( h ⊳ ε G ( b )) y t ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( g ε G ( α ( h )) , h y t ) = ( θ g , ←− Y ( h )) . The result follows. (cid:3)
The right analogue is given by the following proposition.
Proposition 3.6.
Let G ⊲⊳ H be a matched pair of Lie groupoids over a base manifold B . Then,the right invariant vector field corresponding to U ∈ A b (G ⊲⊳ H ) is given by (3.30) −→ U ( g , h ) = (−→ X ( g ) − Y † ( g ) , −−−→ Y ⊳ g ( h )) , where g ∈ G , so that α ( g ) = b , h ∈ H , X ∈ A b G , and Y ∈ A b H .Proof. This time we have for any U ∈ A b (G ⊲⊳ H ) that −→ U ( g , h ) = −−−−−−−−−−−−−−→( X , T ( ε ◦ β )( X ))( g , h ) + −−−−−−−−→( θ ε G ( b ) , Y )( g , h ) . Accordingly, assuming α ( x t ) = b = α ( g ) , −−−−−−−−−−−−−−→( X , T ( ε ◦ β )( X ))( g , h ) = − ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( x t , ( ε H ◦ β )( x t )) − ( g , h ) = − ddt (cid:12)(cid:12)(cid:12)(cid:12) t = (( ε H ◦ β )( x t ) − ⊲ x − t , ( ε H ◦ β )( x t ) − ⊳ x − t )( g , h ) = − ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( x − t , ε H ( b ))( g , h ) = − ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( x − t ( ε H ( b ) ⊲ g ) , ( ε H ( b ) ⊳ g ) h ) = − ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( x − t g , h ) = (−→ X ( g ) , θ h ) , and −−−−−−−−→( θ ε G ( b ) , Y )( g , h ) = − ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( ε G ( b ) , y t ) − ( g , h ) = − ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( y − t ⊲ ε G ( b ) − , y − t ⊳ ε G ( b ) − )( g , h ) = − ddt (cid:12)(cid:12)(cid:12)(cid:12) t = (( ε G ◦ β )( y t ) , y − t )( g , h ) = − ddt (cid:12)(cid:12)(cid:12)(cid:12) t = (( ε G ◦ β )( y t )( y − t ⊲ g ) , ( y − t ⊳ g ) h ) = − ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( y − t ⊲ g , ( y − t ⊳ g ) h ) = (− Y † ( g ) , −−−→ Y ⊳ g ( h )) . The result follows. (cid:3)
Example 3.7.
Let us now derive the left (resp. right) invariant vector fields on ( M × G ) ⊲⊳ ( M × M ) in view of the isomorphism (3.28). ATCHED PAIRS OF DISCRETE DYNAMICAL SYSTEMS 19
Given ( m , g ; m g , n ) ∈ ( M × G ) ⊲⊳ ( M × M ) and ( θ n , ξ ; ξ † ( n ) , Y ) ∈ A n (( M × G ) ⊲⊳ ( M × M )) ,we have ←−−−−−−−−−−−−−( θ n , ξ ; ξ † ( n ) , Y )( m , g ; m g , n )) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( m , g ; m g , n )( n , e t ; ne t , n t ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = (cid:16) ( m , g ) (cid:0) ( m g , n ) ⊲ ( n , e t ) (cid:1) ; (cid:0) ( m g , n ) ⊳ ( n , e t ) (cid:1) ( ne t , n t ) (cid:17) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = (cid:16) ( m , g )( m g , e t ) ; ( m g e t , ne t )( ne t , n t ) (cid:17) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = (cid:16) m , g e t ; m g e t , n t (cid:17) = ( θ m , ←− ξ ( g ) ; ξ † ( m g ) , Y ) . Similarly, for ( m , g ; m g , n ) ∈ ( M × G ) ⊲⊳ ( M × M ) and ( θ m , ξ ; ξ † ( m ) , Y ) ∈ A m (( M × G ) ⊲⊳ ( M × M )) , −−−−−−−−−−−−−−→( θ m , ξ ; ξ † ( m ) , Y )( m , g ; m g , n ) = − ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( m , e t ; me t , n t ) − ( m , g ; m g , n ) = − ddt (cid:12)(cid:12)(cid:12)(cid:12) t = (cid:16) ( me t , n t ) − ⊲ ( m , e t ) − ; ( me t , n t ) − ⊳ ( m , e t ) − (cid:17) ( m , g ; m g , n ) = − ddt (cid:12)(cid:12)(cid:12)(cid:12) t = (cid:16) ( n t , me t ) ⊲ ( me t , e − t ) ; ( n t , me t ) ⊳ ( me t , e − t ) (cid:17) ( m , g ; m g , n ) = − ddt (cid:12)(cid:12)(cid:12)(cid:12) t = (cid:16) n t , e − t ; n t e − t , m (cid:17) ( m , g ; m g , n ) = − ddt (cid:12)(cid:12)(cid:12)(cid:12) t = (cid:16) ( n t , e − t ) (cid:0) ( n t e − t , m ) ⊲ ( m , g ) (cid:1) ; (cid:0) ( n t e − t , m ) ⊳ ( m , g ) (cid:1) ( m g , n ) (cid:17) = − ddt (cid:12)(cid:12)(cid:12)(cid:12) t = (cid:16) ( n t , e − t )( n t e − t , g ) ; ( n t e − t g , m g )( m g , n ) (cid:17) = − ddt (cid:12)(cid:12)(cid:12)(cid:12) t = (cid:16) n t , e − t g ; n t e − t g , n (cid:17) = (− Y , −→ ξ ( g ) ; ( Y − ξ † ( m )) ⊳ g , θ n ) . On the other hand, in view of (3.28), for any ( m , g ; m g , n ) ∈ ( M × G ) ⊲⊳ ( M × M ) , and ( θ n , ξ ; ξ † ( n ) , Y ) = ( θ n , ξ ; ξ † ( n ) , ξ † ( n )) + ( θ n , θ ; θ n , X ) ∈ A n (( M × G ) ⊲⊳ ( M × M )) , we have ←−−−−−−−−−−−−−( θ n , ξ ; ξ † ( n ) , Y )( m , g ; m g , n ) = ←−−−−−−−−−−−−−−−−( θ n , ξ ; ξ † ( n ) , ξ † ( n ))( m , g ; m g , n ) + ←−−−−−−−−−−( θ n , θ ; θ n , X )( m , g ; m g , n ) = ( θ m , ←− ξ ( g ) ; ξ † ( m g ) , ξ † ( n )) + ( θ m , θ g ; θ m g , X ) = ( θ m , ←− ξ ( g ) ; ξ † ( m g ) , ξ † ( n ) + X ) , and for any ( θ m , ξ ; ξ † ( m ) , Y ) = ( θ m , ξ ; ξ † ( m ) , ξ † ( m )) + ( θ m , θ ; θ m , Z ) ∈ A m (( M × G ) ⊲⊳ ( M × M )) , we obtain −−−−−−−−−−−−−−→( θ m , ξ ; ξ † ( m ) , Y )( m , g ; m g , n ) = −−−−−−−−−−−−−−−−−−→( θ m , ξ ; ξ † ( m ) , ξ † ( m ))( m , g ; m g , n ) + −−−−−−−−−−−→( θ m , θ ; θ m , Z )( m , g ; m g , n ) = (− ξ † ( m ) , −→ ξ ( g ) ; θ m g , θ n ) + (− Z , θ g ; − Z ⊳ g , θ n ) = (− ξ † ( m ) − Z , −→ ξ ( g ) ; − Z ⊳ g , θ n ) . Remark 3.8.
We can relate the above calculations to the left (resp. right) invariant vector fieldson the trivial groupoids as follows. The (groupoid) isomorphism (3.14) induces the isomorphism(3.15) on the level of Lie algebroids. Hence, it induces an isomorphism on the level of left (resp.right) invariant vector fields. Indeed, ←−−−−−−−−−−−−−( θ n , ξ ; ξ † ( n ) , Y )( m , g ; m g , n ) = ←−−−−−−−−−−−−A n Φ ( θ n , ξ, Y )( m , g ; m g , n ) ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( m , g ; m g , n )( n , e t ; ne t , n t ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = Φ ( m , g , n ) Φ ( n , e t , n t ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = Φ (cid:0) ( m , g , n )( n , e t , n t ) (cid:1) = T ( m , g , n ) Φ (cid:16) ←−−−−−−−( θ n , ξ, Y )( m , g , n ) (cid:17) . Similarly, −−−−−−−−−−−−−−→( θ m , ξ ; ξ † ( m ) , Y )( m , g ; m g , n ) = −−−−−−−−−−−−→A m Φ ( θ m , ξ, Y )( m , g ; m g , n )− ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( n , e t ; ne t , n t ) − ( m , g ; m g , n ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = Φ (cid:0) ( n , e t , n t ) − (cid:1) Φ ( m , g , n ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = Φ (cid:0) ( n , e t , n t ) − ( m , g , n ) (cid:1) = T ( m , g , n ) Φ (cid:16) −−−−−−−→( θ m , ξ, Y )( m , g , n ) (cid:17) .
4. Discrete dynamics on matched pairs
Discrete Euler-Lagrange equations.
We shall recall briefly the discrete Euler-Lagrange equations from [32, Subsect. 4.1]. Let G be a Lie groupoid, and AG be its associated Lie algebroid. For any fixed g ∈ G and N > G N being the N times cartesian product of G , the set C N g = (cid:8) ( g , ..., g N ) ∈ G N | ( g k , g k + ) ∈ G ∗ G , k N − , g ... g N = g (cid:9) is called the set of admissible sequences with values in G .On the other hand, a discrete Lagrangian is defined as a function L : G → R , and the discreteaction sum associated to it is given by(4.1) S L : C N g → R , ( g , ..., g N ) 7→ N Õ k = L ( g k ) . ATCHED PAIRS OF DISCRETE DYNAMICAL SYSTEMS 21
Now the discrete Hamilton’s principle may be recalled, from [50], as follows. Given g ∈ G and N >
1, an admissible sequence ( g , ..., g N ) is a solution of the Lagrangian system if and only if ( g , ..., g N ) ∈ C N g is a critical point of (4.1). So, along the lines of [32], one arrives at the discreteEuler-Lagrange equations N − Õ k = [←− X k ( g k ) ( L ) − −→ X k ( g k + ) ( L )] = , for any X k ∈ Γ (AG) . In particular, for N =
2, the discrete Euler-Lagrange equations are given by(4.2) ←− X ( g ) ( L ) − −→ X ( g ) ( L ) = X ∈ Γ ( A G) .We review below the examples discussed in [32]. Example 4.1.
Let M × M be the coarse (pair) groupoid of Example 2.3, whose left invariant vectorfields (resp. the right invariant vector fields) correponding to the Lie algebroid of M × M wereobtained in Example 2.9.Now, given a discrete Lagrangian L : M × M → R , the discrete Euler-Lagrange equations(4.2) takes the particular form ←− X ( x , y )( L ) − −→ X ( x , y )( L ) = . (4.3)In terms of the total derivatives on the product manifold M × M , the discrete Euler-Lagrangeequations may be rewritten as D L ( x , y ) + D L ( y , z ) = , (4.4)see also [37]. Example 4.2.
Let G be a Lie group (with the Lie algebra g ), considered as a groupoid over theidentity { e } . We recall that the left invariant and the right invariant vector fields corresponding tothe Lie algebroid g of G coincides with the left invariant and the right invariant vector fields on G .Now, given a discrete Lagrangian density L : G → R the discrete Euler-Lagrange equationsare given by(4.5) ←− ξ ( g k )( L ) − −→ ξ ( g k + )( L ) = , or equivalently h dL ( g k ) , ←− ξ ( g k )i − h dL ( g k + ) , −→ ξ ( g k + )i = h dL ( g k ) , T e ℓ g k ( ξ )i − h dL ( g k + ) , T e r g k + ( ξ )i = h T ∗ e ℓ g k ( dL ( g k )) − T ∗ e r g k + ( dL ( g k + )) , ξ i = , for any ξ ∈ g , and any g k , g k + ∈ G . As such, the discrete Euler-Lagrange equations may be writtenby(4.6) T ∗ e ℓ g k ( dL ( g k )) − T ∗ e r g k + ( dL ( g k + )) = µ k : = (cid:16) r ∗ g k dL (cid:17) ( e ) . Then (4.2) takes the form T ∗ e ℓ g k ( dL ( g k )) − T ∗ e r g k + ( dL ( g k + )) = T ∗ e ℓ g k ( dL ( g k )) − µ k + = T ∗ e ℓ g k T ∗ e ( r g k ◦ r g k − )( dL ( g k )) − µ k + = T ∗ e ℓ g k T ∗ e r g k − T ∗ e r g k ( dL ( g k )) − µ k + = T ∗ e ℓ g k T ∗ e r g k − ( µ k ) − µ k + = Ad ∗ g − k ( µ k ) − µ k + = . In other words, µ k + = Ad ∗ g − k ( µ k ) , called the discrete Euler-Lagrange equations, see also [2, 34, 35]. Remark 4.3.
We note for the adjoint action Ad : G × g → g , ( g , ξ ) 7→ Ad g ( ξ ) = : g ⊲ ξ that h µ, Ad g ( ξ )i = h µ, g ⊲ ξ i = h µ ∗ ⊳ g , ξ i = h g − ∗ ⊲ µ, ξ i = h Ad ∗ g − ( µ ) , ξ i . Example 4.4.
Let M × G be the action Lie groupoid of Example 2.2, with the left invariant andthe right invariant vector fields as in Example 2.10.Given a Lagrangian L : M × G → R , the sequence (( m , g k ) , ( m g k , g k + )) ∈ ( M × G ) ∗ ( M × G ) is a solution of the discrete Euler-Lagrange equations if ←−−−−−( θ m , ξ ) ( m , g k ) ( L ) − −−−−−→( θ m , ξ ) ( m · g k , g k + ) ( L ) = , for any ξ ∈ g . Equivalently, (cid:0) T e ℓ g k (cid:1) ( ξ ) ( L m ) − (cid:0) T e r g k + (cid:1) ( ξ ) ( L m g k ) + ξ † ( m g k ) (cid:0) L g k + (cid:1) = , where L m : G → R is the map given by L m ( g ) : = L ( m , g ) , and similarly L g : M → R is the onegiven by L g ( m ) : = L ( m , g ) .Setting µ k ( m , g k ) = d ( L m ◦ r g k ) ( e ) as above, the discrete Euler-Lagrange equations appearto be µ k + ( m · g k , g k + ) = Ad ∗ g k µ k ( m , g k ) + d (cid:0) L g k + ◦ (( m · g k ) ·) (cid:1) ( e ) , where ( m · g k ) · : G → M is given by ( m · g k ) · ( g ) : = m · ( g k g ) .If, in particular, M is the orbit space of a representation of G on V , then the correspondingequations were first obtained in [2, 3], and they are called the discrete Euler-Poincaré equations. Example 4.5.
Let M × G × M be the trivial groupoid of Example 2.4, whose left invariant andright invariant vector fields are obtained in Example 2.11. Accordingly, given a Lagrangian L : M × G × M → R , together with ( m k , g k , n k ) , ( m k + , g k + , n k + ) ∈ M × G × M with β ( m k , g k , n k ) = n k = m k + = α ( m k + , g k + , n k + ) , the discrete Euler-Lagrange equations are given by ( θ m k , ←− ξ ( g k ) , X )( L ) − (− X , −→ ξ ( g k + ) , θ n k + )( L ) = . Setting µ k = T ∗ r g k d L ( m k , g k , n k ) , the discrete Euler-Lagrange equations appear as(4.7) h X , d L ( n k , g k + , n k + ) + d L ( m k , g k , n k )i + D ξ, Ad ∗ g − k ( µ k ) − µ k + E = , where, for 1 i d i stands for the derivative with respect to the i th variable. ATCHED PAIRS OF DISCRETE DYNAMICAL SYSTEMS 23
Discrete dynamics on matched pairs of Lie groupoids.
In this section, we shall rewrite the discrete Euler-Lagrange equations (4.2) for a matched pair Liegroupoid, that is,(4.8) ←− U ( g k , h k ) ( L ) − −→ U ( g k + , h k + ) ( L ) = , generated by the Lagrangian L : G ⊲⊳ H → R . In view of (3.29) and (3.30), the equation 4.8 takesthe form(4.9) (cid:18) ←−−−−− h k ⊲ X ( g k ) , X † ( h k ) + ←− Y ( h k ) (cid:19) ( L ) − (cid:18) −→ X ( g k + ) − Y † ( g k + ) , −−−−−−→ Y ⊳ g k + ( h k + ) (cid:19) ( L ) = . As such, we arrive at the following proposition.
Proposition 4.6.
Given a matched pair (G , H ) of Lie groupoids, the discrete Euler-Lagrangeequations on the matched pair groupoid G ⊲⊳ H generated by the Lagrangian L : G ⊲⊳ H → R isgiven by (4.10) ←−−−−− h k ⊲ X ( g k )( L )−−→ X ( g k + )( L ) + Y † ( g k + )( L ) + X † ( h k )( L ) + ←− Y ( h k )( L )−−−−−−−→ Y ⊳ g k + ( h k + )( L ) = . In particular, considering the left action of H on G to be trivial, the equation (4.10) reducesto(4.11) ←− X ( g k )( L ) − −→ X ( g k + )( L ) + X † ( h k )( L ) + ←− Y ( h k )( L ) − −−−−−−→ Y ⊳ g k + ( h k + )( L ) = . Similarly, considering this time the right action of G on H to be trivial, the equation (4.10)takes the form of(4.12) ←−−−−− h k ⊲ X ( g k )( L ) − −→ X ( g k + )( L ) + Y † ( g k + )( L ) + ←− Y ( h k )( L ) − −→ Y ( h k + )( L ) = . If both actions are trivial, then the equation (4.10) simplifies to ←− X ( g k )( L ) − −→ X ( g k + )( L ) + ←− Y ( h k )( L ) − −→ Y ( h k + )( L ) = . We conclude the present section with yet another particular case of (4.2) , or of (4.8), for amatched pair of Lie groups (regarded as Lie groupoids) to analyse the discerete dynamics on Liegroups from the matched pair point of view.4.3.
Discrete dynamics on matched pairs of Lie groups.
Let G and H be two Lie groups with mutual actions ρ : H × G → G , ( h , g ) 7→ h ⊲ g , (4.13) σ : H × G → H , ( h , g ) 7→ h ⊳ g , (4.14)where h , h , h ∈ H , g , g , g ∈ G , e H is the identity element in H , and e G is the identity elementin G . If the actions (4.13)-(4.14) satisfy h ⊲ ( g g ) = ( h ⊲ g ) (( h ⊳ g ) ⊲ g ) , (4.15) ( h h ) ⊳ g = ( h ⊳ ( h ⊲ g )) ( h ⊳ g ) , (4.16) then the pair ( G , H ) is called a mathed pair of Lie groups, [28, 29]. In this case, the cartesianproduct G × H may be equipped with the group structure given by the multiplication ( g , h )( g , h ) = ( g ( h ⊲ g ) , ( h ⊳ g ) h ) = ( g ρ ( h , g ) , σ ( h , g ) h ) , and the unit element ( e G , e H ) . This matched pair group is denoted by G ⊲⊳ H . Conversely, if a Liegroup M is a cartesian product of two subgroups G ֒ → M ← ֓ H , and if the multiplication on M defines a bijection G × H → M , then M is a matched pair, that is, M (cid:27) G ⊲⊳ H . In this case, themutual actions are derived from(4.17) h · g = ( h ⊲ g ) ( h ⊳ g ) , for any g ∈ G , and any h ∈ H . Let us also record here the inversion in G ⊲⊳ H as(4.18) ( g , h ) − = (cid:16) h − ⊲ g − , h − ⊳ g − (cid:17) . for later use.We next consider the lifting of the group actions to the Lie algebra level. As for the leftaction, we have H × g −→ g , ( h , ξ ) 7→ h ⊲ ξ : = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = h ⊲ x t , (4.19) h × G −→ T G , ( η, g ) 7→ η † ( g ) : = η ⊲ g : = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = y t ⊲ g , (4.20)where g denotes the Lie algebra of the Lie group G , as h stands for the Lie algebra of H , x t ∈ G isa curve passing through the identity at t = ξ ∈ g , and finally y t ∈ H is a curvepassing through the identity in the direction of η ∈ h .Freezing the group element in (4.19), we arrive at a linear mapping h ⊲ : g → g for any h ∈ H . We shall denote the transpose of this mapping by ∗ ⊳ h : g ∗ → g ∗ , which is given by(4.21) h h ⊲ ξ, µ i = h ξ, µ ∗ ⊳ h i , for any µ ∈ g ∗ , where the pairing is the one between g ∗ and g .Similarly, freezing the group element in (4.20), we arrive at a linear operator b g : h T g G ,given by b g ( η ) : = η ⊲ g . The transpose of this mapping shall be denoted by b ∗ g : T ∗ g G h ∗ , and itis given by(4.22) h η ⊲ g , µ g i = h b g ( η ) , µ g i = h η, b ∗ g ( µ g )i , for any µ g ∈ T ∗ g G .Now on the other hand, for the right G action on H , we have the maps h × G −→ h , ( η, g ) 7→ η ⊳ g : = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = y t ⊳ g , (4.23) H × g −→ T H , ( h , ξ ) 7→ ξ † ( h ) : = h ⊳ ξ : = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = h ⊳ x t . (4.24)Similar to above, freezing the group element in (4.23) we arrive at a linear mapping ⊳ g : h → h for any g ∈ G . The transpose of this map will be denoted by g ∗ ⊲ : h ∗ → h ∗ , and it is defined by(4.25) h η ⊳ g , ν i = h η, g ∗ ⊲ ν i , ATCHED PAIRS OF DISCRETE DYNAMICAL SYSTEMS 25 for any ν ∈ h ∗ , where the pairing is the one between h ∗ and h .Finally, freezing the group element in (4.24) we obtain a mapping a h : g T h H , a h ( ξ ) = h ⊳ ξ for any ξ ∈ g . The transpose a ∗ h : T ∗ h H g ∗ of this linear mapping will be given by(4.26) h h ⊳ ξ, ν h i = h a h ( ξ ) , ν h i = h ξ, a ∗ h ( ν h )i , for any ν h ∈ T ∗ h H .We note also that if G ⊲⊳ H is a matched pair Lie group, then its Lie algebra is the matchedpair Lie algebra g ⊲⊳ h . That is, the induced actions ⊲ : h ⊗ g → g and ⊳ : h ⊗ g → h of the Lie algebras satisfy(4.27) η ⊲ [ ξ , ξ ] = [ η ⊲ ξ , ξ ] + [ ξ , η ⊲ ξ ] + ( η ⊳ ξ ) ⊲ ξ − ( η ⊳ ξ ) ⊲ ξ and(4.28) [ η , η ] ⊳ ξ = [ η , η ⊳ ξ ] + [ η ⊳ ξ, η ] + η ⊳ ( η ⊲ ξ ) − η ⊳ ( η ⊲ ξ ) , for any η, η , η ∈ h , and any ξ, ξ , ξ ∈ g . Such a pair ( g , h ) is called a matched pair of Lie algebras,and the Lie algebra structure on g ⊲⊳ h : = g ⊕ h is given by(4.29) [( ξ , η ) , ( ξ , η )] = ([ ξ , ξ ] + η ⊲ ξ − η ⊲ ξ , [ η , η ] + η ⊳ ξ − η ⊳ ξ ) . It is immediate that both g and h are Lie subalgebras of g ⊲⊳ h via the obvious inclusions. Conversely,given a Lie algebra m with two subalgebras g ֒ → m ← ֓ h , if m (cid:27) g ⊕ h via ( ξ, η ) 7→ ξ + η , then m (cid:27) g ⊲⊳ h as Lie algebras. In this case, the mutual actions of the Lie algebras are uniquelydetermined by [ η, ξ ] = ( η ⊲ ξ, η ⊳ ξ ) . On the other hand, there are integrability conditions under which a matched pair of Lie algebrascan be integrated into a matched pair of Lie groups. For a discussion of this direction we refer thereader to [27, Sect. 4].We shall also need the adjoint action of the matched pair Lie group G ⊲⊳ H on its Lie algebra g ⊲⊳ h . For any ( g , h ) ∈ G ⊲⊳ H , and any ( ξ, η ) ∈ g ⊲⊳ h , we recall from [12, (2.30)] that(4.30) Ad ( g , h ) − ( ξ, η ) = ( h − ⊲ ζ, T h − r h ( h − ⊳ ζ ) + Ad h − ( η ⊳ g )) where ζ : = Ad g − ( ξ ) + T g L g − ( η ⊲ g ) ∈ g .Furthermore, the tangent lifts of the left and right regular actions of G ⊲⊳ H are given in [13,(2.54)&(2.55)] as T ( g , h ) L ( g , h ) (cid:0) U g , V h (cid:1) = (cid:0) T h ⊲ g L g (cid:0) h ⊲ U g (cid:1) , T h ⊳ g R h (cid:0) h ⊳ U g (cid:1) + T h L ( h ⊳ g ) V h (cid:1) , T ( g , h ) R ( g , h ) (cid:0) U g , V h (cid:1) = (cid:0) T g R ( h ⊲ g ) U g + T h ⊲ g L g (cid:0) V h ⊲ g (cid:1) , T h ⊳ g R h (cid:0) V h ⊳ g (cid:1) (cid:1) . We can thus compute the left and right invariant vector fields generated by a Lie algebra element ( ξ, η ) ∈ g ⊲⊳ h as ←−−−( ξ, η )( g , h ) = T ( e G , e H ) L ( g , h ) ( ξ, η ) = (←−−− h ⊲ ξ ( g ) , h ⊳ ξ + ←− η ( h )) , (4.31) −−−→( ξ, η )( g , h ) = T ( e G , e H ) R ( g , h ) ( ξ, η ) = (−→ ξ ( g ) + η ⊲ g , −−−→ η ⊳ g ( h )) . (4.32) Recalling the discrete Euler-Lagrange equations (4.5), discrete dynamics on G ⊲⊳ H generated bya Lagrangian function L : G ⊲⊳ H → R is then given by ←−−−( ξ, η )( g k , h k )( L ) − −−−→( ξ, η )( g k + , h k + )( L ) = . (4.33)Let now the exterior derivative of the Lagrangian L : G ⊲⊳ H → R be a two-tuple ( d L , d L ) ,where d L denotes the derivative with respect to group variable g ∈ G whereas d L denotes thederivative with respect to group variable h ∈ H . Then, in view of the left and right invariant vectorfields (4.31) - (4.32), we arrive at D ←−−−− h k ⊲ ξ ( g k ) , d L ( g k , h k ) E + h h k ⊳ ξ, d L ( g k , h k )i + (cid:10) ←− η ( h k ) , d L ( g k , h k ) (cid:11) − D −→ ξ ( g k + ) , d L ( g k + E − h η ⊲ g k + , d L ( g k + )i − (cid:10) −−−−−−→ η ⊳ g k + ( h k + ) , d L ( g k + , h k + ) (cid:11) = . It is possible to single out ξ ∈ g and η ∈ h from these equations, that is, D ξ, (cid:0) T ∗ L g k · d L ( g k , h k ) (cid:1) ∗ ⊳ h k + a ∗ h k d L ( g k , h k ) − T ∗ R g k + · d L ( g k + , h k + ) E + D η, T ∗ L h k · d L ( g k , h k ) − b ∗ g k + d L ( g k + , h k + ) − g k + ∗ ⊲ T ∗ R h k + · d L ( g k + , h k + ) E = . Proposition 4.7.
In particular, taking the covectors T ∗ R g k · d L ( g k , h k ) = µ k ∈ g ∗ , T ∗ R h k · d L ( g k , h k ) = ν k ∈ h ∗ , the discrete Euler-Lagrange equations on the matched pair Lie group G ⊲⊳ H can be written as (4.34) Ad ∗ g − k ( µ k ) ∗ ⊳ h k + a ∗ h k d L ( g k , h k )− µ k + + Ad ∗ h − k ( ν k )− b ∗ g k + d L ( g k + , h k + )− g k + ∗ ⊲ ν k + = . Furthermore, when the (right) action of G on H is trivial, we have the discrete Euler-Lagrangeequation(4.35) Ad ∗ g − k ( µ k ) ∗ ⊳ h k − µ k + + Ad ∗ h − k ( ν k ) − b ∗ g k + d L ( g k + , h k + ) − ν k + = G ⋊ H .On the other extreme, assuming the (left) action of H on G to be trivial, we arrive at theequation(4.36) Ad ∗ g − k ( µ k ) + a ∗ h k d L ( g k , h k ) − µ k + + Ad ∗ h − k ( ν k ) − g k + ∗ ⊲ ν k + = G ⋉ H .If both actions are trivial, then the equations reduce all the way down to(4.37) Ad ∗ g − k ( µ k ) − µ k + + Ad ∗ h − k ( ν k ) − ν k + = .
5. Examples
Discrete dynamics on the trivial groupoid.
In this subsection we shall illustrate the discrete Euler-Lagrange equation on the trivial groupoidof Example 2.4, regarded as the matched pair groupoid of the coarse (banal) groupoid of Example2.3 and the action groupoid of Example 2.2. To this end, we first recall from Example 3.7 that
ATCHED PAIRS OF DISCRETE DYNAMICAL SYSTEMS 27 given ( m , g ; m g , n ) ∈ ( M × G ) ⊲⊳ ( M × M ) and ( θ n , ξ ; ξ † ( n ) , Y ) ∈ A n (( M × G ) ⊲⊳ ( M × M )) , wehave ←−−−−−−−−−−−−−( θ n , ξ ; ξ † ( n ) , Y )( m , g ; m g , n ) = ( θ n , ←− ξ ( g ) ; ξ † ( m g ) , ξ † ( n ) + X ) , and similarly, for any ( θ m , ξ ; ξ † ( m ) , Y ) ∈ A m (( M × G ) ⊲⊳ ( M × M )) , −−−−−−−−−−−−−−→( θ m , ξ ; ξ † ( m ) , Y )( m , g ; m g , n ) = (− ξ † ( m ) − Z , −→ ξ ( g ) ; − Z ⊳ g , θ n ) . Now, given ( m k , g k ; m k g k , n k ) , ( m k + , g k + ; m k + g k + , n k + ) ∈ ( M × G ) ⊲⊳ ( M × M ) , so that β (( m k , g k ; m k g k , n k )) = n k = m k + = α ( m k + , g k + ; m k + g k + , n k + ) , and a Lagrangian L : ( M × G ) ⊲⊳ ( M × M ) → R , the equation (4.10) yields h X ( n k ) , d L ( n k , g k + ; m k + g k + , n k + ) + d L ( m k , g k ; m k g k , n k )i + D ξ, Ad ∗ g − k µ k − µ k + E + (cid:10) ξ † ( m k g k ) , d L ( m k , g k ; m k g k , n k ) (cid:11)(cid:10) ξ † ( n k ) , d L ( n k , g k + ; m k + g k + , n k + ) + d L ( m k , g k ; m k g k , n k ) (cid:11) + h X ( n k ) ⊳ g k + , d L ( n k , g k + ; m k + g k + , n k + )i = µ k = T ∗ r g k d L ( m k , g k ; m k g k , n k ) , and for 1 j
4, the operator d j denotes the derivativewith respect to the j th variable. Remark 5.1.
Let us note that the equations (5.1) above correspond to the discrete Euler-Lagrangeequations (4.7) on the trivial groupoid M × G × M , under the isomorphism (3.14). More precisely,on one hand we have ←−−−−−−−−−−−−−( θ n , ξ ; ξ † ( n ) , Y )( m , g ; m g , n ) = ( θ m , ←− ξ ( g ) ; ξ † ( m g ) , ξ † ( n ) + X ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( m , g e t ; m g e t , n t e t ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = Φ ( m , g e t , n t e t ) = T Φ (cid:16) θ m , ←− ξ ( g ) , ξ † ( n ) + X (cid:17) = T Φ (cid:18) ←−−−−−−−−−−−−−−−( θ n , ξ, ξ † ( n ) + X )( m , g , n ) (cid:19) , where on the other hand, −−−−−−−−−−−−−−→( θ m , ξ ; ξ † ( m ) , Y )( m , g ; m g , n ) = (− ξ † ( m ) − Z , −→ ξ ( g ) ; − Z ⊳ g , θ n ) = − ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ( m t e t , e − t g ; m t g , n ) = − ddt (cid:12)(cid:12)(cid:12)(cid:12) t = Φ ( m t e t , e − t g , n ) = T Φ (cid:16) − ξ † ( m ) − Z , −→ ξ ( g ) , θ n (cid:17) = T Φ (cid:18) −−−−−−−−−−−−−−−→( θ m , ξ, ξ † ( m ) + Z )( m , g , n ) (cid:19) . The correspondence, then, follows at once.5.2.
Discrete Dynamics on SL ( , C ) = SU ( ) ⊲⊳ K . In this subsection, we shall study the discrete Euler-Lagrange equations on the Lie group SL ( , C ) from the matched pair point of view. To this end, we shall first recall its decomposition(5.2) SL ( , C ) = SU ( ) ⊲⊳ K from [27], see also [12, 13], the group structures, the mutual actions of the groups SU ( ) and K ,together with their lifts. The group(5.3) SU ( ) = (cid:26) (cid:18) ω ϑ − ¯ ϑ ¯ ω (cid:19) ∈ SL ( , C ) : | ω | + | ϑ | = (cid:27) in the matched pair decomposition(5.2) is a universal double cover of the group SO ( ) . As such,for each element A ∈ SU ( ) there exists a unique matrix Rot A ∈ SO ( ) . The Lie algebra su ( ) ofthe group SU ( ) is the matrix Lie algebra su ( ) = (cid:26) − ι (cid:18) t r − ι sr + ι s − t (cid:19) : r , s , t ∈ R (cid:27) of traceless skew-hermitian matrices. Following [27] we fix three matrices(5.4) e = (cid:18) − ι / − ι / (cid:19) , e = (cid:18) − / / (cid:19) , e = (cid:18) − ι / ι / (cid:19) as a basis of the Lie algebra su ( ) . We further make use of this to identify the matrix Lie algebra su ( ) with the Lie algebra R by the cross product;(5.5) r e + se + te ∈ su ( ) ←→ X = ( r , s , t ) ∈ R . We also identify the dual space su ( ) ∗ of su ( ) (cid:27) R with R using the Euclidean dot product.Using this dualization, we can express the coadjoint action of the Lie algebra su ( ) (cid:27) R on su ∗ ( ) (cid:27) R as(5.6) ad ∗ : su ( ) × su ∗ ( ) → su ∗ ( ) , ( X , Φ ) 7→ ad ∗ X Φ : = X × Φ , for any X ∈ su ( ) (cid:27) R , and any Φ ∈ su ∗ ( ) ≃ R .The simply-connected group K , on the other hand, may be represented by(5.7) K = (cid:26) √ + c (cid:18) + c a + ib (cid:19) ∈ SL ( , C ) | a , b ∈ R and c > − (cid:27) where the group operation is the matrix multiplication. The Lie algebra K of the group K is thusgiven by(5.8) K = (cid:26) (cid:18) c a + ib − c (cid:19) ∈ sl ( , C ) | a , b , c ∈ R (cid:27) with matrix commutator being the Lie bracket. The group K can also be realised as a subgroup of GL ( , R ) as(5.9) K = © « + c + c − a − b ª®¬ ∈ GL ( , R ) | a , b ∈ R and c > − , where the group operation is the matrix multiplication. In this case, its Lie algebra K is given by(5.10) K = © « c c − a − b ª®¬ ∈ gl ( , R ) | a , b , c ∈ R , where the Lie bracket is the matrix commutator. The group K can, alternatively, be identified withthe subspace(5.11) K = (cid:8) ( a , b , c ) ∈ R | a , b ∈ R and c > − (cid:9) of R with a non-standard multiplication ( a , b , c ) ∗ ( a , b , c ) = ( a , b , c )( + c ) + ( a , b , c ) , ATCHED PAIRS OF DISCRETE DYNAMICAL SYSTEMS 29 in which case the Lie algebra K is R via the Lie bracket(5.12) [ Y , Y ] = k × ( Y × Y ) , where k is the unit vector ( , , ) ∈ R . In this case, using the dot product, we may identify thedual space K ∗ with R as well. Then, the coadjoint action of the Lie algebra K (cid:27) R on its dualspace K ∗ (cid:27) R can be computed as(5.13) ad ∗ : K × K ∗ → K ∗ , ( Y , Ψ ) 7→ ad ∗ Y Ψ : = ( k · Y ) Ψ − ( Ψ · Y ) k , for any Y ∈ K ≃ R , and any Ψ ∈ K ∗ ≃ R . The group isomorphisms relating (5.7), (5.9), and(5.11) are given by(5.14) 1 √ + c (cid:18) + c a + ib (cid:19) ←→ © « + c + c − a − b ª®¬ ←→ ( a , b , c ) . The Lie algebra isomorphisms(5.15) (cid:18) c a + ι b − c (cid:19) ←→ © « c c − a − b ª®¬ ←→ ( a , b , c ) between (5.8), (5.10) and (5.12) are then obtained by differentiating (5.14).We now move on to the mutual actions of the groups SU ( ) and K on each other. Given any A ∈ SU ( ) , and any B ∈ K ⊂ SL ( , C ) , the left action of K on SU ( ) is given by(5.16) B ⊲ A = (cid:13)(cid:13)(cid:13)(cid:13) B A (cid:18) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) − M (cid:18) B A (cid:18) (cid:19) + B −† A (cid:18) (cid:19) (cid:19) , where B −† stands for the inverse of the conjugate transpose of B ∈ K , and k B k M = tr ( B † B ) refersto the matrix norm on SL ( , C ) . The right action of SU ( ) on K ⊂ R , on the other hand, is(5.17) B ⊳ A = k B k E ( c + ) e + A B − k B k E ( c + ) e ! A − , where k•k E : R → R denotes the Euclidean norm, in view of the identification (5.14) of B ∈ K ⊂ SL ( , C ) with B ∈ K ⊂ R .Differentiating (5.16) with respect to A ∈ SU ( ) , and regarding B ∈ K ⊂ GL ( , R ) via(5.14), we obtain(5.18) ⊲ : K × su ( ) → su ( ) , ( B , X ) 7→ B ⊲ X : = B X , for any X ∈ su ( ) (cid:27) R . Freezing the group element here, we get a linear operator B ⊲ : su ( ) → su ( ) . The transpose of this operator ∗ ⊲ B : su ∗ ( ) → su ∗ ( ) is given by(5.19) Φ ∗ ⊳ B = B T Φ . Similarly, the derivative of (5.17) with respect to A ∈ SU ( ) renders the infinitesimal right actionof the Lie algebra su ( ) on K as(5.20) ⊳ : K × su ( ) → T K , ( B , X ) 7→ B ⊳ X = T e K r B (cid:16) X × e B (cid:17) , where X ∈ su ( ) (cid:27) R , and e B : = c + B − k B k E ( c + ) k identifying once again B ∈ K ⊂ SL ( , C ) with B ∈ K ⊂ R via (5.14). Here, T e K r B is the tangentlift of the right translation r B : K → K by B ∈ K , and it acts simply by the matrix multiplicationregarding X × e B ∈ K (cid:27) R (cid:27) gl ( , R ) via (5.15). Freezing the group element in (5.20) we arriveat a linear operator a B : su ( ) → T B K , the transpose of which is the operator a ∗ B : T ∗ B K → su ∗ ( ) given by(5.21) a ∗ B ( Ψ B ) = T ∗ r B ( Ψ B ) × e B for any Ψ B ∈ T ∗ B K .Next, the derivative of (5.16) with respect to B ∈ K at the identity, in the direction of Y ∈ K ⊂ R , yields ⊲ : K × SU ( ) → T SU ( ) , Y ⊲ A = Tr A (cid:16) Y × (cid:0) Ad A ( e ) − e (cid:1) (cid:17) , (5.22)where we consider Ad A ( e ) − e ∈ R to perform the vector product, then we view the resultingelement in SU ( ) , i.e. as a 2 × b A : K → T A SU ( ) . The transpose of this operator b ∗ A : T ∗ A SU ( ) → K ∗ is given explicitlyby(5.23) b ∗ A ( Φ A ) = (cid:0) Ad A ( e ) − e (cid:1) × Tr ∗ A Φ A . Similarly, the derivative of (5.17) with respect to B ∈ K in the direction of Y ∈ K (cid:27) R produces ⊳ : K × SU ( ) → K , Y ⊳ A = Rot A ( Y ) , and hence defines a linear mapping ⊳ A : K → K , whose transpose A ∗ ⊲ : K ∗ → K ∗ may be given by(5.24) A ∗ ⊲ Ψ : = Rot ∗ A Ψ , for any Ψ ∈ K ∗ .Now we are ready to write the discrete Euler-Lagrange equations on the matched pair Liegroup SL ( , C ) = SU ( ) ⊲⊳ K . Substituting(5.19), (5.21), (5.23), and (5.24) into (4.34), weconclude that B Tk ( Ad ∗ A k Φ k ) + T ∗ r B k ( d L ( A k , B k )) × e B − Φ k + + Ad ∗ B k Ψ k − (cid:0) Ad A k + ( e ) − e (cid:1) × T ∗ r A k + ( d L ( A k + , B k + )) − Rot ∗ A k + Ψ k + = .
6. Conclusion and Discussions
In the present paper, we have studied the discrete dynamics on the matched pairs of Liegroupoids. This enabled us to study the equations of motion governing two mutually interactingdiscrete systems. More precisely, we have presented the discrete Euler-Lagrange equations on amatched pair Lie groupoid as a sum of those over the individual Lie groupoids which are matched -but enriched with the mutual actions. Taking advantage of the Lie groups being the quintessentialexamples of Lie groupoids, in particular we have introduced the discrete Euler-Lagrange equationson (matched pair) Lie groups.In order to illustrate the theory, we have studied two concrete examples; the trivial Liegroupoid, and the group SL ( , C ) . ATCHED PAIRS OF DISCRETE DYNAMICAL SYSTEMS 31
The trivial Lie groupoid is a matched pair of the action groupoid and the coarse groupoid.In view of this observation we have realised the discrete Euler-Lagrange equations of the trivialLie groupoid as a sum of the discrete Euler-Lagrange equations of the action groupoid, and theEuler-Lagrange equations of the coarse groupoid, but decorated with the mutual action terms.As for the second tangible example, we considered the matched pair decomposition ofthe group SL ( , C ) into SU ( ) and a simply connected subgroup K ⊆ SL ( , C ) , which, in turn,corresponds to its Iwasawa decomposition. We then similarly decomposed the discrete Euler-Lagrange equations over SL ( , C ) into those over SU ( ) and K , once again, furnished with theaction terms.We finally note that, the present paper concerns only the discrete dynamics generated byLagrangian functions on Lie groupoids. It is very well known that there exists a theory ofLagrangian dynamics on the Lie algebroid level as well; [38, 50]. We plan to apply the matchedpair strategy to the Lagrangian dynamics from the point of view of the Lie algebroids, whichhowever, deserves a separate paper.
7. Acknowledgments
The first named author (OE) is greateful to Prof. Manuel de León and Prof. David Martín deDiego for their encouragements and kind interests in this present study. Both authors acknowledgethe support by TÜBİTAK (the Scientific and Technological Research Council of Turkey) underthe project "Matched pairs of Lagrangian and Hamiltonian Systems" with the project number117F426.
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