Differential invariants and operators of invariant differentiation of the projectable action of Lie groups

Goncharovskii, M. M.; Широков, И. В.

doi:10.1007/s11232-015-0285-z

Cited by 3 publications

(6 citation statements)

References 14 publications

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“…With this new approach, we can build Entropy by constructing the Casimir Function associated to the Lie group and also in case of non-null cohomology. Igor V. Shirokov [ 71 , 72 , 73 , 74 , 75 ] 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. The idea of the method of constructing coadjoint invariants is to construct the canonical transition to the Darboux coordinates on the orbits of the dual Lie algebra

of maximal dimension dual to the Lie algebra

of the Lie group

.…”

Section: New Entropy Definition As Generalized Casimir Invariant Fmentioning

confidence: 99%

“…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 [ 71 , 72 , 73 , 74 , 75 , 79 , 80 , 81 , 82 , 83 , 84 , 85 , 86 , 87 , 88 , 89 , 90 , 91 , 92 ].…”

Section: New Entropy Definition As Generalized Casimir Invariant Fmentioning

confidence: 99%

“…I. V. Shirokov [ 71 , 72 , 73 , 74 , 75 ] 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. They also have made link with construction of the Lie algebra polarization…”

Section: Appendix A1 Group Of Unit Disk Automorphismsmentioning

confidence: 99%

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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

Barbaresco¹

2020

Entropy

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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 (e.g., 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.

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of maximal dimension dual to the Lie algebra

of the Lie group

.…”

Section: New Entropy Definition As Generalized Casimir Invariant Fmentioning

confidence: 99%

Section: New Entropy Definition As Generalized Casimir Invariant Fmentioning

confidence: 99%

Section: Appendix A1 Group Of Unit Disk Automorphismsmentioning

confidence: 99%

See 1 more Smart Citation

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

Barbaresco¹

2020

Entropy

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“…We construct OID as ( Φ) −1 . The completeness of corresponding system (8) implies that the function Φ( , ) can be found by the system of equations…”

Section: Second Order Equationmentioning

confidence: 99%

“…Various approaches for constructing OIDs were considered in works [2], [7], [8]. In a series of works OIDs were employed for constructing a basis of differential invariants of the admitted algebra in the problems on classification of differential equations, see, for instance, [9]- [11].…”

Section: Introductionmentioning

confidence: 99%

Operator of invariant differentiation and its application for integrating systems of ordinary differential equations

Газизов¹,

Gainetdinova²

2017

Ufimskii Matematicheskii Zhurnal

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We propose an algorithm for integrating -th order ordinary differential equations (ODE) admitting -dimensional Lie algebras of operators. The algorithm is based on invariant representation of the equations by the invariants of the admitted Lie algebra and application of an operator of invariant differentiation of special type. We show that in the case of scalar equations this method is equivalent to the known order reduction methods. We study an applicability of the suggested algorithm to the systems of th order ODEs admitting -dimensional Lie algebras of operators. For the admitted Lie algebra we obtain a condition ensuring the possibility to construct the operator of invariant differentiation of a special type and to reduce the order of the considered system of ODEs. This condition is the implication of the existence of nontrivial solutions to the systems of linear algebraic equations, where the coefficients are the structural constants of the Lie algebra. We present an algorithm for constructing the ( − 1)-dimensional Lie algebra for the reduced system. The suggested approach is applied for integrating the systems of two second order equations.

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Algebraic method for construction of infinitesimal invariants of Lie groups representations

Kurnyavko,

Shirokov

2017

Preprint

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We propose the method for obtaining invariants of arbitrary representations of Lie groups that reduces this problem to known problems of linear algebra. The basis of this method is the idea of a special extension of the representation space, which allows us to regard it as a coalgebra of some Lie algebra. In its turn, this allows us to reduce the problem of constructing invariants of a given representation to the problem of constructing invariants of the coadjoint representation of the corresponding Lie group. In our previous paper it was shown that this problem can always be solved in a natural way by the methods of linear algebra.

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Differential invariants and operators of invariant differentiation of the projectable action of Lie groups

Abstract: We describe the relation between operators of invariant differentiation and invariant operators on orbits of Lie group actions. We propose a new effective method for finding differential invariants and operators of invariant differentiation and present examples.

Cited by 3 publications

References 14 publications

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

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

Operator of invariant differentiation and its application for integrating systems of ordinary differential equations

Algebraic method for construction of infinitesimal invariants of Lie groups representations

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