aa r X i v : . [ m a t h - ph ] A ug Discrete mechanics on unitary octonions
Janusz GrabowskiZohreh Ravanpak
Institute of MathematicsPolish Academy of Sciences
Abstract
In this article we generalize the discrete Lagrangian and Hamiltonianmechanics on Lie groups to non-associative objects generalizing Lie groups(smooth loops). This shows that the associativity assumption is not crucialfor mechanics and opens new perspectives. As a working example we obtainthe discrete Lagrangian and Hamiltonian mechanics on unitary octonions.
Keywords: non-associative geometry; discrete Euler-Lagrange equation; Lie groupoid;smooth loop; octonions
AMSC2010:
The main tool in the theory of geometric integrators (see e.g. [29]) is the discreteLagrangian and Hamiltonian formalism on G = M × M , i.e. the groupoid of pairsof points of a manifold M . This can be generalized to arbitrary Lie groupoid, inparticular any Lie group. This formalism has been studied by Weinstein [43] (seealso [26]).In [32], Moser and Veselov consider the Lagrangian and Hamiltonian formalismsfor discrete mechanics on a Lie group. The Lagrangian function L is defined on a Liegroup G , and the dynamical system is given by a diffeomorphism from G to itself.The corresponding Hamiltonian system is the mapping from the dual Lie algebra g ∗ to itself for which L is the generating function.Lie groupoids have been recently used for a geometric formulation of the La-grangian formalism and information geometry in many papers [4, 9, 15, 16, 26, 30,39, 43]. Infinitesimal parts of Lie groupoids are Lie algebroids and mechanics onLie algebroids has been also extensively studied [17, 22, 23, 31, 43]. More generalalgebroids in which the Jacobi identity is not satisfied ([13, 14]) have been used inthis context as well [6, 7, 8, 11, 12]. The idea is based on the concept of a Tulczyjewtriple [40, 41]. For general theory of Lie groupoid and Lie algebroids we refer toMackenzie [24].Let G ⇒ M be a Lie groupoid, α, β : G → M being its source and target maps,with a multiplication map m : G (2) → G , where G (2) = { ( g, h ) ∈ G × G | β ( g ) =1 Janusz Grabowski & Zohreh Ravanpak α ( h ) } . Denote its corresponding Lie algebroid by AG represented by the normalbundle ν ( M ) = T G | M / T M to the submanifold of units M ⊂ G . Sections of AG are represented by the left-invariant ←− X (or right-invariant −→ X ) vector fields on G associated with X ∈ Sec( AG ).Lagrangian mechanics on a groupoid G for a smooth, real-valued function L on G is defined as follows [43].Let L (2) be the restriction to the set of composable pairs G (2) of the function( g, h ) → L ( g ) + L ( h ) and Σ L ⊂ G (2) be the set of critical points of L (2) along thefibers of the multiplication map m ; that is, the points in Σ L are stationary pointsof the function L ( g ) + L ( h ) when g and h are restricted to admissible pairs with theconstraint that the product gh is fixed. Variations of the constraint are of the form( gu, u − h ) ∈ G (2) .A solution of the Hamilton principle for the Lagrangian function L is a sequence ..., g , g , g , g , g , ... of elements of G , defined on some interval in Z , such that( g i , g i +1 ) ∈ Σ L for each i . The Hamiltonian formalism for discrete Lagrangian sys-tems is based on the fact that each Lagrangian submanifold of a symplectic groupoid(see [42]) determines a Poisson automorphism on the base Poisson manifold. Recallthat the cotangent bundle T ∗ G is, in addition to being a symplectic manifold, a Liegroupoid itself, the base being A ∗ G ; notice that both manifolds are naturally Pois-son. The source and target mappings ˜ α, ˜ β : T ∗ G → A ∗ G are Poisson maps inducedby α and β . In [30], the authors showed that Lagrangian submanifolds of symplecticgroupoids give rise to discrete dynamical system.Discrete Euler-Lagrange equations on Lie groupoids can be derived from thevariational principles. In [26], the discrete Euler-Lagrange equations take the form ←− X ( g i )( L ) − −→ X ( g i +1 )( L ) = 0 (1)on a Lie groupoid G ⇒ M , for every section X of AG . Note that ( g i , g i +1 ) ∈ G (2) and the left and right arrow denotes the right and left-invariant vector field on G associated with X ∈ Sec( AG ) understood as a section on the normal bundle.A development of discrete Lagrangian mechanics on a Lie groups and groupoidshas been developed in many papers (e.g. [4, 15, 16, 26, 27, 28, 39, 43]). Nevertheless,the generalization of the discrete mechanics to non-associative objects is still lacking,and the aim of this paper is to fill this gap by presenting a systematic approach forthe construction of discrete Lagrangian and Hamiltonian formalism on smooth loops[3]. The theory of smooth quasi-groups and loops has already started to find interest-ing applications in geometry and physics. The remarkable development of smoothquasigroups and loops theory since the pioneering works of Mal’cev in 1955 (see [25])was presented by Lev V. Sabinin [35], where the large bibliography on the subjectis given. We refer also to the books [2, 34] and the survey articles [36, 38] if termsand concepts from non-associative algebra are concerned.As a working example we will develop the discrete Lagrangian and Hamiltonianformalism on unitary octonions O (understood as an inverse loop in the algebra ofoctonions O or a subloop in the loop O × of invertible octonions) which as a manifoldis the seven-sphere.It is well known that S , S , S and S are only spheres which are parallelizableand they correspond to elements of unit norm in the normed division algebras of the iscrete mechanics on unitary octonions S = Z , S = U (1), S = SU (2)), but S is the only parallelizablesphere which is not a Lie group since it is not associative.The left and right translations in the loop O act as diffeomorphism, so the leftand right prolongations ←− X and −→ X of X in the tangent space o = T e O = A O atthe neutral element e ∈ O to the loop O are well defined vector fields and theEuler-Lagrange equation (1) makes sense also in this case, although the variationalapproach is not applicable. Note that because of the lack of associativity the vectorfields ←− X and −→ X are no longer left nor right invariant, so that the tangent algebra o is not a Lie algebra. It is enough to observe that the other concepts like the Legendremap, Hamiltonian map, etc. are in the case of a Lie group built on prolongations ←− X and −→ X , so versions of Lagrangian and Hamiltonian formalisms can be formulatedalso for smooth loops. To develop such versions is the main aim of this paper. Let us recall that a loop is an algebraic structure < G, · , e > with a binary operation(written usually as juxtaposition, a · b = ab ) such that r a : x xa (the right trans-lation ) and l a : x ax (the left translation ) are permutations of G , equivalently, inwhich the equations ya = b and ax = b are uniquely solvable for x and y respectively,with a two-sided identity element, e , ex = xe = x . A loop < G, · , e > with identity e is called an inverse loop if it is equipped with a smooth inversion map ι : G → G which we denote simply by ι ( a ) = a − . In other words, to each element a in G therecorresponds an element a − in G such that a − ( ab ) = ( ba ) a − = b for all b ∈ G . It can be then easily shown that in an inverse loop < G, · , − , e > wehave, for all a, b ∈ G , aa − = a − a = e, ( a − ) − = a, and ( ab ) − = b − a − . (2)The above identities imply that ι ( e ) = e , ι = Id G , and ι ◦ l a = r a − ◦ ι . (3)Loops having only one inverse are called left inverse loops (resp. right inverseloops ). Left inverse loops, appear naturally as algebraic structures on transversals or sections of a subgroup in a group. In this case the homogeneous structures areequipped with a binary operation. This observation, going back to R. Baer [1] (cf.also [5, 20]), lies at the heart of much current research on loops, also in differentialgeometry and analysis.A smooth loop G is a smooth manifold equipped with a smooth multiplication, m : G × G → G , ( g, h ) m ( g, h ) = gh , such that the left and right translationsare diffeomorphisms, together with an identity element e ∈ G such that eg = ge = g for every g ∈ G . Let g = T e G . If G is a smooth loop with the smooth inverse ι : G → G (smooth inverse loop), then ι ∗ ( X ) = − X for X ∈ g . (4) Janusz Grabowski & Zohreh Ravanpak
Indeed, if γ : R → G is a curve in G such that γ (0) = e and γ represents X ∈ g ,then ddt | t =0 (cid:0) γ ( t )( γ ( t )) − (cid:1) = X + ι ∗ ( X ) = 0 . Let X e ∈ g be a vector field in T e G . We can left-translate (resp. right-translate)the value of X e by the tangent of the left (resp. right) translation by g . However,although they are not invariant vector fields anymore due to the lack of associativity,we still are able to define the left (resp. right) prolongations of X e to vector fields ←− X (resp. −→ X ) on G using the tangent maps at g ∈ G to the left translation l g andright translation r g : ←− X g = D e ( l g )( X e ) , −→ X g = D e ( r g )( X e ) . Here, D denotes the derivative. This prolongations are smooth vector fields, becausefor a smooth function f defined on G and a smooth curve γ such that γ (0) = e and ddt | t =0 γ ( t ) = X e , we get the following smooth function ←− X f ( g ) = D e ( l g )( X e ) f = ( X e )( f ◦ l g ) = ddt | t =0 ( f ◦ l g ◦ γ ( t ))= ddt | t =0 ( f ( gγ ( t ))) = ddt | t =0 ( f ◦ m ( g, γ ( t )))and similarly for −→ X f ( g ). According to (3) and (4), we have ι ∗ ( ←− X ) = −−→ X ◦ ι , (5)but due to non-associativity we cannot infer that there is [ X, Y ] ∈ g such that[ ←− X , ←− Y ] = ←−−− [ X, Y ] nor [ −→ X , −→ Y ] = −−−→ [ X, Y ]. Moreover, in general, we do not have[ ←− X , −→ Y ] = 0 nor [ −→ X , −→ Y ] = − [ ←− X , ←− Y ].However, the tangent space at the identity T e G ∼ = g inherits a skew-symmetricbilinear product [ · , · ] l from the Lie product of the left prolongations of vector fieldsover the loop. In other words, [ X, Y ] l = [ ←− X , ←− Y ] e . This is indeed a bilinear product,since for a ∈ R we have ←− aX = a ←− X . The Jacobi identity does not hold due to thenon-associativity. So, g it is not a Lie algebra but a skew-algebra , that is, a vectorspace equipped with a skew-symmetric binary operation. A similar bracket [ · , · ] r weobtain from the right prolongations, but in general they do not differ only by sign.The skew-algebra structure on g corresponds to a linear Leibniz structure on g ∗ , i.e.a linear bivector field, exactly like a Lie algebra structure corresponds to a linearPoisson structure on the dual space.In the paper [25] a local diassociative analytic loop G was considered. Dias-sociativity means that any two elements generate a genuine subgroup. Since themultiplication in a loop is a binary operation and the loop is diassociative, one maywrite the analogue of the Campbell-Hausdorff series, which depends only on oneskew-symmetric bilinear operation [ · , · ] in the tangent space g = T e G . In our nota-tion this bracket coincides with [ · , · ] l . This algebra is a binary-Lie algebra , that is,any two of its elements generate a subalgebra which is a Lie algebra. Any binary-Lie algebra generates, by means of the Campbell- Hausdorff formula, a diassociativelocal loop. Thus we get a diassociative smooth loops – binary-Lie algebras theorygeneralizing the Lie groups – Lie algebras theory. iscrete mechanics on unitary octonions Moufang loop if it satisfiesany of the three following equivalent conditions [18](( ax ) a ) y = a ( x ( ay )) , (( xa ) y ) a = x ( a ( ya )) , ( ax )( ya ) = ( a ( xy )) a. The tangent algebra of a smooth Moufang loop is a Mal’cev algebra which is abinary-Lie algebra satisfying [33][[
X, Y ] , [ X, Z ]] = [[[
X, Y ] , Z ] , X ] + [[[ Y, Z ] , X ] , X ] + [[[ Z, X ] , X ] , Y ] , for every X, Y, Z . There is again a sort of Lie’s Third Theorem for smooth Moufangloops and Mal’cev algebras [21, 25, 33].The following is well known.
Theorem 1.
Invertible octonions O × form a smooth inverse Moufang loop underthe octonion multiplication.. It is also well known that the tangent and cotangent bundles of a Lie group areLie groupoids themselves, the tangent bundle T G is a Lie group with the unit ( e, T ∗ G is a smooth groupoid over g ∗ = AG [24, 42]. Themultiplication relation in the first case is the tangent bundle T m ⊂ T G × T G × T G of the multiplication relation m ⊂ G × G × G in G and for T ∗ G it is the annihilator( T m ) ⊂ T ∗ G × T ∗ G × T ∗ G , where the pairing with the third T ∗ G we take with theminus sign. This can be easily extended to smooth loops. Theorem 2.
The tangent bundle T G of a smooth loop G is a smooth loop underthe multiplication D ( g,h ) m ( X g , Y h ) = D g ( r h )( X g ) + D h ( l g )( Y h ) , for X g ∈ T g G and Y h ∈ T h G . However, as we will see later, for the cotangent bundle of a smooth loop themultiplication ( T m ) ⊂ T ∗ G × T ∗ G × T ∗ G gives in general not a smooth loop. Let us first recall the discrete Lagrangian mechanics on Lie groups. A discreteLagrangian system consists of a Lie group G and a smooth, real-valued function L on G . We define a function ( g, h ) → L ( g ) + L ( h ) of elements g, h ∈ G . Asolution of the Lagrange equations for the Lagrangian function L is a sequence g , g , g , ... of elements G such that ( g i , g i +1 ) are the stationary points of the function L ( g i ) + L ( g i +1 ) for every i .Discrete Lagrangian systems on Lie groups can be based on variational principlesas follows. The variational principle for a Lie group G with Lie algebra g is basedon a set of sequences C Ng = { ( g , g , ..., g N ) ∈ G N | g g · · · g N = g ∈ G } . Take a tangent vector at ( g , g ..., g N ) which can be understood as the tangentvector of a curve c ( t ) ∈ C Ng passing through ( g , g ..., g N ) at t = 0. It is easy to seethe following. Janusz Grabowski & Zohreh Ravanpak
Lemma 1.
In a Lie group G we have g i g i +1 = g ′ i g ′ i +1 if and only if there is h ∈ G such that g ′ i = g i h and g ′ i +1 = h − g i +1 . By Lemma 1 the curve c ( t ) is necessarily of the form c ( t ) = ( g γ ( t ) , ( γ ( t )) − g γ ( t ) , ..., ( γ N − ( t )) − g N − γ N − ( t ) , ( γ N − ( t )) − g N ) , (6)such that γ i : t ∈ ( − ǫ, ǫ ) ⊆ R → G are the integral curves of the left invariant vectorfield corresponding to X i ∈ T e G that passes through the identity, that is γ i (0) = e .Therefore the tangent space of C Ng at ( g , . . . , g N ) can be identified with T g ,...,g N C Ng = { ( X , X , ..., X N − ) ∈ g N − | X i ∈ T e G ∼ = g } . The curve c is called a variation of ( g , g ..., g N ) and ( X , X ..., X N − ) is called infinitesimal variational of ( g , g ..., g N ). Now, we define the discrete action sumassociated to the Lagrangian L S L = N X k =1 L ( g k ) . According to the Hamilton’s principle of critical action, the sequence ( g , g ..., g N )is a solution of the Lagrangian system if and only if ( g , g ..., g N ) is a critical pointof S L . Therefore, we calculate ddt | t =0 S L ( c ( t )) = N − X k =1 h ←− X k ( g k )( L ) − −→ X k ( g k +1 )( L ) i = 0 , where X k ∈ g . These equations are called to be discrete Euler-Lagrange equations .In the category of smooth loops, because of the lack of associativity there isno variant of Lemma 1 so not clear variations like (6). But still we can define thediscrete Euler-Lagrange equations using the smooth prolongation of vector fields asfollows. Definition 1.
The d iscrete Euler-Lagrange equations for a discrete Lagrangiansystem on a smooth loop G with Lagrangian L : G → R is given by equations ←− X ( L )( g i ) − −→ X ( L )( g i +1 ) = 0 (7)for every X ∈ T e G , where ←− X and −→ X are the left and right prolongation, respectively.A sequence g , g , ... of elements G is a solution of the Euler-Lagrange equationsif g i and g i +1 satisfy (7) for i = 1 , , . . . .Let γ L : G → G be a smooth map on a smooth loop G for which the couples( g, γ ( g )) are solutions of Euler-Lagrange equations for L . The map γ L : G → G iscalled a discrete flow or discrete Lagrangian evolution operator for L .We have the discrete Legendre transformation for smooth loops similar to whatwe have for Lie groups [27]. Given a Lagrangian L : G → R on smooth loop G withthe skew-algebra g , two discrete Legendre transformations F + L = l ∗ g ◦ dL : G → g ∗ and F − L = r ∗ g ◦ dL : G → g ∗ , where dL : G → T ∗ G , are as follows F + L ( g )( X ) = ←− X ( L )( g ) , F − L ( g )( X ) = −→ X ( L )( g ) , for X ∈ g . Clearly, l ∗ and r ∗ are the pull backs of left and right translations. Directlyfrom the definitions we get the following iscrete mechanics on unitary octonions Proposition 1. γ L : G → G is the discrete flow for the Lagrangian L : G → R ifand only if F − L ◦ γ L = F + L . (8)
Theorem 3.
For an inverse smooth loop G the Legendre map F + L is regular at g if and only if F − L is regular at g − . Moreover, F + L is a diffeomorphism if and onlyif F − L is a diffeomorphism.Proof. According to (5), ι ∗ −→ X ( g ) = ←− X ( g − ) and ι is a diffeomorphism, so they aresimultaneously local diffeomorphisms and injective. Definition 2.
A discrete Lagrangian L : G → R on smooth loop G is said to be regular if and only if the Legendre transformation F + L is a local diffeomorphism. If F + L is global diffeomorphism, L is called to be hyperregular . Theorem 4.
For an inverse smooth loop G the following are equivalent: • A discrete Lagrangian L : G → R on smooth loop G is to be regular; • F − L is a local diffeomorphism;Moreover, L is hyperregular if and only if F − L is a global diffeomorphism. In thiscase the discrete Lagrangian evolution operator is a diffeomorphism.Proof. This follows directly from Theorem 3 and Proposition 1.
In the category of Lie group, the cotangent bundle of a Lie group is a symplecticgroupoid over the dual of the tangent algebra. The Hamiltonian formalism fordiscrete Lagrangian systems on the Lie group G with Lagrangian L : G → R isbased on the fact that L generates a Lagrangian submanifod dL ( G ) ⊂ T ∗ G ofthe cotangent groupoid which, under a hypothesis of non-degeneracy, determines aPoisson map from g ∗ to itself. In this case, if g and h are solutions of the discreteEuler-Lagrange equations for the regular Lagrangian L , (see [26]) then there existtwo open subsets U g and U h of G and a discrete Lagrangian evolution operator γ L : U g → U h such that γ L ( g ′ ) = h ′ whenever ( g ′ , h ′ ) satisfy the corresponding Euler-Lagrange equations, and γ is a unique such diffeomorphism. If L is hyperregular,then γ L = ( F − L ) − ◦ F + L . For a hyperregular Lagrangian function L : G → R ,pushing forward to g ∗ with the discrete Legendre transformations gives the discreteHamiltonian evolution operator ˜ γ L : g ∗ → g ∗ given by˜ γL = F + L ◦ ( F − L ) − . Let now G be a smooth loop with the skew-algebra g and the dual g ∗ . There aretwo projections α, β : T ∗ G → g ∗ such that h β ( µ g ) , X i = h µ g , D e ( l g )( X ) i , for µ g ∈ T ∗ g G and X ∈ g , h α ( ν h ) , Y i = h ν h , D e ( r h )( Y ) i , for ν h ∈ T ∗ h G and Y ∈ g . (9)In other words, h β ( µ g ) , X i = D µ g , ←− X ( g ) E , h α ( ν h ) , X i = D ν h , −→ X ( h ) E . (10) Janusz Grabowski & Zohreh Ravanpak
Now, we consider a discrete Lagrangian function L : G → R on the smooth loop G .The cotangent bundle of T ∗ G is equipped with a canonical symplectic structure butthe lack of associativity is an obstacle for defining a natural loop structure on T ∗ G analogous to the Lie group. In other words in general there is no natural partialmultiplication (’loopoid structure’) on T ∗ G . Setting aside the ’loopoid structure’,for any function L : G → R on manifold G the submanifolds dL ( G ) ⊂ T ∗ G isa Lagrangian submanifold of the cotangent bundle. The discrete Euler-Lagrangedynamics can be equivalently described as follows. Definition 3.
Let G be a smooth loop and L a discrete Lagrangian function on it.A sequence µ , ..., µ n ∈ T ∗ G satisfies the discrete Lagrangian dynamics if µ , ..., µ n ∈ dL ( G ) and they are composable sequence in T ∗ G , that is β ( µ k ) = α ( µ k +1 ) , k = 1 , ..., n − . Theorem 5.
Let G be a smooth loop equipped with a discrete Lagrangian L : G → R . Then a sequence µ , ..., µ n ∈ T ∗ G satisfies the discrete Lagrangian dynamics of dL ( G ) ⊂ T ∗ G if and only if µ k = dL ( g k ) for some g k ∈ G, k = 1 , . . . , n, and the discrete Euler -Lagrangian equations ←− X ( L )( g k ) = −→ X ( L )( g k +1 ) are satisfied, k = 1 , . . . , n − .Proof. It is enough to consider the discrete Legendre transforms of L as F + L = β ◦ dL, F − L = α ◦ dL : G → g ∗ . For more details confront [30].We see that the sequence µ , ..., µ n ∈ T ∗ G of composable pairs satisfies thediscrete Lagrangian dynamics if and only if we have the relation ( µ k , µ k +1 ) ∈ dL ( G ) × dL ( G ), for each pairs of successive elements.Now, if the restricted map α : dL ( G ) → g ∗ is a (local) diffeomorphism, thenthe relation ( µ k , µ k +1 ) ∈ dL ( G ) × dL ( G ) is the graph of an explicit flow map µ k → µ k +1 given by the composition ( α | dL ( G ) ) − ◦ β | dL ( G ) . If the restricted map β : dL ( G ) → g ∗ is also a local diffeomorphism, then the flow is locally reversible withthe inverse ( β | dL ( G ) ) − ◦ α | dL ( G ) . When both the restricted maps α and β arediffeomorphism we say dL ( G ) is a Lagrangian bisection because it is simultaneouslya section of α and β . Obviously, the restricted maps α and β correspond precisely tothe discrete Legendre transforms F ± and the local bisection condition correspondsto regularity of discrete Lagrangian L . Therefore, if L is regular, then the discreteLagrangian flow map ( α | dL ( G ) ) − ◦ β | dL ( G ) is a local diffeomorphism on dL ( G ), thenthe discrete Hamiltonian flow map is given by reversing the order of composition β | dL ( G ) ◦ ( α | dL ( G ) ) − which is a local diffeomorphism on g ∗ . The octonions O are the noncommutative non-associative algebra which is one ofthe four division algebras that exist over the real numbers. The most elementaryway to construct the octonions is to give their multiplication table. Every octonion iscrete mechanics on unitary octonions { e , e , · , · , · , e } where e = 1 representsthe identity element and the imaginary octonion units e i , { i = 1 , ..., } satisfy themultiplication rule e i e j = δ ji + f ijk e k , where δ ji is the Kronecker’s delta and f ijk ’s arecompletely anti-symmetric structure constants which read as [37] f = f = f = f = f = f = f = 1 . The multiplication is subject to the relations ∀ i = 0 [ e i = − , e i e j = − e j e i , for i = j = 0 . and the following multiplication table. Multiplication table e i e j e e e e e e e e e e e e e e e e e e e − e e − e e − e − e e e e − e − e e e e − e − e e e e − e − e e − e e − e e e − e − e − e − e e e e e e e − e e − e − e − e e e e e e − e − e e − e − e e e − e e e − e − e e − e The associator [ g, h, k ] = ( gh ) k − g ( hk ) of three octonions does not vanish ingeneral but octonions satisfy a weak form of associativity known as alternativity,namely [ g, h, g ] = 0. The reason is that, two octonions determine a quaternionicsubalgebra of the octonions, so that any product containing only two octonionicdirections is associative (diassociativity).The octonions are a generalization of the complex numbers, with seven imaginaryunits, so octonionic conjugation is given by reversing the sign of the imaginary basisunits. Conjugation is an involution of O satisfying ( gh ) ∗ = h ∗ g ∗ . The inner producton O is inherited from R and can be rewritten h g, h i = ( gh ∗ + hg ∗ )2 = ( h ∗ g + g ∗ h )2 ∈ R , (11)and the norm of an octonion is just k g k = gg ∗ which satisfies the defining propertyof a normed division algebra, namely k gh k = k g kk h k . The scalar product is invariantwith respect to the multiplication: h ag, ah i = h g, h i for a = 0.Every nonzero octonion g ∈ O has an inverse g − = g ∗ k g k , such that gg − = g − g = 1 , (12)which makes the set of invertible octonions to be an inverse loop with respect to theoctonion multiplication. We remark that the inverse is a genuine one, i.e., g ( g − h ) = g − ( gh ) = h, ∀ g, h ∈ O , Janusz Grabowski & Zohreh Ravanpak which is stronger than the standard property (12) for non-associative algebra.Actually, the set O × of invertible octonions is a smooth Moufang loop underoctonion multiplication.One may represent an octonion as a pair of quaternions Q , then multiplicationcan be defined by( a, b ) · ( c, d ) = ( ac − d ∗ b, da + bc ∗ ) , for a, b, c, d ∈ Q , where the involution, addition and multiplication are those in quaternions. In thiscase the inverse of ( a, b ) is given by( a, b ) − = ( a, b ) ∗ k a k + k b k , where ( a, b ) ∗ = ( a, − b ∗ ) and the norm is in quaternions. Example 1.
The set of all automorphisms of the algebra O , that is the set ofinvertible linear transformations A ∈ Aut ( O ), forms a Lie group called G which isthe smallest of the exceptional Lie groups. We will show that the semidirect product O × ⋉ G is an inverse loop under the multiplication( g, A ) • ( h, B ) = ( gA ( h ) , A ◦ B ) , with identity (1 , Id) and inverse ( g, A ) − = ( A − ( g − ) , A − ). The thing which needsto be checked is the following inverse property,( g, A ) − • (( g, A ) • ( h, B )) = ( A − ( g − ) , A − ) • ( g · A ( h ) , A ◦ B )= ( A − ( g − ) · A − ( g · A ( h )) , B ) = ( h, B ) . Here we use the fact that A − ( g − · g ) = A − ( g − ) · A − ( g ) = 1. Similarly,(( g, A ) • ( h, B )) • ( h, B ) − . Of course, because O × are not associative the above smooth loop is not a Lie group. Example 2.
The manifold of unitary octonions O = { a ∈ O , k a k = 1 } is closed under the octonion multiplication and therefore forms a Moufang loop.The manifold O is diffeomorphic to seven-sphere S , the only paralellizable spherewhich does not carry a Lie group structure. To find the tangent algebra of O ,consider the tangent space o = T e O = span { e , ..., e } to O inside the vector space O . Then, the tangent bundle T O is given by the left(or right) prolongation of imaginary octonions, that is T O = span {←− e , ..., ←− e } ,where ←− e i ( a ) = D e ( l a )( e i ) = ae i ∈ T a O , a ∈ O . Similarly −→ e i ( a ) = D e ( r a )( e i ) = e i a ∈ T a O , a ∈ O . iscrete mechanics on unitary octonions ←− e i , ←− e j ]( a ) = ddt | t =0 ←− e j ( a + tae i ) − ddt | t =0 ←− e i ( a + tae j )= ddt | t =0 ( a + tae i ) e j − ddt | t =0 ( a + tae j ) e i = ( ae i ) e j − ( ae j ) e i = ←−−−−−−−−−−−−−−− a − (( ae i ) e j − ( ae j ) e i )( a ) . In particular [ ←− e i , ←− e j ]( e ) = 2 e i e j . Thus, ( o , [ e i , e j ] = 2 e i e j ) is the skew-algebra(Mal’cev algebra) corresponding to the smooth loop O . The corresponding Leibnizstructure is Λ = X i,j =1 e i e j ∂ e i ∧ ∂ e j . This agrees with the Campbell-Hausdorff formula. The exponential map exp : o → O is exp( e ) = cos( k e k ) + sin( k e k ) e/ k e k . (13)Hence,exp( te i ) exp( te j ) = cos ( t ) + sin( t ) cos( t )( e i + e j ) + sin ( t ) e i e j = exp( e ) . If i = j , then, in view of (13), e = arcsin( p − cos ( t )) p − cos ( t ) (cid:0) sin( t ) cos( t )( e i + e j ) + sin ( t ) e i e j (cid:1) = (1 + o ( t )) (cid:0) t ( e i + e j ) + t e i e j + o ( t ) (cid:1) = (cid:0) t ( e i + e j ) + t e i e j + o ( t ) (cid:1) . Since the Campbell-Hausdorff formula reads te i ∗ tej = t ( e i + e j ) + ( t / e , e ] + o ( t ) , we get again [ e , e ] = 2 e i e j . It is also easily seen that ←− e , . . . , ←− e do not form a Lie algebra over R (i.e., O is not a Lie group).Similarly we obtain [ −→ e i , −→ e j ]( a ) = e j ( e i a ) − e i ( e j a )so that [ e i , e j ] l = − [ e i , e j ] r = 2 e i e j . Note that[ −→ e i , ←− e j ]( a ) = e j ( ae i ) − ( e j a ) e i which is 0 at a = e , but generally not 0 (the left prolongations do not commutewith the right prolongations) due to non-associativity of O . Example 3.
Consider the cotangent bundle T ∗ O of unit octonions. Due to thescalar product (11), T ∗ O = T O as vector bundles. According to the general rulefor the Lie groupoid T ∗ G ⇒ g ∗ in case of a Lie group G , we define the source andtarget projections α, β : T ∗ G → o ∗ where o ∗ is the dual of o : h β ( µ g ) , X i = h µ g , gX i , h α ( ν h ) , X i = h ν h , Xh i . Janusz Grabowski & Zohreh Ravanpak
Here we use a self-explaining notation ←− X g = gX , −→ X h = Xh and we interpret thetangent and cotangent vectors to O as octonions. In this sense the above pairingscan be understood as the scalar products.Two elements µ g ∈ T ∗ g O and ν h ∈ T ∗ h O are composable , i.e., β ( µ g ) = α ( ν h ), if h β ( µ g ) , X i = h α ( ν h ) , X i ⇔ h µ g , gX i = h ν h , Xh i for all X ∈ o . The above holds if there exists an element σ ∈ o ∗ = o such that µ g = D ∗ g ( l g − )( σ ) = gσ and ν h = D ∗ h ( r h − )( σ ) = σh. Let us try to define the product in T ∗ O like it is done for Lie groups. Themultiplication µ g • ν h ∈ T ∗ gh O is then defined by the equation h µ g , X g i + h ν h , Y h i = h µ g • ν h , X g • Y h i , (14)where X g • Y h is the multiplication in the tangent loop as described in Theorem 2.Denote θ gh = µ g • ν h . Using Theorem 2 we can rewrite (14) as h µ g , X g i + h ν h , Y h i = h θ gh , X g h + gY h i . (15)Since X g , Y h are arbitrary, so we get out of (15) a system of equations h µ g , X g i = h θ gh , X g h i , h ν h , Y h i = h θ gh , gY h i . Using the fact that the scalar product on O is invariant with respect to the multi-plication, we get in turn h µ g , X g i = h θ gh h − , X g i , h ν h , Y h i = h g − θ gh , Y h i . Hence, µ g = gσ = θ gh h − and ν h = σh = g − θ gh , so that σ = g − ( θ gh h − ) = ( g − θ gh ) h − . In this way we get the commutativity of the right translation by h − and left trans-lation by g − when acting on θ gh which can be taken arbitrary octonion orthogonalto gh . This is not satisfied in octonions as O is non-associative. This implies thatthe standard way of defining the multiplication in T ∗ O does not give a well-definedproduct. The algebra of octonions O is spanned by { e = 1 } -the unit and 7 additional unitaryelements { e , e , . . . , e } , e i = − e i e j = − e j e i for i = j . The algebra is non-commutative and non-associative (e.g. ( e e ) e = e e = e = e ( e e ) = e e = − e ). In this section we will construct the discrete mechanics on the manifold ofunit octonions O = { a ∈ O , k a k = 1 } , which is an inverse smooth loop under the octonion multiplication. Let L : O → R be a Lagrangian function, then the discrete Euler-Lagrange equations read asrecurrence equation ←− e i ( L )( a n ) = −→ e i ( L )( a n +1 ) , iscrete mechanics on unitary octonions ←− e i ( a ) = D e ( l a )( e i ) = ae i and −→ e i ( a ) = D e ( r a )( e i ) = e i a are the left andthe right prolongation by the element a ∈ O . A solution for those equations is asequence of elements O .If we take the Lagrangian as a linear function, for instance take L = e = h e , ·i defined by the inner product (11), then ←− e i ( L )( a n ) = ( a n e i )( L )( a n ) = h e , a n e i i . The right-hand side of the above relation is obtained by taking the integral curve γ ( t ) = a n + ta n e i for the tangent vector a n e i and then we have ddt | t =0 L ( a n + ta n e i ) = ddt | t =0 h e , a n + ta n e i i . Note that we interpret the tangent vector at the points of O as an element ofoctonions.Therefore, by the definition the Euler-Lagrange equations are h e , a n e i − e i a n +1 i = 0 , for i = 1 , ..., . Every element a n ∈ O can be written as a n = α n + α kn e k such that P s =0 | α sn | = 1,so the above equations turn to X k =1 (cid:10) e , α n e i − α n +1 e i + ( α kn + α kn +1 ) e k e i (cid:11) = 0 , for i = 1 , ..., . (16)Now, if i = 1, since h e , e i = 1 and h e , e k e i = 0 for k = 0, we get α n − α n +1 = 0.If i >
1, the two first expressions of (16) are zero because h e , e i i = 0 for i = 1and thus we left by the third expression, that is X k =1 (cid:10) e , ( α kn + α kn +1 ) e k e i (cid:11) = 0 , for i = 1 , ..., . But for each k , there is some i such that e k e i = ± e and all i ′ s are used. Conse-quently, we get the Euler-Lagrange equations α n − α n +1 = 0 , α kn + α kn +1 = 0 , k = 1 , ..., . It is obvious from the equations that the solution of Euler-Lagrange equations arejust the conjugate pairs in O .Next step is to check whether the Lagrangian L is regular or hyperregular. So,we would need to find the Legendre maps associated with L . Consider the tangentskew-algebra o and the its dual o ∗ with the basis { e , ..., e } . The correspondingLegendre maps F + L, F − L : O → o ∗ are F + L ( a ) = X i =1 ←− e i ( a )( L ) e i , F − L ( a ) = X i =1 −→ e i ( a )( L ) e i , a ∈ O . Let us remark that there is no hyperregular Lagrangian on unit octonions O , be-cause the Legendre maps are F + L, F − L : S → R which cannot be diffeomorphisms.Thus we can only find Lagrangians which are (locally) regular.4 Janusz Grabowski & Zohreh Ravanpak
Consider the linear Lagrangian L = h e , ·i = e . We have −→ e i ( a )( e ) = h e , e i a i and corresponding Legendre map F − L ( a ) = X i =1 −→ e i ( e )( a ) e i = X i =1 h e , e i a i e i , a ∈ O . The Lagrangian L = e is not regular at e because F − L ( e ) = X i =1 h e , e i i e i = e and D e ( F − L )( e ) = ddt | t =0 F − L ( e + te ) = X i =1 h e , e i e i e = 0 . If we take the Lagrangian L = ( e ) −→ e i ( a )( L ) = e ( a ) −→ e i ( a )( e ) = h e , a i h e , e i a i and F − L ( a ) = P i =1 h e , a i h e , e i a i e i . We have F − L ( e s ) = 0 for every s = 0 , ..., L is not regular in a neighbourhood of e . Indeed, D e ( F − L )( e s ) = ddt | t =0 F − L ( e + te s ) = X i =1 h e , e s ih e , e i i e i = 0 , for s = 1 . The discrete Euler-Lagrange equation is h e , a n i h e , e i a n i = h e , a n +1 i h e , a n +1 e i i , i = 1 , . . . , . If we write a n = P s =0 α sn e s and a n +1 = P s =0 α sn +1 e s , then this reduces to thequadratic recurrence equation α n α n = α n +1 α n +1 , and α n α jn = − α n +1 α jn +1 , for j = 2 , . . . . Now, we take the Lagrangian L = X k =1 m k ( e k ) , (17)where m k >
0, as the ‘total kinetic energy’ of the system. Then −→ e i ( a )( L ) = X k =1 m k e k ( a ) −→ e i ( a )( e k ) = X k =1 h m k e k , a i h e k , e i a i , and ←− e i ( a )( L ) = X k =1 m k e k ( a ) ←− e i ( a )( e k ) = X k =1 h m k e k , a i h e k , ae i i , iscrete mechanics on unitary octonions F − L ( a ) = X i,k =1 h m k e k , a i h e k , e i a i e i . (18)and F + L ( a ) = X i,k =1 h m k e k , a i h e k , ae i i e i . (19)The discrete Euler-Lagrange equation is X k =1 h m k e k , a n i h e k , e i a n i = X k =1 h m k e k , a n +1 i h e k , a n +1 e i i , i = 1 , . . . , . If we write a n = P s =0 α sn e s and a n +1 = P s =0 α sn +1 e s , then this reduces to thequadratic recurrence equation α in α n = α in +1 α n +1 , i = 1 , ..., m k . Since P i =0 ( α in ) = 1, we have X i =1 (cid:0) α in α n (cid:1) = ( α n ) (cid:0) − ( α n ) (cid:1) = ( α n +1 ) (cid:0) − ( α n +1 ) (cid:1) (20)which have non-trivial different solutions giving rise to non-trivial solutions of theEuler-Lagrange equation. If, however, α n and α n +1 are close to 1, thus a n and a n +1 are close to e , then α n = α n +1 , since the function f ( x ) = x (1 − x ) is monotonicon the interval [1 / √ , a n = a n +1 . Theorem 6.
The discrete Euler-Lagrange equation for the Lagrangian (17) on thesmooth loop O admits in a neighbourhood of e only trivial solutions. We easily see see that F − L ( e s ) = 0 for all s ∈ { , ..., } , hence, changing the baseof imaginary octonions, we infer that F − L ( a ) = 0 for any imaginary octonion, thatsupports once more the fact that the Lagrangian is not hyperregular. We have also D e F − L ( e s ) = X i,k =1 h m k e k , e s i h e k , e i i e i = m s e s = D e F + L ( e s ) . Under identification T e O = o = o ∗ the differential of F − L : O → o at e , andsimilarly for F + L : O → o , can be identified with the diagonal automorphism on o for which e s m s e s . In particular F − L and F + L are regular in a neighbourhood of e . According to Theorem 6 and (8), F − L = F + L in a neighbourhood of e Actually, F − L ( a ) = F + L ( a ) for all a ∈ O , because they are quadratic (thus analytic) in a .Hence, the discrete Lagrangian evolution operator in a neighbourhood U of e is γ L = ( F − L ) − ◦ F + L = Id U and the local discrete Hamiltonian operator in a neighbourhood V of 0 in o ∗ reads˜ γ L = F + L ◦ ( F − L ) − = Id V . Janusz Grabowski & Zohreh Ravanpak
Note however that there are nontrivial solutions lying outside the neighbourhoodof e . For instance, ( a n , a n +1 ) = (cid:0) (0 , a ′ n ) , (0 , a ′ n +1 ) (cid:1) , where a ′ n and a ′ n +1 representarbitrary imaginary and unitary octonions. We can also take A = α n = α n +1 = B , | A | < | B | <
1, which are different solutions of (20). Note that in this case B isnot close to 1, as | A | < / √
2. Then, for any a n = ( A, a ′ n ), where a ′ n representsan imaginary octonion of length √ − A , the pair (( A, a ′ n ) , ( B, ua ′ n )), with u =(1 − B ) / (1 − A ), is a solution of the Euler-Lagrange equation. Acknowledgements
J. Grabowski acknowledges that his research was funded by the Polish National Sci-ence Centre grant HARMONIA under the contract number 2016/22/M/ST1/00542.
We have shown how the discrete Lagrangian and Hamiltonian formalism on Liegroupoids can be extended to non-associative objects like smooth inverse loops. Theworking example was the inverse loop of unitary octonions O on which we defineda ‘linear’ Lagrangian for which solutions of the discrete Euler-Lagrange equationsare pairs of conjugate octonions. We also gave an example of a regular Lagrangianand discuss the corresponding Euler-Lagrange equations.Since the discrete mechanics has its version on Lie groupoids (with fundamentalfor our paper works [26, 43]), a natural question is to find the non-associative gen-eralizations of Lie groupoids and to construct discrete mechanics on them. The firstideas of such objects, smooth loopoids , can be found in [10, 19]. We will discuss thisproblem in a separate paper. References [1] R. Baer,
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