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Article Lie Group Statistics and Lie Group Machine Learning based on Souriau Lie Groups Thermodynamics & Koszul-Souriau-Fisher Metric: New Entropy Definition as Generalized Casimir Invariant Function in Coadjoint Representation
Frédéric Barbaresco 1
1 Thales Land & Air Systems; [email protected]
Abstract: In 1969, Jean-Marie Souriau introduced a “Lie Groups Thermodynamics” in Statistical Mechanics in the framework of Geometric Mechanics. This Souriau’s model considers the statistical mechanics of dynamic systems in their "space of evolution" associated to a homogeneous symplectic manifold by a Lagrange 2-form, and defines in case of non null cohomology (non equivariance of the coadjoint action on the moment map with appearance of an additional cocyle) a Gibbs density (of maximum entropy) that is covariant under the action of dynamic groups of physics (eg, Galileo's group in classical physics). Souriau Lie Group Thermodynamics was also addressed 30 years after Souriau by R. F. Streater in the framework of Quantum Physics by Information Geometry for some Lie algebras, but only in the case of null cohomology. Souriau method could then be applied on Lie Groups to define a covariant maximum entropy density by Kirillov representation theory. We will illustrate this method for homogeneous Siegel domains and more especially for Poincaré unit disk by considering SU(1,1) group coadjoint orbit and by using its Souriau’s moment map. For this case, the coadjoint action on moment map is equivariant. For non-null cohomology, we give the case of Lie group SE(2). Finally, we will propose a new geometric definition of Entropy that could be built as a generalized Casimir invariant function in coadjoint representation, and Massieu characteristic function, dual of Entropy by Legendre transform, as a generalized Casimir invariant function in adjoint representation, where Souriau cocycle is a measure of the lack of equivariance of the moment mapping.
Keywords: Lie Groups Thermodynamics, Lie Group Machine Learning, Kirillov Representation Theory, Coadjoint Orbits, Moment Map, Covariant Gibbs Density, Maximum Entropy Density, Souriau-Fisher Metric, Generalized Casimir Invariant Function.
La thèse de Kirillov, parue en 1962, a suscité immédiatement beaucoup d’intérêt…En outre, quantité de notions naturelles concernant les représentations s’interprètent géométriquement en terme d’orbites coadjointes : restriction à un sous-groupe, induction unitaire, produit tensoriel, mesure de Plancherel, la topologie de l’ensemble représentations unitaires irréductibles... Kirillov s’est vite convaincu, et il a convaincu la communauté mathématique que cette « méthode des orbites » devait être applicable à des groupes bien plus généraux que les groupes nilpotents. Il n’a pas hésité à aborder le cas des groupes de Lie connexes quelconques. Evidemment, des difficultés considérables ont surgi immédiatement. Néanmoins, Kirillov a indiqué une voie d’accès, qui ensuite a été largement utilisée. Jacques Dixmier, Brèves remarques sur l’œuvre de A.A. Kirillov
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On comprend ainsi comment Lagrange a pu développer les lois de la Mécanique des systèmes formés de solides sans s’occuper des variations de la température de ces corps et Fourier traiter des variations de la température de ces mêmes corps solides sans s’occuper de leur mouvement ; comment on peut étudier le mouvement de la Terre, assimilée à un solide rigide, sans se préoccuper de la température de cet astre et étudier le refroidissement du globe terrestre sans se préoccuper de son mouvement. Une telle indépendance entre les problèmes qui ressortissent à la Mécanique et les problèmes qui ressortissent à la Théorie de la chaleur n’existe plus lorsque les systèmes auxquels on a affaire ne sont plus des systèmes classiques ; si, par exemple, au lieu de regarder la Terre comme un solide rigide, d’état invariable, on tient compte des changements de volume, de forme, d’état physique et chimique qui accompagnent son refroidissement, on ne peut plus séparer le problème du mouvement de la Terre et le problème du refroidissement terrestre. … On sait que cette forme de relations supplémentaires avait été introduite par Newton et les géomètres du XVIIIème siècle dans la théorie du son. Ces considérations montrent que les questions qui ressortissent à la Thermodynamique ont dû solliciter l’attention des physiciens dès qu’on a voulu aborder l’étude des systèmes autres que des systèmes classiques ; et, en fait, c’est la théorie de la propagation du son dans l’air qui a provoqué Laplace à créer la Thermodynamique - P. Duhem, L'intégrale des forces vives en thermodynamique, JMPA 4:5-19, 1898 [1-4]
Sous cette aspiration, la physique qui était d’abord une science des “agents” doit devenir une science des “milieux”. C’est en s’adressant à des milieux nouveaux que l’on peut espérer pousser la diversification et l’analyse des phénomènes jusqu’à en provoquer la géométrisation fine et complexe, vraiment intrinsèque…Sans doute, la réalité ne nous a pas encore livré tous ses modèles, mais nous savons déjà qu’elle ne peut en posséder un plus grand nombre que celui qui lui est assigné par la théorie mathématique des groupes. Gaston Bachelard, Etude sur l’Evolution d’un problème de Physique –La propagation thermique dans les solides, 1928
The classical simple gradient descent used in Deep Learning has two drawbacks: the use of the same non-adaptive learning rate for all parameter components, and a non-invariance with respect to parameter re- encoding inducing different learning rates. As the parameter space of multilayer networks forms a Riemannian space equipped with Fisher information metric, instead of the usual gradient descent method, the natural gradient or Riemannian gradient method, which takes account of the geometric structure of the Riemannian space, is more effective for learning. The natural gradient preserves this invariance to be insensitive to the characteristic scale of each parameter direction. The Fisher metric defines a Riemannian metric as the Hessian of two dual potential functions (the Entropy and the Massieu Characteristic Function). In Souriau’s Lie groups thermodynamics, the invariance by re-parameterization in information geometry has been replaced by invariance with respect to the action of the group. In Souriau model, under the action of the group, the entropy and the Fisher metric are invariant. Souriau defined a Gibbs density that is covariant under the action of the group. The study of exponential densities invariant by a group goes back to the work of Muriel Casalis in her 1990 thesis. The general problem was solved for Lie groups by Jean-Marie Souriau in Geometric Mechanics in 1969, by defining a "Lie groups Thermodynamics" in Statistical Mechanics. These new tools are bedrocks for Lie Group Statistics and Lie Group Machine Learning. Souriau introduced a Riemannian metric, linked to a generalization of the Fisher metric for homogeneous Symplectic manifolds. This model considers the KKS 2-form (Kostant-Kirillov-Souriau) defined on the coadjoint orbits of the Lie group in the case of non-null cohomology, with the introduction of a Symplectic cocycle, called "Souriau's cocycle", characterizing the non-equivariance of the coadjoint action (action of the Lie group on the moment map). We can observe that Souriau Entropy SQ( ) defined on coadjoint orbit of the group is invariant # with Souriau affine definition of coadjoint action #* where S( Adg () Q) = S( Q) Adgg() Q= Ad () Q +θ ( g ) θ ( g ) is called Souriau cocyle. Based on Souriau Lie groups Thermodynamics, we will propose a new geometric definition of Entropy that could be built as a generalized Casimir invariant function in coadjoint representation, and Massieu characteristic function, dual of Entropy by Legendre transform, as a generalized Casimir function in adjoint representation. This geometric structure of Entropy is a foundation for a new geometric theory of information, where the Entropy is no longer defined axiomatically as Shannon
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or von Neumann Entropies, but built as Casimir Invariant function in coadjoint representation and solution to the Casimir equation extended **∂S k , with Souriau cocycle ad∂ Q +Θ =C adi Q +Θ =0 S ij ∂S k j ∂Q ∂Q j j ∂Q where in θ ( g ) Θ=()X Teθθ( Xe ()) with Θ( XY ,) =Θ (), X Y = J[XY, ] −{ JXY, J} =− d (), X Y , XY,∈g case of non null cohomology. The dual Lie algebra foliates into coadjoint orbits that are also the level sets on the entropy. The KKS (Kostant- Kirillov Souriau) 2-form, and the Souriau-Koszul-Fisher metric make each orbit into homogeneous Symplectic manifold. The information manifold foliates into level sets of the entropy that could be interpreted in Thermodynamics: motion remaining on this complex surfaces is non-dissipative, whereas motion transversal to dQ ∂H these surfaces is dissipative, where the dynamic is given by: = =* +Θ with stable {Q, H}Θ ad∂H Q dt ∂Q ∂Q dQ ∂S equilibrium given when = ⇒ = =* +Θ = . H S{ Q,0 S}Θ ad∂S Q dt ∂Q ∂Q We will introduce the link between Koszul geometry of homogeneous bounded domains, Souriau "Lie Groups Thermodynamics", Information Geometry and Kirillov representation theory [12] to define probability densities as Souriau covariant Gibbs densities (density of Maximum of Entropy). We will illustrate this case for the matrix Lie group SU(1,1) (case with null cohomology) through the computation of Souriau’s moment map, and Kirillov's orbit method. We will also indicate application for SE(2) Lie group (case with non-null cohomology) where a Souriau cocycle should be take into account for the default of equivariance of the coadjoint action on moment map.
0. Historical Preamble
Lie groups Thermodynamics is a “knot of high denisity” between different disciplines and make links between different branches of statistical sciences from statistical physics, to Information Geometry and theory of inference. We give in the following the main contributions and tools that are covered by this topic: o Statistical Physics, Massieu & Poincaré Characteristic Functions • 1724, Alexis Claude Clairaut introduced Clairaut Equation. In 1787, Adrien-Marie Legendre introduced Legendre transform to solve a minimal surface problem given by Gaspard Monge • In 1869, François Massieu introduced in Thermodynamics dual (Massieu) Characteristic Functions, related by Legendre transform. • In 1912, Henri Poincaré, inspired by Massieu, introduced a Characteristic Function in Probability by Laplace Transform (logarithm of Poincaré characteristic function if Massieu characteristic function). o Fisher Metric and Information Geometry • In 1939, Maurice Fréchet gave a Lecture at IHP introducing (Cramer-Rao) Bound and Clairaut-Legendre equation of Information Geometry (for « densités distinguées », distributions where the variance of estimators reached the Cramer-Rao-Fréchet-Darmois bound). Fréchet also observed that we have to consider hessian of a real function that was link to characteristic function. • In 1945, C. R. Rao has rediscovered Cramer-Rao Bound and used Fisher Information Matrix to define a metric in space of densities of probabilities. • In 1982, N. Chentsov has axiomatized Information Geometry and characterized the Fisher information metric as the only Riemannian metric that is invariant under sufficient statistics, based on category theory. o Learning « Natural Gradient » of Information Geometry • In 1998, Sun-Ishi Amari in the framework of Information geometry, introduced Natural Gradient in percepton parameter space, proved to be Fisher efficient. The parameter space of multilayer networks forms a Riemannian space equipped with Fisher information metric. Instead of the usual gradient descent method, the natural gradient or Riemannian gradient
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method, which takes account of the geometric structure of the Riemmanian space, is more effective for learning. • In 2015, Yann Ollivier and in 2016 with Gaëtan Marceau-Caron, mitigating difficulty of Fisher information matrix inversion, introduced a Practical Natural Gradient for Deep Learning on very deep neural networks with many layers and parameters. o Thermodynamics & statistical Physics • With J.C. Maxwell and Ludwig Boltzmann, J.W. Gibbs founded statistical mechanics. • Influenced by J. W. Gibbs, in 1891, Pierre Duhem extended Thermodynamics Potentials and Heat Capacities, and introduced Clausius-Duhem Inequality. • In 1909, C. Carathéodory formulated the Laws of Thermodynamics axiomatically. • Influenced by V. Bargmann [5], in 1969, Jean-Marie Souriau introduced Symplectic Geometry in (Statistical) Mechanics, and extension of Fisher Metric in Lie Groups Thermodynamics by studying non-equivariance of coadjoint operator on moment map (KKS 2 form in non-null cohomology and Souriau Symplectic Cocycle). Souriau gave a new definition of Entropy with a new geometric thermodynamics that is fully covariant under the action of the group acting on the system. o Geometry of Homogeneous Convex Cones • Jean-Louis Koszul, introduced the Koszul-Vinberg Characteristic Function, and a Koszul 2 form on sharp convex cones, that constitutes the algebraic foundation of Fisher Metric. • Based on J. Diximier work, in 1972, Alexandre A. Kirrilov introduced Representation Theory and Coadjoint Orbit Method for Harmonic Analysis on Lie Groups
Figure 1. Joint Structures and Common Foundation of Statistical Physics, Information Geometry and Inference for Learning.
The study of exponential densities invariant by a group goes back to the work of Muriel Casalis in her 1990 thesis supervised by Gérard Letac, and more recently by Ishi and Tojo [30-32]. This problem was previously studied in a geometric framework by Jean-Louis Koszul in the 1960s, in parallel with the work of Ernest Vinberg in Russia, to define Riemannian metric on sharp convex cones, invariant by the automorphisms of these cones. The general problem was solved for Lie groups by Jean-Marie Souriau in Geometric Mechanics in 1969, by defining a "Lie groups Thermodynamics" in Statistical Mechanics (http://souriau2019.fr/) [6-11, 13, 42-45]. This Souriau’s model considers the statistical mechanics of dynamic systems in their "space of evolution" associated with a symplectic manifold, and defines in case of non null cohomology (non equivariance of the coadjoint operator on the moment map with appearance of a cocyle) a density (of Gibbs) that is covariant under the action of dynamic groups of physics (eg, Galileo's group in classical physics) [14]. The family of exponential densities invariant by a group is a special case associated with the affine group. Koszul and
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Souriau's approach uses the affine representation of Lie groups and Lie algebra (https://fgsi2019.sciencesconf.org/). V. Arnorld [47-48] studied Geometric approach of Thermodynamics with Contact and Symplectic geometries in Fluid Mechanics and R. Balian [49-58] in Quantum Physics. Jean-Louis Koszul returns to this model of Jean-Marie Souriau in 1987 in Book “Introduction to Symplectic Geometry” (work that has just been translated into English). It is the tool by excellence for dealing with symmetries in Symplectic Geometry. These geometric structures associated with the exponential families make it possible to define generalizations of the Fisher metric in Information geometry (Koszul-Fisher metric related to the Koszul’s 2-form on the sharp convex cones and Souriau-Fisher metric tied to coadjoint orbits and the moment map). Maurice Fréchet concomitantly with his discovery of the Fréchet-Darmois bound (called Cramer-Rao Bound) discovered the equation of Clairaut-Legendre at the foundation of these geometric structures in 1943. These new tools are bedrocks for Lie Group Machine Learning, due to Souriau Theorem relating to each Coadjoint Orbits of Lie Group an homogeneous Symplectic Manifold structure. In the domain of Calculus of Variations, Jean-Marie Souriau studied the manifold of system movment in his book “Structure des systèmes dynamiques” published in 1969. When the system is Hamiltonian, the manifold of movements has a natural symplectic structure discovered in 1809 by Lagrange by introducing the Lagrange brackets. Jean-Marie Souriau gave a global geometric formulation of this Lagrange model. When a Lie group G acts by symplectomorphisms on a symplectic manifold under certain cohomological conditions, there is a natural map defined on the manifold and with values in the dual of the Lie algebra of G. Souriau called this map the “moment map”. If we chose a base of the Lie algebra, the components of this map in the dual base are functions having for associated Hamiltonian vectors field the infinitesimal generators of the action of the group. The moment map is a geometrization of Emmy Noether’s first theorem. This theorem asserts that every differentiable symmetries of the action of a physical system has a corresponding conservation law. If we consider the associated Hamiltonian system, the notion of moment map gives to this theorem a more geometric form: if the Hamiltonian of the system has a Lie algebra of infinitesimal symmetries, the corresponding moment map is constant on each integral curve of the system. The usual form of the theorem is obtained by considering each component of the moment application separately. In chapter IV of his book, Jean-Marie Souriau has applied this model for statistical mechanic and build a geometric model of thermodynamics that he called “Lie Groups Thermodynamics. This is the main achievement of Souriau in the domain of Calculus of Variations for a geometrization of statistical physics.
Figure 2. Jean-Marie Souriau contribution in the domain of Calculus of Variations.
We will develop Souriau Lie Groups Thermodynamics model and expression of covariant Gibbs density and Souriau-Koszul-Fisher Metric. After mathematical details about Souriau Moment map, we will illustrate the model for Poincaré Unit Disk considered as a symplectic manifold where the Lie Group SU(1,1)/K acts transitively (case of null cohomology). We will give expression of SU(1,1)/K Moment map to compute by mean of Kirillov representation theory and kirillov character, Souriau covariant Gibbs density. By this way, we can define this Gibbs density as a generalization of “Gauss density” in Poincaré Unit disk through representation theory. This “Gaussian density” and its statistical moments will be
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invariant under the action of the group. We will indicate also method to extend this approach to non-null cohomology case for particular use-case of SE(2) Lie Group.
1. Lie Groups Thermodynamics and Covariant Gibbs Density We identify the Riemanian metric introduced by Souriau based on cohomology, in the framework of “Lie groups thermodynamics” as an extension of classical Fisher metric introduced in information geometry. We have observed that Souriau metric preserves Fisher metric structure as the Hessian of the minus logarithm of a partition function, where the partition function is defined as a generalized Laplace transform on a sharp convex cone. Souriau’s definition of Fisher metric extends the classical one in case of Lie groups or homogeneous manifolds. Souriau has developed this “Lie groups thermodynamics” theory in the framework of homogeneous symplectic manifolds in geometric statistical mechanics for dynamical systems, but as observed by Souriau, these model equations are no longer linked to the symplectic manifold but equations only depend on the Lie group and the associated cocycle [77-78]. This analogy with Fisher metric opens potential applications in machine learning, where the Fisher metric is used in the framework of information geometry, to define the “natural gradient” tool for improving ordinary stochastic gradient descent sensitivity to rescaling or changes of variable in parameter space. In machine learning revised by natural gradient of information geometry, the ordinary gradient is designed to integrate the Fisher matrix. Amari has theoretically proved the asymptotic optimality of the natural gradient compared to classical gradient. With the Souriau approach, the Fisher metric could be extended, by Souriau-Fisher metric, to design natural gradients for data on homogeneous manifolds. Information geometry has been derived from invariant geometrical structure involved in statistical inference. The Fisher metric defines a Riemannian metric as the Hessian of two dual potential functions, linked to dually coupled affine connections in a manifold of probability distributions. With the Souriau model, this structure is extended preserving the Legendre transform between two dual potential function parametrized in Lie algebra of the group acting transentively on the homogeneous manifold.
1.1. Inference by natutal gradient and Legendre structure
Classically, to optimize the parameter θ of a probabilistic model, based on a sequence of observations yt , is an online gradient descent: T ∂ly( ) θθη←− tt (1) ttt−1 ∂θ
with learning rate ηt , and the loss function lt= −log py( tt / yˆ ) . This simple gradient descent has a first drawback of using the same non-adaptive learning rate for all parameter components, and a second drawback of non invariance with respect to parameter re-encoding inducing different learning rates. Amari has introduced the natural gradient to preserve this invariance to be insensitive to the characteristic scale of each parameter direction. The gradient descent could be corrected by I()θ −1 where I is the Fisher information matrix with respect to parameter θ , given by: ∂2 logpy( /θ ) θ = = − (2) I( ) gij with gEij y p(/) y θ ∂∂θθij ij with natural gradient: T ∂ly( ) θ←− θ ηθI()−1 tt (3) ttt−1 ∂θ Amari has proved that the Riemannian metric in an exponential family is the Fisher information matrix defined by: 2 ∂Φ − θ g =− with Φ=− (θ ) log e, y dy (4) ij ∂∂θθ ∫ ijij and the dual potential, the Shannon entropy, is given by the Legendre transform: ∂Φ()θη∂S() (5) S(η )= θη , −Φ ( θ ) with ηii= and θ = ∂∂θηii
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We can observe that Φ=−(θ ) log ∫ e− θ , y dy =−logψθ( ) is linked with the cumulant generating function.
J.L. Koszul and E. Vinberg have introduced an affinely invariant Hessian metric on a sharp convex cone through its characteristic function: Φθ =− − θ , y =− ψθ θ∈Ω ΩΩ( ) log ∫ e dy log ( ) with sharp convex cone Ω* (6) ψθ= − θ , y Ω () ∫ e dy with Koszul-Vinberg Characteristic function Ω*
Jean-Louis Koszul has introduced the following forms
st 1 Koszul form α :α=Φ=−ddΩΩ() θ log ψθ () (7)
nd 2 Koszul form γ : γ=D α = Dd log ψθΩ ( ) (8)
with the following property of positive definitiveness:
2 1 Ddlogψ ()() x u =F()ξξ22 d . G () ξξ d −> F ().() ξ G ξξ d 0 ( Ω ) 2 ∫∫ ∫ ψ ()u ** * (9) Ω ΩΩ Ω 11 −−xx,,ξξ with Fe (ξ )= 22 and Ge ( ξξ )= u ,
Koszul has defined the following Diffeomorphism:
e− ξθ, ηα==−=dlog ψθ () ξ pd () ξξ with p () ξ= (10) Ω ∫ θθ− ξθ, * ξ Ω ∫ ed Ω*
with preservation of Legendre transform:
SΩΩ(η )= θη , −Φ ( θ ) with η=d Φ Ω ( θ ) and θ=dS Ω ( η ) (11)
1.2. Souriau Lie Groups Thermodynamique and Souriau-Koszul-Fisher metric
This relations have been extended by Jean-Marie Souriau in geometric statistical mechanics, where he developed a “Lie groups thermodynamics” of dynamical systems where the (maximum entropy) Gibbs density is covariant with respect to the action of the Lie group. In the Souriau model, previous structures of information geometry are preserved: ∂Φ2 I(ββ )=− with Φ=− ( ) log e− βξ,U () d λ and UM : →* (12) 2 ∫ ω g ∂β M ∂Φ()β ∂SQ() SQ( )=ββ , Q −Φ ( ) with Q= ∈gg* and β = ∈ (13) ∂∂β Q In the Souriau Lie groups thermodynamics model, β is a “geometric” (Planck) temperature, element of Lie algebra g of the group, and Q is a “geometric” heat, element of dual Lie algebra g* of the group. Souriau has proposed a Riemannian metric that we have identified as a generalization of the Fisher metric: β= ββ= Θ β (14) I( ) gββ with g([ , Z1] ,[ , Z 2]) β( ZZ 12,,[ ]) with ΘZZ , =Θ+ ZZ , QadZ , ( ) where adZ ( )= ZZ , (15) β ( 12) ( 12) ZZ112 2[ 12] Souriau has proved that all co-adjoint orbit of a Lie Group given by *− 1 ** Ο=Fg{Ad F = g Fg, g ∈ G} subset of gg , F ∈ carries a natural homogeneous symplectic structure by a
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* * * closed G-invariant 2-form. If we define = = − and K() X= − ad with K Adg ( Adg 1 ) * ( X )
**−1 =−1 ∀∈ ∈ ∈ where if X ∈g, Ad() X= gXg ∈g, the G-invariant 2-form AdFYg , FAdY ,g , gGY ,,gg F g
is given by the following expression σ Ω (adFadFXY, ) = B F( XY,) = F ,[ XY ,] ,, XY∈g . Souriau Foundamental Theorem is that « every symplectic manifold is a coadjoint orbit ». We can observe that for Souriau model, Fisher metric is an extension of this 2-form in non-equivariant case . gβ ([ββ, Z1] ,,[ Z 2]) =Θ+( ZZ 12 ,,[ β]) QZZ , 12 ,,[ β] The Souriau additional term Θ(ZZ12,,[β ]) is generated by non-equivariance through Symplectic cocycle. The tensor Θ used to define this extended Fisher metric is defined by the moment map Jx(), application from M (homogeneous symplectic manifold) to the dual Lie algebra g* , given by: (16) Θ=−(,)XY J[XY, ] { JXY , J} * with Jx ( ) : M→=∈gg such that JX ( x ) Jx ( ), X , X (17) This tensor Θ is also defined in tangent space of the cocycle θ ( g )∈g* (this cocycle appears due to the * non-equivariance of the coadjoint operator Adg , action of the group on the dual lie algebra; the action of the group on dual Lie algebra is modified with a cocycle so that the momentu map becomes equivariant relative to this new affine action): * (18) Q( Adgg()βθ) = Ad () Q + ( g ) θ ( g )∈g* is called nonequivariance one-cocycle, and it is a measure of the lack of equivariance of the moment map. Θ ( XY,) :gg × →ℜ with Θ (X ) = Tθ ( Xe ( )) e (19) X,Y Θ (XY ), The cocycle should verify: * θ (st )= J (( st) .) x − Adst J () x θ =−+− * * ** (st ) Jstx (.( .) ) AdJtxs( .) AdJtxs( .) AdAdJxst() (20) θθ=+−** (st ) () s Adst Jtx( .) AdJx () * θθ(st )= () s + Ads θ () t We can also compute tangent of one-cocycle θ at neutral element, to compute 2-cocycle Θ : ζ∈==− θ θζ ζ* ζ = ζ − ζ g , ζ (s) ( s), Jsx( .,) AdJxs ( ), Jsx( .,) JxAd( ) , s−1 θξ=+= ξ ζ ζ ξ Teζξ( ) Txp J. ( x ), J( x) , ad with pX J ,ξ (21) θ ξ = ζ + ξζ Te ζ ( ) XJx( ),ξ Jx( ), J( x),,[ ]
Teθζ ( ξ) =−{ J, ξ , J , ζ} += Jx ( ),[ ξζ , ] Θ( ξ) * We can also write: Txp J(ξξ() x) =− adξ J () x +Θ( ,.) By differentiating the equation on affine action, we have:
dJ( Xx )= adX J ( x ) + dθ ( X ) , x ∈∈ M , X g (22)
dJ( Xx ), Y= adX J ( x ), Y + dθ ( X ), Y , x ∈∈ M , X , Y g dJXxY( ),= Jx ( ),[ XY ,] += dθ ( X ), Y { JX, , JY ,} ( x ) (23) Jx( ),[ XY ,] −={ J , X , JY ,} ( x )− dθ ( X ), Y It can be then deduced that the tensor could be also written: (24) Θ=−(,)XY J[XY, ] { JXY, J} =− dθ ( X ), Y , XY , ∈g with the cocycle property: Θ([ XY,,] Z) +Θ([ XY ,,] Z) +Θ([ XY ,,] Z) = 0 , XYZ,, ∈g (25) By noting the action of the group on the dual Lie algebra: ** * G×→gg,( s ,ξ ) s ξ = Ads ξθ +() s (26)
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Associativity is also derived: ()ssξ= Ad* ξθ + () ss = AdAd**ξθ ++() s Ad* θ () s 12 ss12 12 s1 s 2 1 s1 2 (27) ** * ()ss12ξ= AdAdss( ξθ +( s2 )) + θ ( sss 1 ) = 1( 2 ξ) , ∀∈∈ ssG1 , 2 ,ξg 12 This study of the moment map J equivariance, and the existence of an affine action of G on g* , whose linear part is the coadjoint action, for which the moment J is equivariant, is at the cornerstone of Souriau theory of geometric mechanics and Lie groups thermodynamics.
1.3. Souriau Entropy and Souriau-Fisher-Koszul metric Invariance under the action of the group and Covariant Souriau Gibbs density
In Souriau’s Lie groups thermodynamics, the invariance by re-parameterization in information geometry has been replaced by invariance with respect to the action of the group. When an element of the group g acts
on the element β ∈g of the Lie algebra, given by adjoint operator Adg . Under the action of the group
Adg ()β , the entropy SQ( ) and the Fisher metric I (β ) are invariant: S Q Ad()β = S( Q) ( g ) (28) ββ∈→g Adg () ⇒ ββ= I Adg () I ( ) In the framework of Lie group action on a symplectic manifold, equivariance of moment map could be studied to prove that there is a unique action a(.,.) of the Lie group G on the dual g* of its Lie algebra for which the moment map J is equivariant, that means for each x ∈ M : * J (Φ g (x))= a(g, J (x)) = Ad g (J (x))+θ (g) (29) When coadjoint action is not equivariant, the symmetry is broken, and new “cohomological” relations should be verified in Lie algebra of the group. A natural equilibrium state will thus be characterized by an element of the Lie algebra of the Lie group, determining the equilibrium temperature β . The entropy s(Q) , parametrized by Q the geometric heat (mean of energy U , element of the dual Lie algebra) is defined by the Legendre transform of the Massieu potential Φ(β ) parametrized by β ( Φ(β ) is the minus logarithm of the partition
function ψ Ω (β )).
Souriau has then defined a Gibbs density that is covariant under the action of the group: − βξ,U () Φ−(ββξ ) ,UU () e −βξ, () p()ξ= e = , with Φ=− (βλ ) log ed Gibbs − βξ,U () λ ∫ ω ∫ edω M M (30) − βξ,U () Ue()ξλ dω ∂Φ()β ∫ Q= = M = U()()ξ pd ξλ ∂β − βξ,U () λ ∫ ω ∫ edω M M
Figure 1. Fondamental Equation of Souriau Lie Groups Thermodynamics.
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Figure 2. Souriau Model of Lie Groups Thermodynamics.
Souriau completed his “geometric heat theory” by introducing a 2-form in the Lie algebra, that is a Riemannian metric tensor in the values of adjoint orbit of β , [β,Z] with Z an element of the Lie algebra. This metric is given for (β,Q):
g β ([β,Z1 ],[β,Z 2 ]) = Θ(Z1 ),[β,Z 2 ] + Q,[Z1,[β,Z 2 ]] (31)
where Θ is a cocycle of the Lie algebra, defined by Θ = Teθ with θ a cocycle of the Lie group defined by * θ (M ) = Q(Ad M (β ))− Ad M Q . We observe that Souriau Riemannian metric, introduced with symplectic cocycle, is a generalization of the Fisher metric, that we call the Souriau-Fisher metric, that preserves the property to be defined as a Hessian ∂ 2Φ ∂ 2 logψ of the partition function logarithm g = − = Ω as in classical information geometry. We will β ∂β 2 ∂β 2 establish the equality of two terms, between Souriau definition based on Lie group cocycle Θ and parameterized by “geometric heat” Q (element of dual Lie algebra) and “geometric temperature” β (element of
Lie algebra) and hessian of characteristic function Φ(β ) = − logψ Ω (β ) with respect to the variable β: ∂ 2 logψ g ([β, Z ],[β, Z ]) = Θ(Z ),[β, Z ] + Q,[Z ,[β, Z ]] = Ω (32) β 1 2 1 2 1 2 ∂β 2 * If we differentiate this relation of Souriau theorem Q(Ad g (β )) = Ad g (Q) +θ (g), this relation occurs: ∂Q ~ ~ (− [Z , β ],.) = Θ(Z ,[β,.]) + Q, Ad ([β,.]) = Θ β (Z ,[β,.]) (33) ∂β 1 1 .Z1 1 ∂Q ~ ~ − ([Z , β ], Z .) = Θ(Z ,[β, Z ]) + Q, Ad ([β, Z ]) = Θ β (Z ,[β, Z ]) (34) ∂β 1 2 1 2 .Z1 2 1 2 ∂Q ⇒ − = g ([β, Z ],[β, Z ]) (35) ∂β β 1 2
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Figure 3. Souriau-Fisher metric as extension of KKS 2-form in case of non-null Cohomogy.
As the entropy is defined by the Legendre transform of the characteristic function, a dual metric of the Fisher metric is also given by the hessian of “geometric entropy” SQ() with respect to the dual variable given ∂2 by Q: SQ(). ∂Q2 ∂ 2 logψ For the maximum entropy density (Gibbs density), the following three terms coincide: Ω that ∂β 2 ∂2 logp (ξ ) β = − β describes the convexity of the log-likelihood function, IE() 2 the Fisher metric that ∂β describes the covariance of the log-likelihood gradient, whereas I(β ) = E[(ξ − Q)(ξ − Q)T ]= Var(ξ ) that ∂Q describes the covariance of the observables. We can also observe that the Fisher metric I(β ) = − is exactly ∂β the Souriau metric defined through symplectic cocycle: ~ I(β ) = Θ β (Z1,[β,Z 2 ]) = g β ([β,Z1 ],[β,Z 2 ]) (36) ∂ 2Φ(β) ∂Q The Fisher metric I(β ) = − = − has been considered by Souriau as a generalization of “heat ∂β 2 ∂β capacity”. Souriau called it K the “geometric capacity”.
1.4. Covariant Souriau Gibbs density and Information Manifold Foliation
R.F. Streater has studied in 1999, Information Geometry for some Lie algebra where for certain unitary representation of a Lie algebra, he has defined the statistical manifold of states as convex cone for which the partition function is finite, making reference to Bogoliubov-Kubo-Mori metric. But Streater has only developed the case with null cohomology for so(3) and sl(2,R) Lie alebras. Nevertheless, as observed by R.F. Streater in his paper “Information Geometry for some Lie algebras” [83], referring to Kirillov work and Roger Balian paper, “We can expect further natural structures to arise in this case. Indeed, it is known (*) that the dual to the Lie algebra, which parametrizes the state-space in this case, foliates into coadjoint orbits; there are also the level sets on the entropy; Kirillov form, and the BKM (Bogoliubov-Kubo-Mori) metric, together make each orbit into kähler space, along the lines proposed by Kostant. Motion along these holomorphic directions is nondissipative. The transversal to the orbits is a real half-line, which represents the dissipative direction…We study the case of sl(2,R) in the discrete series of representations. We show the information manifold foliates into level sets of the entropy, each being isometric to H, the Poincaré upper half-plane… The states of constant entropy are the hyperboloids and β is the dissipative coordinate…For an integrable system described by a Lie algebra in a traceable representation, we find that the information manifold foliates into
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complex spaces; the level sets of entropy can be given a complex structure by the method of Kostant. Motion remaining on the complex surfaces is nondissipative, whereas motion transversal to these surfaces is dissipative. In information geometry, the state is parametrized by the canonical coordinates. Which function of them is measured by a thermometer? In our models, it is reasonable to designate 1/β to be the temperature; it is a dissipative coordinate, and it increases with time, showing that the system is thermalizing”.
2. Mathematical definition of Souriau Moment map
In previous chapter, we have introduced Souriau moment map. In this chapter, we provide mathematical definition of Souriau moment map, as defined in Souriau's book "Structure of Dynamical System: A Symplectic View of Physics" [6] with modern notations [105-106,108]. More details could be also find in Jean-Louis Koszul Book “Introduction to Symplectic Geometry”, that we have recently supervised translation [109].
2.1. Operations on Vector Fields
Consider a map FX: ⊂ RMN →⊂ Y R, y= Fx() , the derivative of F at xX∈ , DF: X→ R NM× is given by: ∂∂yy11 11 δδyx∂∂xx1 M Fx(+− tδ x ) Fx () (37) = =DF( x )(δ x ) = Lim t→0 t NMNN δδyx∂∂yy ∂∂xx1 M
2 2 NMM×× ∂∂yy Second derivative is given by the linear map DF: X→ R :δδδ=()x = DFx2 ()() x (38) ∂∂xx2 Consider a vector Field V on XR⊂ M defined by : VX: ⊂→ RMM R, operations on vector fields are given by adjoint action and Lie bracket: d = tV −−11= = (39) AdF V() y F e F() y DF () x( V () x) with x F () y dt t=0 d = = − (40) [UV,] () x AdVxesU () DUx ()( Vx ()) DVxUx ()( ()) ds s=0
0-form is a scalar, 1-form are row ωωω= ( 1 M ) in dual space. 2-forms can be regarded as antisymmetric
ωω11 1M matrices ω with t . m-forms are all scalar multiples of the standard volume ( ij ) ω (uv, ) = u v ωωM 1 MM
form vol, defined by Vol( v11, , vmm) = det( matrix with columns v ,..., v ) .
2.2. Derivative rules by Sophus Lie, Elie Cartan and Henri Cartan
With the following classical definitions: • Pull back: F *ω is a p-form on X
* Fωω( v1,, vp) = Fx()( DF( x )( v1 ), , DF ( x )( vp )) (41)
• Interior product: iV ω is the (p-1)form on M obtained by inserting Vx()as the first argument of ω
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iVpωω( v22, v) = ( Vx( ), v , , vp) (42) • Exterior product: θω∧ is the (p + 1)-form on X where ω is a p-form and θ is a 1-form on M (where the hat indicates a term to be omitted).: p i (43) θω∧=−(vv00,,p) ∑( 1) θ () vvvvi ω( ,,,,ˆ ip) i=0 • Lie derivative : LV ω is a p-form on M , and LV ω = 0 if the flow of V consists of symmetries of ω : d tV * (44) LvvVpωω( 11,,) = e( vv,,p) dt t=0 dω is the (p+1)-form on M defined by taking the ordinary derivative of ω and then antisymmetrizing: • Exterior derivative: p i ∂ω (45) dvvω ( 00,,p) =∑( − 1) ()(,,,,) vvvviˆ ip i=0 ∂x =ω =∂ ω = ω =∂ ωω −∂ = ω =∂ ω +∂ ω +∂ ω (46) p0,[ d]i i ; pd 1,[ ]ij i j j i ; p 2,[ d]ijk i jk j ki k ij From these definitions, the properties of the exterior and Lie Derivative were established by Sophus Lie, Elie Cartan, and Henri Cartan: • LVωωω= di VV + i d (lie. Cartan) (47) • (Henri Cartan) (48) i[UV, ]ωωω= iLV U − Li UV • (Sophus Lie) (49) L[UV, ]ωωω= LLVU − LL UV
2.3. Souriau Moment Map Considering Manifolds and Lie groups, We define the tangent bundle TX of X as the disjoint union of the δ x , or the set of all pairs with ∈ and δ ∈ . If → is a smooth map between TXx xX x TXx FX: Y x δδ manifolds, its tangent map is the map: x DF( x )( x ) (50) F* = x Fx() A Lie group is a group G with a manifold structure such that the product ( g, h) gh and the inversion −1 gg are smooth maps from GG× (resp. G) to G . Its Lie algebra is the tangent space g= TGe at the identity element. A smooth action of G on a manifold X is a group morphism: Φ:G ×→ X Diff () X (51) ( g ,. x) gx The orbit of xX∈ is G() x={ gx .: g ∈ G}. The tangent space to an orbit at x :
d tZ TGxx() x={ Z (): x Z ∈g} =g / g with Zx()= e () x and where ggx =∈={Z: Zx () 0} (52) dt t=0 Let (M ,σ ) be a connected symplectic manifold. A vector field η on M is called symplectic if its flow
preserves the 2-form : Lησ = 0 . If we use Elie Cartan's formula, we can deduce that Lησσσ=+= di ηη i d 0
but as dσ = 0 then diησ = 0. We observe that the 1-form iησ is closed. When this 1-form is exact, there is a smooth function xH on M with:
iησ = − dH (53)
This vector field η is called Hamiltonian and could be defined as symplectic gradientη = ∇Symp H . Let a Lie group G that acts on M and that also preserve σ . A moment map exists if these infinitesimal generators are actually hamiltonian, so that a map JM: → g* exists with: iσ = − dH where H= Jx( ), Z (54) ZZX Z The Poisson bracket of two functions H , H ' is defined by :
{HH,','} =σ( ηη) =∇∇ σ( SympHH', Symp ) with iησ = − dH and iη 'σ = − dH ' (55)
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If is connected, then the moment map is G-equivariant if and only if it satisfies . G {HHZZ, '} = H[ZZ,'] Souriau has proved thet every coadjoint orbit of a Lie group is a homogeneous symplectic manifold when endowed with the KKS 2-form σ (Zx(),'() Z x) = x ,[ Z ', Z] , and conversely, every homogeneous symplectic manifold of a connected Lie group G is, up to a possible covering, a coadjoint orbit of some central extension of G. σ is G-invariant.
3. Poincaré Unit Disk, SU(1,1) Lie Group and Souriau Moment map We will introduce Souriau moment map for SU(1,1)/K group that acts transitively on Poincaré Unit Disk.
3.1. Poincaré Unit Disk and SU(1,1) Lie Group
The group of complex unimodular pseudo-unitary matrices SU (1,1) , is the set of elements u such that [20-22,27-29, 79-82]:
+ +10 uMu= M with M = (56) 01− We can show that the most general matrix u belongs to the Lie group given by: ab 22 (57) G= SU (1,1) =**/a −= b 1, a, b ∈ C ba Its Cartan decomposition is given by:
ab 1 z aa 0 −1 −1/2 = =* = − 2 (58) ** a * * with z ba( ) , a( 1 z ) ba z1 0 aa
a' = bz* + a ab 1 z 1' z aa'' 0 = (59) ** * a ' * * with az+ b ba z1 z '1 0aa '' z ' = bz**+ a SU (1,1) is associated to group of holomorphic automorphisms of the Poincaré unit disk D={ z =+∈ x iy C/1 z <} in the complex plane, by considering its action on the disk as g() z=++( az b) /( b** z a ) . The following measure on Unit disk:
* * 1 dz∧ dz (60) dµ0 ( zz, ) = π 2 2 2 i (1− z )
is invariant under the action of SU (1,1) captured by the fractional holomorphic transformation: dz''∧∧ dz** dz dz = (61) 2222 (1'−−zz) ( 1 )
The complex unit disk admits a Kähler structure determined by potential function: Φ( z', z**) =−− log( 1zz ' ) (62)
2* 11∂Φ( zz, ) dz∧ dz* The invariant 2-form is: Ω= dz∧ dz* = (63) ∂∂* 2 2 i zz i(1− z )
which is closed dΩ=0 . This group SU (1,1) is isomorphic to the group SL(2, ) as a real Lie group, and the Lie algebra g= su(1,1) is given by:
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− η ir (64) g=* /,rR ∈∈η C η ir 0−−ii 01 0 with the bases ∈ : 1 11 (uuu123,,) g u1= , uu 23= , = 2ii0 22 10 0
with the commutation relation:[uu32,] ==−=− u 1 ,,[ uu 31] u 2 ,,[ uu 21] u 3 (65)
*** * ** Dual base on dual Lie algebra is named (uuu1,, 23)∈g . The dual vector space g= su (1,1) can be identified with the subspace of sl (2,C ) of the form:
* +z x iy 01 0i 10 g = =x ++ y z/,, xyz ∈ R (66) −+x iy − z −10i 0 0− 1
Coadjoint action of gG∈ on dual Lie algebra ξ ∈g* is written g.ξ .
3.2. Coadjoint Orbit of SU(1,1) and Souriau Moment Map We will use results of C. Cishahayo and S. de Bièvre [15] and B. Cahen [16,17] for computation of ∈ *+ * * moment map of SU (1,1) . Let rR, orbit Ο(ru3 ) of ru3 for the coadjoint action of gG∈ could * * * 2222 be identified with the upper half sheet x3 > 0 of {ξ =xu11 + xu 2 2 + xu 33/ −− x 1 x 2 + x 3 = r }, the two- * sheet hyperboloid. The stabilizer of ru3 for the coadjoint action of G is torus eiθ 0 = θ ∈ . K induces rotations of the unit disk, and leaves 0 invariant. The stabilizer K −iθ , 0 e for the origin 0 of unit disk is maximal compact subgroup K of SU(1,1). We can observe [16] that * * Ο=(ru3 ) G/ K . On the other hand Ο=(ru3 ) G/ K is diffeomorphic to the unit disk D=∈<{ zCz/1}, then by composition, the Souriau moment map is given by:
* J: D→Ο( ru3 )
2 (67) zz+−** zz 1+ z =* ++ ** z Jz () r2 u1 22 u 23 u 1−z iz 11 −− z ( ) ( ) ( )
J is linked to the natural action of G on D (by fractional linear transforms) but also the coadjoint * −1 action of G on Ο=(ru3 ) G/ K . J could be interpreted as the stereographic projection from the
2 n + two-sphere S onto C ∪∞[46]. In case r = where nNn∈≥,2 then the coadjoint orbit is given 2
n ** by Ο=Ο(ζ ) with ξ =u ∈g , with stabilizer of ξn for coadjoint action the torus nn n 2 3 eiθ 0 = θ ∈ with Lie algebra Ru . Ο=Ο(ζ ) is associated with a holomorphic KR−iθ , 3 nn 0 e
discrete series representation π n of G by the KKS (Kirillov-Kostant-Souriau) method of orbits.
JD: →Οn
** 2 (68) n zz+− zz 1+ z =* ++ ** z Jz () 2u1 22 u 23 u 2 1−z iz 11 −− z ( ) ( ) ( )
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Group G act on D by homography ab az+ b . This action corresponds with gz..=** z = ba bz**+ a
coadjoint action of G on Οn . The Kirillov-Kostant-Souriau 2-form of Οn is given by:
Ωnn(ζ)( X( ζζ), Y( )) = ζ ,[ XY ,] , XY ,∈g and ζ ∈Ο (69)
and is associated in the frame by J with: in * (70) ωn = dz∧ dz 2 2 (1− z )
2 2 ∂∂fg ∂∂ fg with the corresponding Poisson Bracket: { fg,1} =−− i z (71) ( ) ∂∂zz** ∂ z ∂ z
It has been also observed that there are 3 basic observables generating the SU (1,1) symmetry on classical level:
DR→ DR→→ DR 2 ** (72) 1+ z , 1,zz−+ zz z kz()= zkz()= zkz()= 3 2 12i 22 1− z 11−−zz
With the Poisson commutation rule: {kk31,} = k 2 ,,{ kk 32} =−=− k 1 ,,{ kk 12} k 3 (73)
3 (kkk123,,) vector points to the upper sheet of the two-sheeted hyperboloid in R given by 222 kkk31−−= 21 , whose the stereographic projection onto the open unit disk is: + (kkk123,,)∈Η → D (74) k+ ik k −1 izarg ze=21 = 3 11++kk33
Under the action of ab 22 : g∈= G SU (1,1) =**/a− b = 1, a, b ∈ C ba
2 k− k3 k 21+ ik k 3 1 21zz+ = = is transform in: k k k k− ik 2 2 * 3+ 3 211− z 12+ zz
−− ''kgz11.. kgz kk − ( ) 3 ( ) −−kk− 3 t − 3 = = gg11 (75) '' −−11( ) kk+ kk3 + 3 kgz3 ( ..) kgz+ ( )
This transform can be viewed as the co-adjoint action of SU (1,1) on the coadjoint orbit identified 222 with kkk31−−= 21.
4. Covariant Gibbs Density by Souriau Thermodynamics for Poincaré Unit Disk
4.1. Covariant Gibbs densité for Poincaré Unit Disk
Representation theory studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and algebraic objects (Lie groups, Lie algebras) by describing its elements by matrices and the algebraic operations in terms of matrix addition and matrix multiplication, reducing problems of abstract algebra to problems in linear algebra. Representation
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theory generalizes Fourier analysis via harmonic analysis. The modern development of Fourier analysis during XXth century has explored the generalization of Fourier and Fourier-Plancherel formula for non-commutative harmonic analysis, applied to locally compact non-Abelian groups. This has been solved by geometric approaches based on “orbits methods” (Fourier-Plancherel formula for G is given by coadjoint representation of G in dual vector space of its Lie algebra) with many contributors (Dixmier, Kirillov, Bernat, Arnold, Berezin, Kostant, Souriau, Duflo, Guichardet, Torasso, Vergne, Paradan, etc.) [23-26, 33-38, 40-41].
For classical commutative harmonic analysis, we consider the following groups:
GRZ=Τ=n nn/ for Fourier series, GR= n for Fourier Transform G group character (linked to eikx ) :χ : G→ U with U =∈={ zCz / 1} ˆ G= {χ/ χχ12 . ( g )= χ 1 ( gg ) χ 2 ( )} and Fourier transform is given by:
ϕ :GC→ ϕˆ :GCˆ → −1 and (76) g ϕ( g) = ∫ ϕχχˆ()() gd χ χ ϕχˆ ( ) = ϕ()()g χ g dg ˆ ∫ G G
For non-commutative harmonic analysis, Group unitary irreductible representation is
U:GU→Η( ) with H Hilbert space and character by χU ()g= tr Ug . Fourier transform for non- commutative group is = ϕ with character χ ()g= tr U. If we describe group element Uϕ ∫ (g )Ug dg U ϕ G with exponential map = ψ , we have: Uψ ∫ (X )Uexp(X ) dX g
∧ * −1 µ Ο= ∈ = τµ ψ Gf. : Liouville meas. on Gf . , f g trUψ dim .Gf. . j with ∧∧ (77) −−11 ∧ µψGf...jj : Integral of ψ . wrt µGf ψ .j−1* :gg→ , Four. Transf.
∞ 2n where 1/2 1 x xx (78) j( X) = (det s( ad X )) with s ( x ) = ∑ = sh n=0 (2n + 1)! 2 22
Kirillov Character formula is: χµ= = −1 ifX, (79) UX(exp(X )) trUexp( ) jX( ) ∫ e dΟ () f Ο
1/2 adXX/2− ad /2 ifX, ee− µ = with = (80) ∫ e dΟ ( f ) jX ( )trUexp(X ) jX( ) det Ο ad X /2
We will use Kirillov representation theory and his character formula to compute Souriau covariant Gibbs density in the unit Poincaré disk. For any Lie group G , a coadjoint orbit Ο⊂g* has a
canonical symplectic form ω0 given by KKS 2-form. As seen, if G is finite dimensional, the corresponding volume element defines a G -invariant measure supported on Ο , which can be interpreted as a tempered distribution. The Fourier transform (where d is the half of the dimension of the orbit O) :
−ix,λ 1 * (81) ℑ=()xe dωλd with ∈gg and x ∈ ∫ O Ο⊂g* d !
is Ad G -invariant. When Ο⊂g* is an integral coadjoint orbit, Kirillov formula, given previously,
expresses Fourier transform ℑ()x by Kirillov character χΟ :
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sinh(ad( x / 2)) ℑ= χ x =1/2 (82) (x ) jx ( )Ο ( e) where jx ( ) det ad( x /2)
χΟ is, as defined previously, the “Kirillov character” of a unitary representation associated to the orbit. We will consider the universal covering of PSU (1,1) , the Lie algebra is:
iE p* g**= su(1,1) =/,E ∈∈ Rp C (83) p− iE
2 As observed in [18-19], the Ad-invariant form mE22= − p allows to identify the following operator Ad and Ad * , m could be considered analogously as rest mass, E as energy, and
p= p12 + ip as the momentum vector. The coadjoint orbits are the rest mass shells. Let D=∈<{ wCw/1} Poincaré unit disk, for any m > 0, there is a corresponding action of the universal covering of PSU (1,1) on κ m/2 (with κ the holomorphic cotangent bundle of unit disk), with the invariant symplectic form:
* 2 dw∧ dw ωκ=curv( ) = − i ∂∂* log dw = 2 i (84) 2 2 (1− w )
+ 2 The moment map is an equivariant isomorphism ( Οm coadjoint orbit for m > 0 and E > 0 ): m 2 J:, w∈ D curvκ m/2 p, E = 2iw ,1+ w ∈Ο+ (85) ( ( )) ( ) 2 ( ) m (1− w )
In case m >1, the Kirillov character formula is given by:
x . iE p* −i , − x . −1 . x p− iE (86) χωm exp= jx() e + ∫ Οm−1 . −x + Οm−1
1/2 xx/2 /2 sinh(x ) where j( x )= det sinh ad / ad = (87) −−xx/2 /2 x
2 1+ w − mx (mx 1) 2 e 1− w 1 which reduces to : j() x= e dw∧ dw* (88) 1− e2x ∫ 2 2 D (1− w )
Finally, the Souriau-Gibbs density is given by:
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xiη iE p* −i , * iη −− x p iE em2 p w = with p , E = 2iw ,1+ w ∈Ο+ Gibbs ( ) xiη ( ) 2 ( ) m 1− w ixη* − ( ) jx()χ e m (89) + 2 * 1 w w im222 m xiη 11−−ww − , 2 * 2 2 ηηww+ ** x−+ηη w** w + xw ixη − w 1+ w 1+ w ( ) ( ) −− −−2mx −2mx 2m22 im 22 2 11−−ww 11−−ww 1− w e ee pw= = = Gibbs ( ) xiη xiηηxi * ** jx()χ eixη − jx()χχ eixηη−− jx() eix m mm
ix −η ∂Φ β with −1*( ) and ββ=g* =Λ∈(QQ) where = =Λ∈( ) g −−η ix ∂β
2 1+ w w m22− im2 11−−ww Mean moment map : QE= where wD∈ 2 w 1+ w − im2 22 m 11−−ww
Nota 1: Localization for Fourier-Laplace transform
We can observe that there is another way to compute the Laplace or Fourier transform:
ixλ, 1 ℑ= ωλn →* ∈ (90) ()xe∫ n d with : Mgg and x M (2!π ) n
Let λ :Mx→∈gg* and be the moment map, Duistermaat and Heckman have used the method of exact stationary phase to prove a formula that expresses this integral explicitely in term of local invariant. When x is purely imaginary, the integral is the partition function of a statistical system with phase space M. This approach is also valid for Im(x )> 0 belonging to a special cone in g. The non-abelian measure was first evaluated by Duflo and Vergne. For each regular x∈∈gC , Im( x ) Int( C), the integral exist and:
iλ ( px ), ixλ, 1 e ℑ= ω nn= (91) ()xe∫ nn di∑ M (2!π ) n p p ∏αi ()x i=
Nota 2: Extension for SU(p,q) Unitary Group for Siegel Unit Disk To address computation of covariant Gibbs density for Siegel Unit Disk, we can consider SU(,) p q Unitary Group:
A 0 G= SUpq( , ) and K = SUp( ( ) ×= Uq ( )) /A ∈ Up ( ), D ∈ Uq ( ),det( A )det( D )= 1 (92) 0 D
We can use the following decomposition for gG∈ C :
−1 −1 AB I BD A− BD C 0 I p 0 = ∈=C p (93) g Gg, −1 CD 0 Iq 0 D DC Iq
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and consider the action of C on Siegel Unit Disk + given by: gG∈ SD=∈{ Z Mpq( C) /0 I p −> ZZ }
−1 −1 AB I BD A− BD C 0 I p 0 = ∈=C p (94) g Gg, −1 CD 0 Iq 0 D DC Iq Benjamin Cahen has study this case and introduced the moment map by identifing G-equivariantly g* with g by means of the Killing form β on gC : g* G-equivariant with g by Killing form β ( X , Y) = 2( p + q) Tr( XY ) The set of all elements of g fixed by K is h:
−qI p 0 h= {element of GK fixed by } , ξ∈=h , ξλ i 00 (95) 0 pIq ⇒ξλ++ =− +2 ∀∈ 0 ,,Z Z 2 i( p q) Tr( ZZ), Z D
Then, we the equivatiant moment map is given by:
C * ++ ∀∈X g , Z ∈ D , ψ( Z) = Ad(exp( − Z) ζξ( exp Z exp Z )) 0 * (96) ∀∈g G, Z ∈ D then ψψ( g . Z) = Adg ( Z )
ψξ is a diffeomorphism from SDO onto orbit ( 0 )
with:
++−−11+ (Ip− ZZ) ( −− pZZ qI pq) ( p + q) Z( I − Z Z ) ψλ(Zi) = (97) −−11 −+ −++ ++ − + ()p q( Iq ZZ) Z( pIqq qZZI)( ZZ)
+ −1 + Ipq ZI( − ZZ) and ζ (expZZ exp ) = (98) 0 Iq
5. New Entropy Definition as Generalized Casimir Invariant Functions for Coadjoint and Adjoint Representation
5.1. Casimir Invariant and Generalized Casimir Invariant
Hendrik Brugt Gerhard Casimir, a Dutch physicist, studied what is called Casimir operators and Casimir invariants (H. Casimir and Van der Waerden studied the SU(2) group, the group of isospin/angular momentum, as the model of the algebraic approach to the study of the unitary representations of semi-simple compact Lie groups). Kirillov has explained that Casimir operators are in one-to-one correspondence with polynomial invariants characterizing orbits of the coadjoint representation. Solutions are not necessarily polynomials and the nonpolynomial solutions are called generalized Casimir invariants. For certain classes of Lie algebras, all invariants of the coadjoint representation are functions of polynomial ones. In physics, Hamiltonians and integrals of motion of classical integrable Hamiltonian systems are not polynomials in the momenta [84- 100,103-104].
5.2. Souriau Entropy as Generalized Casimir Invariant in Coadjoint representation
In Souriau Lie groups Thermodynamics, we will see that coadjoint orbits lie on level sets of the Entropy that could be considered as a Casimir invariant function:
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SR: * → g (99) Q SQ( ) We will consider first the case of null-cohomology, Entropy as Casimir invariant function is a conserved quantity, because Casimir function has null Lie Poisson brackets functions [100,101]:
∂∂SH ** {SHQQ, }( ) =, , =∀→ 0 , H :gg R , Q ∈ , AB , = B( AB ,) Cartan-Killing form ∂∂QQ (100) dS∂ with ∂= S(Q) SQ( +δδ Q) = Q, dQε ε =0 ∂ ∂S We can observe that β = , then: ∂Q ∂∂ HH ** (101) Q,β ,=Q , adβ =∀→ 0 , H :gg R , Q ∈ , ada b =[ a , b] ∂∂QQ We can also write:
∂∂ ∂ ∂ SH HH** (102) Q,,,=Q ad∂∂SS = ad Q, =∀→ 0 , HR :g ∂∂QQ ∂∂QQ∂QQ ∂
∂ It means that ** S . We can remark that if we note **k ad∂S Q= adβ Q =0 , β = ad∂ Q= C adi Q = 0 ∂ S ij ∂S k ∂Q Q ∂Q j ∂Q
k with Cij the structure tensor, we observe that this equation is in fact the Casimir condition for invariant
function in coadjoint representation as we will see hereafter. The restriction of the Lie-Poisson bracket to an orbit generates a symplectic structure on the orbit, called the KKS (Kirillov-Kostant-Souriau) structure, or the canonical symplectic structure. Casimir function is characterized as a quantity which commutes with each linear functional on the Poisson manifold, and then it is conserved by dynamics of any Hamiltonian. Given a Hamiltonian HR:g* → , the equation of motion for Q ∈g* is:
dQ **dQ (103) ={Q, H} = ad∂∂HS Q with H =⇒= S{ Q, S} = ad Q =0 dt ∂∂QQdt In case of non-null cohomology, the Lie Poisson brackets functions are given by:
∂∂SH ∂∂ SH = +Θ = ∀** → ∈ {SH, }Θ ( Q) Q,,, 0 , HQ :gg , ∂∂QQ ∂∂ QQ (104) with Θ=− (XYJ , ) [XY, ] { JJXY,} where JxX ( )= JxX ( ), Θ( XY,) :gg × →ℜ with Θ (X ) = Teθ ( Xe ( )) X,Y Θ (XY ),
That we can develop in the following:
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∂∂SH ∂ S ∂ H = +Θ = {SH,}Θ ( Q) Q ,,, 0 ∂∂QQ ∂ Q ∂ Q ∂H ∂∂ SH = +Θ = {S,, H}Θ ( Q) Q ad ∂S ,0 ∂Q ∂Q ∂∂ QQ (105) ∂H ∂∂ SH = * +Θ = {S, H}Θ ( Q) ad∂S Q , ,0 ∂Q ∂Q ∂∂ QQ ∂∂SH ∂S ∀ =** +Θ + = ⇒ +Θ = H,,{ S H}Θ ( Q) ad∂∂SS Q , 0 ad Q 0 ∂∂QQ∂∂QQ ∂Q
We have found the generalized Casimir equation for Entropy in the non-null cohomology case:
= (106) {SH,0}Θ ( Q)
That coulbe also written: ∂ * S (107) ad∂S Q +Θ =0 ∂Q ∂Q This equation was observed by Souriau in his paper of 1974, where he has written that geometric temperature β is a kernel of Θβ , that is written:
β∈Ker Θβ ⇒ Q,,[ ββ Z] +Θ( , Z ) = 0 (108)
That we can develop to recover the Casimir equation:
* ⇒Q, adββ Z +Θ(ββ ,0 Z) = ⇒ ad Q , Z +Θ( ,0 Z ) = (109) ∂S** ∂∂ SS β = ⇒ad∂∂SS Q, Z +Θ, Z = ad Q +Θ , Z = 0, ∀ Z ∂Q∂∂QQ ∂∂ QQ
* ∂S ⇒ad∂S Q +Θ =0 ∂Q ∂Q
Then the generalized Casimir Equation in non-null cohomogy is given by:
**∂S k (110) ad∂ Q +Θ =C adi Q +Θ =0 S ij ∂S k j ∂Q ∂Q j j ∂Q
Given a Hamiltonian HR:g* → , the equation of motion for Q ∈g* is: dQ ∂∂H dQ S = =** +Θ = ⇒ = = +Θ = (111) {Q, H}ΘΘ ad∂∂HS Q with H S{ Q, S} ad Q 0 dt ∂∂QQ∂∂Q dt Q Level sets of the Casimir Entropy function, on which the coadjoint orbits lie, are symplectic manifolds.
5.3. Souriau Entropy invariance in coadjoint representation
If we note An( g* ) the space of analytic function on the dual Lie agebra g* , a function F **∈An( g ) is a
* ** * Casimir invariant if for any g∈∈ GX, g , we have F( Adg X) = F() X . We have observed previously that
Souriau’s Entropy analytic function SQ() defined on dual Lie algebra g* by Legendre transform of Massieu Characteric analytic function Φ()β (minus logarithm of Laplace transform) defined on Lie algebra g was an
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invariant function under the affine coadjoint action βθ=* +=. In case of SQAd( gg()) SAdQ () ( g) SQ( )
null-cohomology, Souriau cocycle cancels θ ( g ) = 0, and we recover Casimir invariant function in coadjoint
representation * = . S Adg () Q S( Q)
We can then claim that Souriau Entropy is an extended Casimir invariant function in case of non-null cohomogy. This characteristic of Souriau Entropy could be a new general definition of Entropy. In Souriau Lie groups Thermodynamics, Entropy SQ() is a generalized Casimir invariant function for coadjoint representation in case of non-null cohomology, and Massieu Characteristic function by Legendre duality is a generalized Casimir function for adjoint representation. We will explain how to prove that Souriau Entropy is invariant under the action of the group, starting from its definition: ∂Φ()β ∂SQ() SQ( )=ββ , Q −Φ ( ) with Q= ∈gg* and β = ∈ (112) ∂∂β Q
with Φ=−βλ− βξ,U () →* (113) ( ) log ∫ e dω and UM : g M ∂Φ()β Considering Souriau Entropy SQ( ) where the heat Q = ∈g* an element of the dual Lie algebra is ∂β parameterized by β ∈g an element of the Lie algebra, the Lie group G acts through gG∈ by adjoint
operator , the entropy is given by β with β given by fundamental Souriau Adg S Q( Adg ()) Q( Adg ())
equation:
* (114) Q( Adgg()βθ) = Ad () Q + ( g )
The invariance of Souriau Entropy is deduced from the following developments: − β βββ∈ → ⇒Ψ = Adg ( ), U λ g Adgg() ( Ad ()) ∫ e d ω M
* −1 − β ,Ad U −−β,U Ad− βθ g −1 ( g 1 ) ( ) Ψ=βλg = λ (Adg ( )) ∫∫ e dωω e d MM βθ, (g −1 ) (115) Ψ=Ψ(Adg (ββ )) e ( )
* − βθ,Ad− () g θθ−1*=− ⇒Ψ β =g 1 Ψ β ( g) Adg−1 ( g ) ( Adg ( )) e ( ) Φ=−Ψ⇒Φβ β β =Φ− β βθ−1* =Φ+ β β θ ( ) log()( Adg ()) ( ) ,( g ) ( ) ,Adg −1 () g Based on this expression of Massieu Characteristic function transform by action of the group, we can use Legendre transform to study how Souriau Entropy is changed:
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SQ()=β , Q −Φ () β ⇒ SQAd( ( g β)) = AdQAd gg ββ, ( ) −Φ( Ad g β) βθ=* + Q( Adgg()) Ad () Q( g ) −1 Φ( Adgg()β ) =− log( ΨAd ())β =− βθ ,( g ) +Φ( β) (116) *1− ⇒SQAd( ( gβ)) = AdAdQ gg β, ()+ θ( g) + βθ ,( g ) −Φ( β) ⇒ β= β**+ θ − βθ−Φ β S( Q( Adg)) Ad gg, Ad () Q( g) , Adg −1 () g ( ) ⇒ββ =** +− *θ β* θβ−Φ S( Q( Adgg)) , Adgg−−11 Ad () Q Ad( g) , Adgg −1 () ( ) **= ⇒ ββ= −Φ β = β AdAdQQSQAdg −1 gg() ( ( )) , Q( ) S ( )
We finally prove that Souriau Entropy is invariant in coadjoint representation * in S( Adg () Q+=θβ( g)) S ( )
general case of non-null cohomology, that we could write # , if we note affine coadjoint S( Adg () Q) = S (β )
#* action Adgg() Q= Ad () Q +θ ( g ) . This is also true in case of null-cohomology when the Souriau cocycle cancels
θ ( g ) = 0, and we recover classical generalized Casimir invariant function definition on coadjoint representation
for Entropy * generalized Casimir invariant function definition on adjoint representation for S( Adg () Q) = S (β )
Massieu Characteristic function . Φ=Φ( Adg ()ββ) ( )
5.4. Souriau Entropy given by Casimir Invariant Functions Equations
Based on development given in the following we can state that: As the Entropy S is a generalized Casimir invariant function in the coadjoint representation,
* ξ = , then S should be solution of the following differential equation: S( Adet h) S() h
k ∂SQ( ) CQij k= C ij () Q = Bij CQk =0 , i , jk , = dimg , with (117) ij k ∂Q =ij = j BQ ( xy,) Bxyij Q,,[ xy]
k Where Cij is the structure tensor of the Lie algebra g in the basis (ee12, ,..., en ) , while X k are the coordinates
* 12 n j in g in the basis (ee, ,..., e) defined by ee, i= δ ij . The structure tensor s given by
∂ φφ= k φ with k . (ei), ( e i) Ce ij( k ) φ (ei) = CX ij k , i= 1,..., n ∂X j
5.5. Characterization of Generalized Casimir Invariant Functions in Coadjoint Representation
We will describe recent characterization of generalized Casimir invariant functions by Oleg L. Kurnyavko and Igor V. Shirokov [85,89,96] who have proposed Algebraic method for construction of Casimir invariants of Lie groups coadjoint representations. Modern invariant theory based on geometric methods, which was credited classically as non-constructive, has some exception admitting a constructive solution related to the constructing invariants of Lie Groups representations.
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Let T be a connected Lie group, TG() a representation of the group G in the linear space V , Tg the
operators associated to the representation of the group G on the linear space V , then the invariants are given by:
∞ (118) FTx( gg) = Fx() , x ∈∈ Vg , GT , ∈ TG ( ),() Fx ∈ C ( V )
With the properties that:
−1 T= I , T = TT , T−1 = T (119) e ggab g a g b g ( g)
Solution is given by the following differential equation:
i dimV ∂ ∂Fx() (Tg ) −===ij i j (120) ∑ txkj ik0 with tkj and k 1,...,dim G ij, ∂∂xg ge=
i tkj are elements of the matrices of the Lie algebra representation basis of G .
ij∂ That we can write t= − tx and tFxk ()= 0. k kj ∂xi If we consider the dual space V * , the co-tangent representation is given by:
T* () gXTgx ,()= Xx , (121)
And co-represnetation invariants are given by: ∂ ** * i (122) tFk ( X )= 0 with tk = t kj X i ∂X j They have underlined the relationship between invariants of representations and conjugate representations, where the algebraic construction of Lie groups representations invariants are given by invariants of the conjugate representation with respect to the invariants of the original representation. Shirokov Theorem 1: Let Fx() be a non-degenerate invariant of the representationTG(), then conjugate representation invariant can be found by Legrendre tranform: ∂∂Fx() Fx() F* ( X )=−= xXi Fx ( ) xX , − Fx ( ) with X = such that X= (123) ii∂∂xxi and also the converse problem: ∂∂** ii**FX() FX() (124) Fx( )=−=− xXFXi ( ) xX , FX ( ) with x = such that x= ∂∂XXi Shirokov has considered Fx() the representation invariant TG(), and FX* ()the representation invariant TG* ()conjugate to TG(), with the conditions : ∂Fx() ∂ * −=ij and i FX() (125) txkj i 0 tXlj i = 0 ∂x ∂X j
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∂FX* () ∂ ∂xkk∂X ∂∂Fx() x i =i k −=i +ikk − i tXlj i tXlj i x() XXk F( xX ()) tXlj i Xk tXx lj i tXlj i k ∂XXjj ∂ ∂Xj ∂ X j ∂∂ xX j ∂F* () X ∂∂ xkk Fx() ∂ Fx () ∂ Fx () ∂ Fx () ∂ x (126) i =i +−i kjδ i tXlj i tXlj i ktlj i xk t lj ik ∂Xjj ∂ Xx ∂ ∂ x ∂ x ∂∂ xXj ∂∂FX* () F (x) ii= = tXlj i txlj j i 0 ∂X j ∂x Invariant Casimir Functions of the coadjoint representation has been studied for completely integrable Hamiltonian systems, as classical systems on the orbits of the coadjoint representation. Oleg L. Kurnyavko and Igor V. Shirokov have considered the relationship between invariants of representations of Lie groups and their conjugate dual representations. Considering the coadjoint action given by:
** = − ∈ ∈∈ (127) AdXxg , XAdx ,g 1 , gGX ,gg , x
Invariants of a coadjoint representation are called Casimir functions, with the property:
** * (128) F( Adg X) = F( X )
the infinitesimal invariance is given by the equations: ∂FX* ( ) k (129) CXij ( )= 0 with CXij ( )= CXij k , ijk , ,= dimg ∂X j
The number of functionally independent invariants is given by the rank of the matrix CXij (), called the index
of the Lie algebra * g: indgg=dim − suprankCij ( X ) X∈g*
From these adjoint and coadjoint representation, Shirokov has introduced the following theorem: Shirokov Theorem 2:
Let F( Adg x) = F() x be a non-degenerate invariant of the adjoint representation AdG , then conjugate
* representation invariant, invariant of coadjoint representation AdG can be found by formula: ∂∂Fx() Fx() F* ( X )=−= xXi Fx ( ) xX , − Fx ( ) with X = such that X= (130) ii∂∂xxi
** * and also the converse problem , let F( Adg X) = F( X ) , invariant of coadjoint representation AdG is given
∂∂** by: ii**FX() FX() (131) Fx( )=−=− xXFXi ( ) xX , FX ( ) with x = such that x= ∂∂XXi
* k ∂FX( ) CXij k= C ij () X = Bij Nota: CXk =0 , i , jk , = dimg , with (132) ij k ∂X =ij = j BX ( xy,) Bxyij X,,[ xy]
5.6. Constructing Generalized Casimir Invariant Functions in Coadjoint Representation
I. V. Shirokov has proposed a method for constructing invariants of the coadjoint representation of Lie groups with an arbitrary dimension and structure based on local symplectic coordinates on the coadjoint orbits. Oleg L. Kurnyavko and Igor V. Shirokov have also proposed a general method for constructing Casimir invariants.
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We will give some other developments of Casimir Invariant Functions by A.T. Fomenko and V.V. Trofimov, related to Orbits of the coadjoint representation and the associated canonical symplectic structure
* The coadjoint orbit Οh passing through the point h∈g is given by
** (133) Ο∈hg={Ad h / g G} where h ∈g
* ** ThhΟ=/{ adρ h ρ ∈⊂gg} ,h ∈ g
* d Adtρ h exp vT= ∈Ο , ρ ∈g (134) dt hh t=0
1*ni Let (e1, ,..., en) basis of gg , (e ,..., e) basis of , with ee , j= δ ij
** d hii,, Adttρρ h d Ad h e iiexp exp h= hei ⇒= v = dt dt (135) tt=00= d h, Ad−tρ ei d Ad−tρ e i exp exp i vh= = , dt dt t=0 t=0 i * * v= h,, −[ρ ei] =− adρ h , e ii = v , e ⇒=− v adhρ
Kirillov, Kostant and Souriau have introduced a KKS 2-form on co-adjoint co-orbits that then inherit a structure of homogeneous symplectic manifold:
* ** ξη,∈ΟThh =/{ adχ h ρ ∈gg} ⊂ ,h ∈ g (136) ** * * ωh ( ξη, ) = ω(adξη h, ad h) = h ,[ξ11 , η] with ξ =adξ h and η= adη h 11 1 1
ω ** This KKS 2-form is invariant with respect to the coadjoint action ωgf( Ad ξ,, Ad f η) = ω h( ξη) :
** ** ** ωgf( Ad ξ,, Ad f ηω) = gf( AdadhAdadhξη f ) 11 with g= Ad** h , ξη = ad h , = ad * h and f ∈∈ G , g , h g* f ξη11 ** * * ** * * Adadhf ξξ= adAd( Adh f ) and Adadhf ηη= adAd( Adh f ) 11ff11 ωAd** ξ,, Ad ηω= ad * Adhad * * Adh * (137) g( f f) g( Adffξη11( f) Ad( f )) ωξAd* , Ad*ηω= ad ** gad, g= gAd ,,ξ Ad η gf( f) g( Ad ffξη11Ad ) f11 f ω** ξ η= ξη = *ξη = * gf( Ad,,, Ad f) g Ad f[ 11] Adff−−11 g,[ 11 ,] with h Ad g
** ωgf( Ad ξ, Ad f η) = h ,,[ ξη11]
** ωgf( Ad ξ,, Ad f η) = ω h( ξη)
The symplectic structure is given due to the property that dω = 0 , that could be proved making link with Jacobi identity.
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∂m ω = = i Let gradskew m such that (v , gradskew m) v( m )∑ v i smooth vector field on M i ∂x ij (138) {m,, n} = ωω( gradskew m gradskew n) = ∑ ij( grad skew m) ( gradskew n) ij<
i ∂m ∂∂mn =ωωij ⇒=ij with ( gradskew m) ∑∑j{m, n} ij j ∂x ij< ∂∂xx
We can compute terms of Jacobi identity:
{m,,{ np}} =−=( grad m){ np, } − L{ np,} with Lζ : Lie derivative Skew gradSkew m
If ξ = gradSkew m (139) ij ∂∂npij ∂∂ npij ∂∂(ξξnp) ∂ pij ∂ n( ) Lξξ{ np, } = Lω =Lξ( ωω) ++ ω ∂∂xxij ∂∂ xxij ∂ x i ∂ x j ∂ x i ∂ x j
ij ∂∂np ij ∂∂np Lξξ{ np,} = L(ω) ++={ ξnp,,} { n ξω p} Lξ( ) −{{mn,},p}−{ n ,,{ mp}} ∂∂xxij ∂∂xxij ij ∂∂np ⇒++={m,,{ np}} {{ mn ,,} p} { n ,{ mp ,}} Lξ (ω) ∂∂xxij
Using Elie Cartan formula Lξω= i( ξ) d ω + di( ξω) . If ξ is a Hamiltonian vector field, di(ξω) = 0 and
then Lξω= id( ξω) .We can the observe that if dω = 0 , then the Jacobi identity is satisfied
{m,,{ np}}++={{ mn ,,} p} { n ,{ mp ,}} 0 and conversely.
Let consider the Berezin Bracket: ∂∂mn {mn, } =−= Cxkk with e , e Ce ij k ∂∂xxij i j ij k 12 n * (140) where (ee12 , ,..., en ) basis of Lie algebra g, (ee , ,..., e) basis of dual Lie algebra g 1*n of corresponding coordinates xx ,..., for gg , xx1 ,...,n for This Berezin Bracket is given by:
= **= {m, n}x dmx( ad dn() x () x) ( addn() x () x)( dmx ) (141) ∂∂nm ∂n ∂ m = = k = = {m,,, n}x x[ dnx dm x] C ij x k ij with dnx ei , dm x e j ∂∂xx ∂xij ∂ x
∂∂mn By developping Berezin Bracket {mn, } =−= Cxkk with e , e Ce, we can prove that the bracket ij k ∂∂xxij i j ij k
verify jacoby identy {m,,{ np}}++={{ mn ,,} p} { n ,{ mp ,}} 0 and then dω = 0 .
We will see that differential equation for (semi-)invariants of the coadjoint representations could be established.
We will note An( g* ) the space of analytic function on the dual Lie agebra g* . A function F **∈An( g ) is
* ** * an invariant if for any g∈∈ GX, g , we have F( Adg X) = F() X , and is semi-invariant if
** * F( Adg X) = χ() g F ( X ) where χ()g is a character of the Lie group G .
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We have a representation of Lie algebras φ :g →ΓVec( ) defined on basis (ee12, ,..., en ) in g,where Vec(Γ)
is the space of vector fields on Γ an open subset in g* , given by: ∂ k (142) φ (ei) = CX ij k , i= 1,..., n ∂X j
k Where Cij is the structure tensor of the Lie algebra g in the basis (ee12, ,..., en ) , while X k are the coordinates
* 12 n j in g in the basis (ee, ,..., e) defined by ee, i= δ ij . The representation is not dependent of the choice of
the basis, with the property: k φφ(ei), ( e i) = Ce ij φ( k ).
We have the property, that:
n ** d F( Adtξ h) n e =( −φξ( )) Fh* ( ) (143) dt n t=0 This result is obtained by the following development:
** *** dF( Adtξ h) ∂∂FFd Adttξξ h,, X iid h Ad− e e = (hh ). ee= ( ). dt∂∂ Xii dt X dt t=0 tt=00=
d h, Ad−tξ ei e =−=−h,[ξ , e] hC ,kj ξ e with [ ξξ , e] =j e , e = Ckj ξ e dt i ij k i j i ji k (144) t=0
d h, Ad−tξ ei e =−=−hek, C kjξξ e C kj h dt k ji k ji k t=0
** * dF( Adetξ h) k j ∂F * = −C jiξ hk (ξ) =( − φξ( ) Fh)() dt ∂X i t=0
** We use then Taylor expansion of ξ given by: F( Adet h)
n ∞ (−φξ()) F * **= * + n (145) F( Adetξ h) F() h ∑ ().h t n=1 n!
* ** * n * * We can observe that F is invariant if tξ = and then −=φξ()F 0 or φξ()F = 0 that F( Ade h) F() h ( )
∂ * could be written kj F Chjiξξ k ( ) = 0 . ∂X i If F * is semi-invariant of the coadjoint representation of group if and only if:
** φ(eFi )=−= λ i F with λχ ii d( e) (d χ : derivative of χ at the group G identity element
ξξtχξ( ) **= χχtt* = * F( Adetξ h) ( e) F( h ) with ( e) e n (146) −=φξF** h χ ξ Fh() ( ( )) ( ) * ( ) n ∞ χξ( ) ⇒=+** * n * F( Adetξ h) 1 ∑ t. F () h = n! n 1
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6. Lie Groups Thermodynamics for SE(2) Lie group
After SU (1,1) Lie group with null cohomology and then without Souriau one-cocycle, we will consider Souriau model for SE(2) Lie group with non-null cohomology and then with introduction of Souriau one- cocycle [107].
We will consider first SO(2) Lie group:
−cosϕϕ sin (147) SO(2) = Rϕ = /ϕ ∈ R sinϕϕ cos
A vector at the identity to SO(2) is given by:
dR 01 tη = −η ℑ with ℑ =, ℑT = ℑ−1 = −ℑ (148) − dt t=0 10
We consider the special Euclidean group SE(2)= SO( 2) × R2 .
R τ ϕ 2 (149) SE(2) =/Rϕ ∈∈ SO( 2) ,τ R 01
the group operation is given by:
Rϕτ1 R ϕ τ 2 RR ϕϕϕ Rττ21++ Rϕϕϕ+ Rττ21 1 2 = 121 = 121 01 01 0 1 0 1 (150)
⇒=+(R,.τ1) ( Rϕ , τ 2) ( RRϕϕ+ , ϕ ττ21) 1 2 12 1
−1 Rϕττ RR−− ϕϕ− −1 11111 (151) = ⇒=−(Rϕ,,ττ11) ( RR−− ϕϕ) 01 0 1 1 11
The Lie algebra se(2) of SE(2) has underlying vector space R3 and Lie bracket:
2 −ℑξ u (ξ,u)∈ se (2) =×⇒ R R ∈se(2) (152) 00
Lie bracket is given by:
(ξ,uv) ,( η ,) =( 0, ξη ℑ+ℑ vu) (153)
Adjoint action of SE(2) is given by: R τ −ℑ ξ u R − R τ −ℑ ξ ξτ ℑ +Ru Ad(ξ, u) = ϕ −−ϕϕ= ϕ (Rϕ ,τ ) 01 0 0 0 1 0 0 (154)
Ad(ξ,, u) =( ξ Rϕ u +ℑ ξτ) (Rϕ ,τ )
Coadjoint action of SE(2) is given by:
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* Ad( m,ρ) =( m +ℑ Rϕϕρτ., R ρ) (155) (Rϕ ,τ )
The moment map J:2 R2*→ se ( ) of SE(2) is defined by:
(156) J(ξ ,u) () x= Jx ().,(ξ u)
with the right action of SE(2) on R2 :
xR.,( ϕτ) = R−ϕ ( x − τ) (157)
the infinitesimal generator of (ξ,u)∈ se (2) has the expression: d x., Rtξ tu d R( x− tu) ( ) −tξ (158) (ξξ,u) 2 () x= = =ℑ−xu R dt dt t=0 t=0
Let 2* be the moment map of this action relative to the symplectic form, we can J(ξ ,u) ( x ): R→ se ( 2) compute it from its definition:
= − ωξ dJ(ξ ,u) ( x ). y 2(( , u)R2 , y)
with ω(( ξ ,u) 2 , y) = ωξ( ℑ− xuy, ) =( ξ ℑ− xu). ℑ= y(ξ x +ℑ uy). R (159)
⇒dJ(ξ ,u) ( x ). y =− 2(ξ x +ℑ u) . y
1122 ⇒Jξ ,u () x = − 2ξξ x + ℑ ux. = − 2 x , −ℑ x .( , u) ( ) 22
1 2 2 JxJxuJxξ ,u ()= ().,(ξ ) ⇒ () = − 2 x , −ℑ xxR , ∈ ( ) 2
We then compute the one-cocycle of SE(2) from the moment map:
θτ= −=−* τ ((Rϕτ, )) J(0.( Rϕ , )) Adτ J(0) J( R−ϕ ) (Rϕ , ) (160) 1122 θRϕτ, =−ℑ=−2, ττ RR−ϕ 2, τπ τ (( )) −−ϕ 222
Coadjoint orbit of SE(2) are generated by:
Ο=* ρθ + τ τ∈ (m,ρ ) A( m,) (( Rϕϕ ,/,)) ( R) SE ( 2) { (Rϕτ, ) } (161) 2 Ο=−ρ x R πρτ. − τ ,R− ϕ ρ − 2 R πϕ τ /, R τ ∈ SE ( 2) (m, ) − −−ϕ ( ) 22
The Souriau Symplectic form in this case of non-null cohomology is given by:
ω **ρ−ℑ ρ−ℑ=−ℑ+ℑ+ℑ ρξ η (mm,ρρ)( ', ') (ad( ξ , u) ( m', ') ( 0, 2u) , ad(η,v) ( m ', ') ( 0, 2v)) '.( v u) 2 u . v (162) 2 3 with (m ',ρ ') = xR −π ρτ. − τ , R− ϕπ ρ − 2 R τ ∈Ορ ⊂ R − −−ϕ (m, ) 22
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With the expression of moment map and Fourier Transform for SE(2) by Kirillov Orbit method and Kirillov character, we can compute Souriau Gibbs density, in the same way that we have developed the case for SU (1,1) .
7. Conclusion: Lie Groups Thermodynamics for Machine Learning
Lie Group tools based on Representation Theory and Orbits Methods could be used with Souriau-Fisher Metric on Coadjoint Orbits that is an extension of Fisher Metric for Lie Group through homogeneous Symplectic Manifolds on Lie Group Co-Adjoint Orbits. In appendix C, we provide an algorithm invented by Jean-Marie Souriau to compute Exponential Map of matrices, that we can use for “geodesic shooting”.
Different tools developed based on Souriau Lie Groups Thermodynamics and Kirillov Representation Theory for :
• Supervised Machine Learning
o Geodesic Natural Gradient on Lie Algebra: Extension of Neural Network Natural Gradient from Information Geometry on Lie Algebra for Lie Groups Machine Learning.
o Souriau Maximum Entropy Density on Co-Adjoint Orbits: Covariant Maximum Entropy Probability Density for Lie Groups defined with Souriau Moment Map, Co-Adjoint Orbits Method & Kirillov Representation Theory
o Symplectic Integrator preserving Moment Map: Extension of Neural Network Natural Gradient to Geometric Integrators as Symplectic integrators that preserve moment map
• Non-Supervised Machine Learning
o Souriau Exponential Map on Lie Algebra: Exponential Map for Geodesic Natural Gradient on Lie Algebra based on Souriau Algorithm for Matrix Characteristic Polynomial
o Fréchet Geodesic Barycenter by Hermann Karcher Flow: Extension of Mean/Median on Lie Group by Fréchet Definition of Geodesic Barycenter on Souriau-Fisher Metric Space, solved by Karcher Flow.
o Mean-Shift on Lie Groups with Souriau-Fisher Distance: Extension of Mean-Shift for Homogeneous Symplectic Manifold and Souriau-Fisher Metric Space
Figure 4. Supervised/Non-Supervised Machine Learning based on Lie Groups Thermodynamics
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La notion classique d’ensemble canonique de Gibbs est étendue au cas d’une variété symplectique sur laquelle un groupe de Lie possède une action symplectique (“groupe dynamique”). La définition rigoureuse donnée ici permet d’étendre un certains nombre de propriétés thermodynamiques classiques (la température est ici un élément de l’algèbre de Lie du groupe, la chaleur un élément de son dual), notamment des inégalités de convexité. Dans le cas de groupes non commutatifs, des propriétés particulières apparaissent : la symétrie est spontanément brisée, certaines relations de type cohomologique sont vérifiées dans l’algèbre de Lie du groupe – Jean-Marie Souriau, Mécanique Statistique, Groupes de Lie et Cosmologie, colloque CNRS n°237 – Géométrie Symplectique et physique mathématique
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Appendix A: Coadjoint orbits and Moment Map for SU(1,1)
We give more details developpements to obtain SU(1,1)/K coadjoint orbit and moment map [39].
If we consider hyperbolic Group SL(2, R)
mp SL(2, R ) = ∈GL(2, R) / mn −= pq 1 (A1) qn
Elements of SL(2, R) could be written with Iwasawa decomposition:
g= kθθ. antb . with k ∈∈∈ K , at An , b N (A2)
−cosθθ sin −iθ m− iq Kk=θ =/ 0≤<θπ 2 with e= sinθθ cos mq22+ et 0 (A3) ==∈=+2t 22 A at −t / tR with e m q 0 e 1 b mp+ qn N==∈= nb / bR with b 01 mq22+
K is a maximal compact sub-group of GSL .
Group of Unit Disk automorphisms
Consider SU (1,1) sub-group of SL(2, R) given by ab 22 (A4) SU(1,1) == A **/ det(A ) =−= a b1, a, b ∈ R ba
The following interior automorphism that transforms SL(2, R ) to SU (1,1) , inducing an isomorphism between them:
SL(2, R )→ SU (1,1) (A5) −1 1 1 −i g CgC with C = 2 1 i
SU (1,1) acts on the Poincaré unit disk D=∈<{ zCz/1}:
+ ab −−11az b (A6) g. z = with g =∈∈**SU(1,1) and z D ba**+ ba
Valentine Bargmann has parameterized SU (1,1) :
γ <1 b γ = −1/2 iω 2 (A7) a ⇒=ae(1 −γ ) ωα= arg( ) mod 2 π − 2 1/2 be=iωγγ1 − ( )
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Then SU (1,1)={(γω ,) / γ < 1, ω ∈−] π , + π]} with Group composition law given by:
(γω,) .( γ ', ω ') = ( γ '', ω '')
−1 γ''=++ γ ' γee−2'iiωω 1 γγ *'2' (A8) 1 −1 ω''=++ ω ω ' log 1 +γγ*'ee−− 2'iiωω 1 + γγ '* 2' mod 2π 2i ( ) g=(γω,,) ⇒=−− ge−12( γiω ω)
SU (1,1) is topological product of unit disk and circle.
Universal covering of SL(2, R )
If we consider GR={(γω,) / γ <∈ 1, ω }, the following mapping:
Θ→:G SU (1,1) with KerΘ={(0, 2 kπ ) / k ∈ Z} (A9) Θ= (γω , )( γω , mod 2 π)
Topological product of unit Disk D dans C and real straight line R is the universal covering of SU (1,1) .
e−iθ 0 A maximal compact subgroup of SU (1,1) is −1 = θ ∈ and the subgroup for CKC iθ / R 0 e G is Θ=∈−−11(CKC ) {(0,θθ) / R}.
Pukanszky and Sally have defined irreductible unitary representation of SL (2, R) , classified in principal serie, discrete serie and complemantary serie.
The Lie algebra g of G and SU (1,1) is given by: − 11 0 11 01 0 1 (A10) l1= ,, ll 20= = 20− 1 22 10 1 0
with the commutation relation:[ll01,] ==−= l 2 ,,[ ll 12] l 0 ,,[ ll 20] l 1 .
Dual Lie algebra g* of g to g thanks to Killing form:
(A11) B∑∑ xlii,2 yl ii =( x11 y +− x 2 y 2 x 0 y 0) ii=1,2,0= 1,2,0
Coadjointes orbits of SU (1,1)
Considering adjoint representation Adg :gg→ , and coadjointe representation as transpose linear * * ** * mapping of Ad −1 , written by = −1 , Ad :gg→ . g Adg ( Adg ) g
Let f = (η,0,0) in base {lll120,,} with η > 0 coadjoint orbit of f is:
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−1 ac Ο=η gfg/ g = ∈SL(2, R ) bd (A12)
Ο=η {ηη(1 + 2bc) ,( bd − ac) , η( bd + ac) / a , b , c , d ∈ R with ad − bc = 1}
et 0 Stabilizer of f is = ∈ = = ∪− = = ∈ (A13) Gf1()/.{ gGgffA} { A} with Aat −t / t R 0 e
1 b = = ∪− where and SL(2, R ) K ' NA ' with A ' A{ A} N= nb =/ bR ∈ 01 −θθ cos sin and as is a bijection with Ο , then Ο is in Kk'=θ =/0≤<θπ SL(2, R ) / G1 ( f ) η η sinθθ cos
bijection with KN' , diffeomorph to KI/{ d }× N. Then all element x ∈Οη can be written through * this bijection x= knθθbb, k ∈∈ K ', n N. Οη is set of points l=( xxx120,,) ∈g such that:
222 2 * xxx1+−= 20η >0 a one sheet hyperboloid in g .
We will study discrete sequence.
Quantization of Kähler Manifold
Let (M ,ω) a Kalherian manifold of dimension 2n (complex manifold with complex structure J) with 2-form g( X,, Y) = ω ( X JY ) , a Riemannian structure on M .
We have seen that SU (1,1) is conjugate of SL2 () R in SL(2, R )
αβ ab ** ∈∈SU (1,1), SL2 () R βα cd (A14) αβ −i 1 −i ab i i ** = βα 2 1 i cd −11
1 a =(αα +**) ++( ββ) 2 αα+=+* ad 1 1 ** α =(a ++ d) ib( − c) * b =αα − −− ββ αα−=ib( − c) ( ) ( ) (A15) 2 ⇒⇒2i * 11 ββ+=−ad − ** β=(a −− d) ib( + c) c=(αα −) +−( ββ) 22* i ββ−=−+ib( c) 1 d =(αα +**) −+( ββ) 2
1 1 0 01 0− 1 1 Let in base where 1 11with > fh=0,0, − {lll120,,} l1= ,, ll 20= = h 2 20− 1 22 10 1 0 2 1 1 (respectively fh=0,0, + with h <− ) , the coadjoint orbit of f is: 2 2
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−1 ac Ο=h gfg/ g = ∈SL(2, R ) bd 11 (A16) hh−−sign hhsign 1 2222 22 22 22 Ο=h h −sign h( ac + bd ) ; (d+− c( a + b )); (d+++ c a b ) 22 2
* Οh appears as points l=( xxx120,,) ∈g such that:
1 >> 2 xh0 0 if 222 2 1 2 (A17) xxx1+−=−− 20 h <0 with 2 1 xh<0 if <− 0 2
* Οh is one of hyperboloid sheets of g , associated to representation π h of discrete serie of G .
x1 = h'( ac + bd ) h'12222 (A18) x2 =( dacb −+−) with h ' = h − sign h 22 h' 2 222 x0 =( dacb +++) 2
Then, we obtain:
** x1 = ih'(αβ − α β ) ** (A19) xh2 =−+'(αβ α β ) =αα** + ββ xh0 '( )
If we set:
− * ih'( z z ) x1 = 2 eiθ 1− z α = 2 * 1− z −+hz'( z) (A20) ⇒=x1 2 ze−iθ 1− z β = 2 2 1− z hz'1( + ) x0 = 2 1− z
We use the parametrization of Οh by unit disk D :
D= zCz ∈/1 < →Ο { } h (A21)
z ( xxx120 ,,)
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1 Im(z ) xh1 =−−2 sign h2 2 1− z 1 Re(z ) (A22) xh1 =−−2 sign h2 2 1− z 2 1 1+ z xh= − sign h 0 2 2 1− z
This parametrisation provides a kahlerian structure for Οh , inherit from D .
We have Ο=h G/ K = SL (2, R ) / K , and as SL(2, R ) is isomorph to SU (1,1) , Οh is identified with α 0 SU(1,1) / K= D . Stabilizer of the origin is . Then Ο is globally 0 K0 = * /1α = h 0 α diffeomorph to D for all h .
' ωh ' We can compute Liouville measure on Οh . This measure is ω = where ωh is the canonical 2- h 2π
form on orbit. Parameterization on Οh gives:
' ω −−ih(21) 2 ω =h = (A23) h 2 signh dz 2π 21π ( − z )
We can observe that:
i dx =z − z* 1 − zz* dh ' +− h '1 z*2 dz −− h '1 z2 dz * 1 2 ( )( ) ( ) ( ) 1− z ( ) −1 (A24) dx =z + z* 1 − zz* dh ' ++ h '1 z*2 dz −+ h '1 z2 dz * 2 2 ( )( ) ( ) ( ) (1− z ) 1 ** * * dx =1 +−zz 1 zz dh '2 + z h ' dz − 2 zh ' dz 0 2 ( )( ) 1− z ( )
*22 2 2 zz−11 − z −−( z) −−ih'1 z ( ) * *2 2 (A25) dx120 dx dx = detz++ z 1 z 1 + z 2 6 (1− z ) 2 * 122+ zz z
2 −2'ih 2 2 = = ω (A26) dx120 dx dx2 dh' dz h '' dh h (1− z ) π
* 222 This the measure defined on open set of g given by lll1+−< 200 .
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Appendix B: Bargman Parameterization of SU(1,1)
SU (1,1) is isomorphic to SL(2, R) = Sp( 2, R) through the complex unitary matrix W : ab SL(2, R) = g =/ det g = ad −= bc 1 (A27) cd
ab T 01+ Sp(2, R) = g = /, gJg= J J = (A28) cd −10 −−11 11ωω + −1 π W= = ( W) with ω = eii /4 =(1 + ) (A29) 22−ωω If we observe that W−1 JW= − iM , the isomorphism is given explicitely by:
ab −1 Re(αβ+−) Im ( αβ −) =g() u = WuW = (A30) cd Im(αβ+−) Re( αβ) αβ a+− d ib − c a −+ d ib + c −1 1 ( ) ( ) ( ) ( ) (A31) **=u() g = W gW = βα 2 (a−− d) ib( + c) ( a ++ d) ib( − c) We can also make also a link with SO(2,1) of “1+2” pseudo-orthogonal matrices: +10 0 T (A32) SO(2,1)= Γ∈ GL( 3, 3) / det(Γ) = 1, ΓKK Γ =Γ , = 0 − 1 0 00− 1
11222 2 222 2 (abcd+++) ( abcd −+−) −− cdab 22 11 (A33) Γ()g =( abcd222 +−− 2) ( abcd 222 −−+ 2) cdab − 22 −−bd ac bd− ac ad + bc with −1 −1 Γ( g1) Γ( g 2) =Γ( gg 12), Γ=Γ () I I , ( g) =Γ( g) The SO(2,1) matrix corresponds to any SU (1,1) : α22+ β αβ **αβ 2Re 2Im Γ=(u ) 2Reαβ Re( α22+ β) Im( α 22 − β ) (A34) 22 22 −2 Imαβ −+ Im( α β) Re( α − β ) 11 and αβ=±( Γ+Γ+Γ−Γ) ii( ), =( Γ−Γ) 2211 12 12 21 α 10 20 The properties of connectivity of Sp(2, R ) is described by its isomorphy with SU (1,1) . Using unimodular condition: 22 222 2 α− β =⇒+−=+≥1 αRI α β R1 β I 1 with αα =+RIRIii α and ββ =+ β
If βI is fixed, (ααβRIR,, ) are constrained to define a one-sheeted revolution hyperboloid, with its circular waist in the α plane. To SU (1,1) , we can associate the simply-connected universal covering group, using the maximal compact subgroup U (1) and corresponding to the Iwasawa decomposition (factorization of a noncompact semisimple group into its maximal compact subgroup times a solvable subgroup). 1 ω=arg α = i ln ( αα*1− ) ω 2 α β ei 0 λµ (A35) = λα= > ** iω * with 0 βα 0 e µλ * − ω α µβ=e i = β α
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22 2 β=eiω µα, −=−= β λ2 µ1 so µ < λ (A36) Bargmann has generalized this parameterization for Sp(2, N R) , more convenient but difficult to generalize to N dimensions. For SU (1,1) :, Bargmann has used (ωγ, ) : µβ 1 γ γ==<= ( γλ 1) , , µ = (A37) 22 λα 11−−γγ For SL(2, R) = Sp( 2, R) , the Bargman, parameterization is given by this decomposition of a non- singular matrix into the product of an orthogonal and a positive definite symmetric matrix: ab cosω−+ sin ωλ Re µ Im µ = (A38) cd sinω cos ω Im µλ− Re µ −iω Conversely: ωµ=arg (a +− d) ib( − c) , = e( a −+ d) ib( + c) ω is counted modulo 2π , ωω≡ (mod 2 π) . SU (1,1) and SL(2, R) = Sp( 2, R) are described when ω is counted modulo 2π , ωω≡ (mod 2 π) . Valentine Bargmann has proposed the covering of the general symplectic group Sp(2, N R): AB 0 I TT N (A39) Sp(2, N R) == g /,,gJ2N g = J 22 NN J =−= J 22 NN J CD −I N 0 with relations: T TT TT TT TT T AB= BA,,, AC = CA BD = DB CD = DC , AD −= BC I N (A40) DBTT− ∈ ⇒=−1 T = (A41) g Sp(2, N R) g M22NN g M TT −CA Bargmann has observed that although Sp(2, N R) is not isomorphic to any pseudo-unitary group, its inclusion in U( NN, ) will display the connectivity properties through its unitary UN( ) maximal compact subgroup, generalizing the role of U(1) = SO (2) in Sp(2, R). ωω−−11 = ⊗ a 22NN× matrix where 1 ππ/4 /4 with iπ /4 1 , which WNN WI WW=1 = ω =ei =(1 + ) 2 −ωωππ/4 /4 2 gives the NN× block coefficients. A+− D iB − C A −+ D iB + C αβ −1 1 [ ] [ ] [ ] [ ] (A42) u( g) = WNN gW = = ** 2 [ A−− D] iB[ + C] [ A ++ D] iB[ − C] βα αα++−= ββII, α + α − βT β * = With NN (A43) TT T* + αβ−= βα0, α β −= β α 0 αβ+ − T and −11= +−= (A44) u M22NN uM + T −βα The symplecticity property of g becomes: I 0 +−1 N (A45) uM2N u= M 22 N, M N = iW N J 2 NN W = 0 −I N AB Reαβ+− Im αβ − −1 ( ) ( ) (A46) =g() u = WNN uW = CD Im(αβ+−) Re( αβ) Valentine Bargmann has extended the well-know theorem that any real matrix R may be decomposed into the product of an orthogonal Q and a symmetric positive definite matrix S , uniquely as R= QS . Bargmann has shown that if R∈ Sp(2, N R) , then R= QS with Q, S∈ Sp( 2, N R) where Q maps onto unitary matrix and S maps onto Hermitian positive definite matrix: α 0 0 ξ + and T (A47) uQ( ) = * ,αα=IUNN , α ∈ () uS( ) = exp* ,ξξ= 0 α ξ 0
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We can generalize Bargmann parameterization of SU (1,1) to Sp(2, N R) : eIiω 0 λµ ωλµ =N ⊕>λ (A48) u{ ,, } −iω **,det 0 0 eIN µλ Then the Bargmann parameters are: 1 detα ω= arg det αλ ,=e−−iω αµ , = eei ωω β , iN = ,detλ= det α > 0 (A49) N detα The Sp(2, N R) matrices in terms of the Bargmann parameters are:
cosωIINN− sin ω Re( λµ +−) Im ( λµ −) g{ωλµ,,} = (A50) sinωIINN cos ω Im( λµ+−) Re( λµ) V. Bargmann has proposed the covering of the general symplectic group Sp(2, N R): AB 0 I TT N (A51) Sp(2, N R) == g /,,gJ2N g = J 22 NN J =−= J 22 NN J CD −I N 0 T TT TT TT TT T AB= BA,,, AC = CA BD = DB CD = DC , AD −= BC I N (A52) Bargmann has observed that although Sp(2, N R) is not isomorphic to any pseudo-unitary group, its inclusion in U( NN, ) will display the connectivity properties through its unitary UN( ) maximal compact subgroup, generalizing the role of U(1) = SO (2) in Sp(2, R) : ωω−−11 = ⊗ ,2NN× 2 matrix where 1 ππ/4 /4 with iπ /4 1 . WNN WI WW=1 = ω =ei =(1 + ) 2 −ωωππ/4 /4 2 A+− D iB − C A −+ D iB + C αβ −1 1 [ ] [ ] [ ] [ ] (A53) u( g) = WNN gW = = ** 2 [ A−− D] iB[ + C] [ A ++ D] iB[ − C] βα ++ +T * TT T* + with αα−= ββIINN, α α − β β = and αβ−= βα0, α β −= β α 0 (A54) The symplecticity property of g becomes: I 0 +−1 N (A55) uM2N u= M 22 N, M N = iW N J 2 NN W = 0 −I N AB Reαβ+− Im αβ − −1 ( ) ( ) (A56) =g() u = WNN uW = CD Im(αβ+−) Re( αβ)
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Appendix C: Souriau Algorithm for Exponential Map
The algorithm to compute characteristic polynomial of a matrix was discovered by Urbain Jean Joseph Leverrier in 1840, and was rediscovered in 1948 by Jean-Marie Souriau and modified to its present form, but published only in French. Other authors, P. Horst, D. K. Faddejew and Sominski, J.S. Frame, U. Wegner and L. Csanky, were credited with rediscovering the technique. As soon as 1955, Souriau algorithm was tested and benchmarked by the National Bureau of Standards, Los Angeles, under the sponsorship of the Wright Air Development Center, U. S. Air Force, and the Office of Naval Research, and was concluded at the University of California, by the Office of Naval Research. As observed and illustrated by Souriau, for n=10, his algorithm uses only 8 thousands of additions and multiplications, compared to 37 million of additions and 62 million of multiplications for classical approach (Gaussian elimination). Main drawback of most efficient classical algorithm based on Krylov iterates cannot be parallelized. Souriau algorithm has a complexity O(n4) or O(n +1) in sequential computation, and so cannot compete with Krylov-based algorithm, but Souriau algorithm has been parallelized by L. Csanky, proving that characteristic polynomial computation could be solved in parallel time log2n with a polynomial number of processors. Souriau algorithm parallelization by Czansky has been improved more recently by Preparata & Sarwate using fast matrix product, and by Keller-Gehrig using matrix reduction. Reduction to complexity O(n ) is given for generic matrices, but for non-generic ones, only O(n logn) complexity could be achieved. A disadvantage of both algorithms (Le verrier and Gaussian elimination) is the presence of divisions. The computation of the matrix exponential is a classical problem in numerical mathematics as explained in 1978 paper of Moler & van Loan and many efficient algorithms described in 1998 paper Hochbruck, Lubich & Selhofer 1998. But this problem is very far from being fully solved , especially to approximate an exponential of a matrix which resides in a Lie algebra, a central problem in geometric integration as studied by Iserles, Munthe-Kaas, Nørsett, Zanna & Celledoni.
From Le Verrier to Souriau Algorithm
The algorithm to compute characteristic polynomial of a matrix was discovered by Urbain Jean Joseph Leverrier in 1840, and was rediscovered in 1948 by Jean-Marie Souriau and modified to its present form, but published only in French. Other authors, P. Horst, D. K. Faddejew and Sominski, J.S. Frame, U. Wegner and L. Csanky, were credited with rediscovering the technique. As soon as 1955, Souriau algorithm was tested and benchmarked by the National Bureau of Standards, Los Angeles, under the sponsorship of the Wright Air Development Center, U. S. Air Force, and the Office of Naval Research, and was concluded at the University of California, by the Office of Naval Research. As observed and illustrated by Souriau, for n=10, his algorithm uses only 8 thousands of additions and multiplications, compared to 37 million of additions and 62 million of multiplications for classical approach (Gaussian elimination). Main drawback of most efficient classical algorithm based on Krylov iterates cannot be parallelized. Souriau algorithm has a complexity O(n4) or O(nω+1) in sequential computation, and so cannot compete with Krylov-based algorithm, but Souriau algorithm has been parallelized by L. Csanky, proving that characteristic polynomial computation could be solved in parallel time log2n with a polynomial number of processors. Souriau algorithm parallelization by Czansky has been improved more recently by Preparata & Sarwate using fast matrix product, and by Keller-Gehrig using matrix reduction. Reduction to complexity O(nω) is given for generic matrices, but for non-generic ones, only O(nωlogn) complexity could be achieved. A disadvantage of both algorithms (Le verrier and Gaussian elimination) is the presence of divisions. The computation of the matrix exponential is a classical problem in numerical mathematics as explained in 1978 paper of Moler & van Loan and many efficient algorithms described in 1998 paper Hochbruck, Lubich & Selhofer 1998. But this problem is very far from being fully solved , especially to approximate an exponential
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of a matrix which resides in a Lie algebra, a central problem in geometric integration as studied by Iserles, Munthe-Kaas, Nørsett, Zanna & Celledoni [63-74].
Souriau matrix characteristic polynomial computation
Jean-Marie Souriau introduced his algorithm in the framework of his lecture on Multilinear Algebra by consideration on volume form [59-62]. In a vector space E of dimension n, we can prove that vector space of
n-forms (n-form as an anti-symmetric n-linear operator with scalar value). After Selecting a frame (ee12, ,..., en )
of E , we can define an n-form called “volume form” with: vol( e12)( e)...( en ) = 1 (A57)
Volume of parallelepiped generated by vectors ( xx12, ,..., xn ) is given by: vol( x12)( x)...( xn ) (A58)
Souriau called “espace jaugé (jauged space)”, all vector space E , of finite size, where we have selected a “unit- jauge” defined by vol . If we define a linear operator AE: → E, considered as “affiner” in a jauged space, we can then give definition of:
• Determinant of A, det ( A)by:
det ( A) vol( v12)( v)...( vnn) = vol( Av1)( Av 2)...( Av ) (A59)
• Adjoint linear operator of A, adj( A), by:
vol( adj( A ) v12)( v) ...( vnn) = vol( v1)( Av 2)...( Av ) (A60)
• Trace number of A, tr( A) , by:
trAvolv v... v= volAv v... v+ volv Av... v ( ) ( 12)( ) ( n) ( 12)( ) ( nn) ( 1)( 2) ( ) (A61)
...+ vol( v12)( v)...( Avn )
By using the following relation deduced from previous equations:
vol( adj( A ) Av12)( v) ...( vnn) = vol( Av1)( Av 2)..( Av) = det ( A) vol( v12)( v)...( v n) (A62)
If A is invertible, we recover classical equations:
−1 −1 adj( A) A= det ( A) I and A= det ( A) adj( A) (A63)
Using these formulas, we can try to invert [λIA− ] assuming that det(λIA−≠) 0 . If we use previous
determinant definition, we have:
det (λIAvolv−) ( )( v)...( v) =−− vol(λλ v Av)( v Av)...( λ v − Av ) 12 n 1 1 2 2 nn (A64) n =λ vol( v12)( v)...( vn ) + ...
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where det (λIA− ) is the characteristic polynomial of A, a polynomial in λ of degree n, with:
adjIA(λ−)[ λ IA −=] det ( λ IAI −) ⇔ AQ. (λ) = λλ Q( ) − P( λ) I (A65)
(if λ is an eigenvalue of A , the nonzero columns of Q(λ) are corresponding eigenvectors). We can then
observe that adj(λ I− A) is a polynomial of degree n-1. We can then define both P(λ) and Q(λ) by
polynomials: n n−1 ni− and ni−−1 (A66) P(λλ) =det ( IA −=) ∑ ki λ Q(λλ) = adj( I −= A) ∑ λ Bi i=0 i=0
n n−1 With k0 =1 , kAn =( −1) det ( ) , BI0 = and Bn−1 =( −1) adj( A) (A67)
By developing equation adjIA(λλ−)[ IA −=] det ( λ IAI −) , we can write:
nn−−11n ni− ni−−1 ni− (A68) ∑∑kiλ I= λλ Bi[ I −= A] λ Bn−1 + ∑ λ[ Bii − BA−−11] − B n A ii=00= i=1
By identification term by term, we find the expression of matrices Bi :
BI0 = (A69) Bii=+=− B−1 A kI i , i 1,..., n 1 Bnn−1 A+= kI 0 B We can observe that A−1 = − n−1 and also the Cayley-Hamilton theorem: kn
nn−1 kA01+ kA ++... knn− 1 A + kI =0 (A70)
To go further, we have to use this classical result from analysis on differentiationgiven by d δδdet (G) = tr( adj( G) G) . If we set G=()λ IA − and δ = , we then obtain dλ d tr( adj(λλ I−= A)) det ( IA −) providing: dλ n−−11nn ni−−11d ni− ni−− (A71) ∑λtr( Bii) = ∑∑ k λλ=() n − i k i i=0 dλ ii=00=
We can then deduce that tr( Bii) =−=−( n i) k , i 0,..., n 1.
tr( Bi−1 A) As Bii= B−1 A + kI i, tr( Bii) = tr( B−1 A) + n. k i, and then k = − (A72) i i We finally obtain the Souriau Algorithm:
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k00=1 and BI = 1 A= B A , k =−=− tr( A), i 1,..., n 1 ii−1 ii i (A73) 1 B=+=− A kI or B BA trBAI( ) iii ii−−11i i 1 A= B A and k= − tr( A ) nn−1 nn n
Souriau Algorithm to compute exponential map of matrix
Souriau approach of Exponential computation is based on algebraic analogy:
−1 Q(λ) [λIA−] = ⇔−[λIAQ] ( λλ) = P( ) I (A74) P(λ) d d dd and the differential property (with λ = ): I−= AQ P I (A75) dt dt dt dt d If a numeric function γ verifies Pγ = 0 , then: dt
n d (ni− ) (nn ) (− 1) (1) (A76) Pγ=∑ ki γ =++++= kk01 γγ... knn− 1 γγ k0 dt i=0 dtnγ () with γ ()n = n-th derivative of function γ , with initial conditions: dt n γγ(0)=(1) (0) = ... = γ(nn−− 2) = 0 and γ( 1) (0)= 1 (A77) d dtΦ() In this case, the matrix function Φ=Qγ is solution of the differential equation =At Φ(), with dt dt
initial condition Φ=(0) I :
n−1 d (ni−− 1) (n− 1) (n− 2) (A78) Φ=Qγγ =∑ BBBBin = γ01 + γ +... + γ− 1 dt i=0 We can then observe that the exponential map of matrix tA is given by:
n−1 tA (n−− i 1) (n− 1) (n− 2) e=∑γ BBBBin = γγ01 + ++... γ− 1 i=0 (A79) tr( Bi−1 A) with B= I and B = BA − I 01ii− i
(nn ) (− 1) (1) γ such that kk01 γγ++++=... knn− 1 γγ k0 (A80) tr( B− A) −− with k =−=i 1 , γγ (0) ...===(nn 2) 0 and γ( 1) (0) 1 i i
The solution γ ()t of characteristic ordinary differential equation is obtained in [0, h] of the spectral interval
of integration. In the remaining part, the exponential function Φ()t is computed by:
p Φ=Φ(ph )( () h ) (A81)
Souriau Algorithm for Exponential Map of Matrix is given by:
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tr( Bi−1 A) B01= I and Bii = BA− − I i 1) tr( Bi−1 A) k==−=1 , k in 1,..., 0 i i γ integrated on [ 0,h] such that (nn ) (− 1) (1) 2) kk01γγ++++=... knn− 1 γγ k0 (nn−− 2) ( 1) with γγ (0)= ... = = 0 and γ (0)= 1 n−1 tA (n−− i 1) (A82) 3) Computation of Φ= (t ) e =∑γ (tB )i on [ 0, h] i=0 p 4) Extension of Computation on [ 0,ph] by Φ=Φ( pt) ( ( t ))
5) Xt ( )=Φ= ( tX )00 with X X (0) 0 −−11 s sAln( ) If we observe that ln(A ) =∫ [ sI − A] −−[ sI I] ds , this algorithm could be used also to compute Ae= −∞
1 dtΦ() tA such as A 2 . This Souriau algorithm to solve =At Φ() by computation of exponential Φ=()te dt d2ΦΦ() t dt () could be extended to solve L+ M +Φ Nt() = 0 by substituting dt2 dt
2 λλ++ λ = λto λ−= λλ through the following algebraic relations: L M NQ( ) P( ) I [ I AQ] ( ) P( ) I
22 2 (λλL++ MNadjL) ( λλ ++= MN) det ( λλ L ++ MNI) (A83)
2n 22n− 2 2 λ= λλ + += λ2ni− , λ= λλ + += λ22ni−− (A84) P( ) det ( L MN) ∑ ki Q( ) adj( L M N ) ∑ Bi i=0 i=0
Examples of Souriau exponential map algorithm
We can illustrate Souriau algorithm with:
01− tJ cos(tt )− sin( ) J = and e=+=cos( t ) I sin( t ) J ∈SO(2) (A85) 10 sin(tt ) cos( )
BI00= and k = 1
B10=−= BJ trBJ( 0) I J and k1 =−= trBJ( 0) 0 2 (A86) tr( B1 J ) 2 tr( J ) −1 0 10 00 B21=−=−= BJ I J I += 220− 1 01 00 tr( J 2 ) k =−=1 2 2
dt2γ () γγ on [ 0,h] such that +=0 dt 2 ⇒=γ ()tt sin() (A87) (1) with γγ (0)= 0 and (0)= 1
dtγ () Φ=()t B +γ ( tB ) = cos( tI ) + sin( tJ ) on [ 0, h] dt 01 (A88) dΦ =Jt Φ() dt Another example is given by harmonic oscillator:
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d pq−− 01 p p 2 = = = J with JI= − (A89) dt qp 10 q q
tJ pcos t− sin tp p then e = , rotation in -plane. (A90) qsin t cos tq q Next example, is given for skew-symmetric matrix, corresponding to exponential map for so(3) , the Lie
Algebra of Lie group SO(3)={ R / R−1 = RT }:
0 −ωω32 ω= ω −= ω ω + ω + ω ∈ω = ωωω ∈ 3 (A91) × 3 0 1 11LLL 2 2 33 so(3) and ( 1 , 2 , 3) −ωω210 The generators of so(3) correspond to the derivatives of rotation around the each of the standard axes, evaluated at identity. The exponential map that takes skew symmetric matrices to rotation matrices is simply the matrix exponential over a linear combination of the generators. We compute this exponential map by Souriau algorithm:
ω× (2) (1) e=γ B0 ++ γγ BB 12 (A92)
Souriau algorithm provides:
BI00= and k = 1 Tr( I.ωω) Tr( I ) BI=−=.ωω×× I and k =−=0 (A93) 11××11
Tr (ωω..) 22Tr (ωω) BB=−=+=−=.ω ××Iωω. ω I and k ×× ω (A94) 21× 22×× 2
2 T We can observe that BI2 =+=⊗ωω××. ω ω ω and k3 = 0 , and we obtain:
ω× (2) (1) T eI=γ + γ ω× +⊗ γω ω (A95)
The function γ ()t should verify:
(3) (2) (1) 2 k01γγ() tk+++= () tk 23 γγ () tkt () 0 with kkk01=1, = 0, 2 = ω , k 3=0 (A96)
2 γ(3) (tt )+= ωγ(1) ( ) 0 with γ(2) (0) === 1, γ(1) (0) 0, γ (0) 0 (A97)
We can then deduce that:
(1) 11 (A98) γ(t )= sin( ωγ tt) and ( )=2 ( 1 − cos( ω t)) ω ω We can then deduce the exponential map of so(3) :
ω 1 1− cos( ω t) t. × T (A99) e=++⊗cos( ω tI) sin ( ωt) ω× ωω ω ω 2
T 2 But using the relation ω⊗= ω ωω××. + ω I , we recover Rodrigues formula:
ω 1 1− cos( ω t) t. × 2 (A100) eI=++sin ( ωω t) ××ω ω ω 2
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Rt 3 The exponential map from se(3) to SE(3)== C / C , R ∈∈ SO (3), t R is the matrix exponential on 01
a linear combination of the generators:
ω× u δ= =++++uG11 uG 2 2 uG 33ωωω 14 G 2 G 5 + 3 G 6 00 (A101) T δω=(u)∈∈se(3) and ( uR ω) 6
ω u ω× δ × e Vu 112 (A102) e=exp = with VI=+++ωω××( ) 00 01 2! 3!
3 T 2 By using the identity, (ω×) =−=−( ωω). ω ×× ω ω:
ii ∞ωω21ii++ 22 ∞∞−−θθ22ii ×× ( 11) ( ) 2 (A103) VI=++=++Iωω×× ∑++ ∑∑ + + i=0(2ii 2!) ( 2 3!) ii=00( 2 i 2!) = ( 2 i 3!)
1−− cos( ω) ωωsin ( ) VI= + ωω + 2 (A104) 23 ×× ωω
We can apply Souriau formula for exponential map of su(2) , the Lie Algebra of Lie group SU (2) through a linear combination of the generators given by the Pauli spin matrices: a++ id b ic 4 (A105) aI....+ i( cσσxy ++ b d σ z) = with (abcd , , , )∈ R −+b ic a − di
Last example deals with “Geodesic Shooting” for multivariate Gaussian densities ℵ(mR, ) . Information
Geometry provides an invariant Koszul-Fisher metric and geodesic by Euler-Lagrange equations: R + m m T − RR −1R = 0 −1 m − RR m = 0 (A106) Using Souriau theorem of moment map (geometrization of Noether theorem): R−−11 R +== R mm T B cste ⇒ −1 R m = b = cste (A107) This moment map could be computed if we consider the following Lie group action in case of Gaussian densities: (m, R)∈ R n × Sym+ (n) Y 1/ 2 X 1/ 2 + R m R X m 1/ 2 = = , R m (A108) 1 0 1 1 1 M = ∈Gaff 0 1 X ≈ ℵ(0, I) → Y ≈ ℵ(m, R) With R1/2 , square root of R, is given by Cholesky decomposition of R . R1/ 2 is the Lie group of triangular matrix with positive elements on the diagonal. Euler-Poincaré equations, reduced equations from Euler- Lagrange equations, are then given by: m = Rb (A109) T R = R(B − bm ) Geodesic shooting is obtained by using equations established by Eriksen for “exponential map” using the following change of variables [75-76]:
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∆=− B ∆+ bmT −1 ∆=()t Rt () − ∆=−(0) B ⇒δ =−++Bb δ1 δδT ∆1 with −1 ( ) δ ()t= R () tmt () δ (0) = b ∆=δ = (0)I p , (0) 0 (A110) The method based on geodesic shooting consists in iteratively approaching the solution by geodesic shooting in direction (δ(0),∆ (0)), using Souriau exponential map: ∆Φδ ∞ n ()tA TT Λ=(t ) exp( tA) =∑ =δ εγ n=0 n! T ΦΓγ −Bb 0 TT with Ab= 0 − b 0 −bB (A111) 2 −B b 0 B2 +− bbTT Bb − bb 2 T T T TT (A112) A= b02 −=− b bB bb − bB TT2 0 −b B −− bb Bb B + bb
k00=1, B = I and k 11 = 0, B = A because tr( A) = 0
tr( A22) tr( A ) tr( B A) tr( B A) B=−=−=−=− A2 Ik,, B B Aii−−11 Ik , 2222ii− 1iii (A113)
(nn ) (− 1) (1) (n−− 2) ( n 1) kk01γγ++++=... knn− 1 γγ k0 with γ (0) === ... γ 0, γ (0) = 1 n−1 tA (n−− i 1) e = ∑γ ()tBi i=0 (A114)
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