Geodesic
The structure of differential invariants for a free symmetry group action
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 6 (2023), pp. 31-40 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In the paper, we consider the problem of describing the general structure of differential invariants for transformation groups that act freely and reguralry. We formulate two theorems describing the structures of differential invariants for intransitive and transitive free actions, respectively. In both cases it is shown that the differential invariants can be expressed in terms of the symbols of right-invariant vector fields. Finally, we discuss prospects for solving the problem considered for more general group actions.
Keywords: symmetry group, differential invariant, free action. 
@article{IVM_2023_6_a2,
     author = {A. A. Magazev and I. V. Shirokov},
     title = {The structure of differential invariants for a free symmetry group action},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {31--40},
     year = {2023},
     number = {6},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2023_6_a2/}
}
TY  - JOUR
AU  - A. A. Magazev
AU  - I. V. Shirokov
TI  - The structure of differential invariants for a free symmetry group action
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2023
SP  - 31
EP  - 40
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/IVM_2023_6_a2/
LA  - ru
ID  - IVM_2023_6_a2
ER  - 
%0 Journal Article
%A A. A. Magazev
%A I. V. Shirokov
%T The structure of differential invariants for a free symmetry group action
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2023
%P 31-40
%N 6
%U http://geodesic.mathdoc.fr/item/IVM_2023_6_a2/
%G ru
%F IVM_2023_6_a2
A. A. Magazev; I. V. Shirokov. The structure of differential invariants for a free symmetry group action. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 6 (2023), pp. 31-40. http://geodesic.mathdoc.fr/item/IVM_2023_6_a2/

[1] Lie S., “Classification und Integration von gewöhnlichen Differentialgleichungen zwischen $xy$, die eine Gruppe von Transformationen gestatten”, Math. Ann., 32:2 (1888), 213–281 | DOI | MR

[2] Olver P. J., “Differential Invariants and Invariant Differential Equations”, Lie Groups and Their Appl., 1994, no. 1, 177–192 | MR | Zbl

[3] Ovsyannikov L. V., Gruppovoi analiz differentsialnykh uravnenii, Nauka, M., 1978 | MR

[4] Olver P., Prilozheniya grupp Li k differentsialnym uravneniyam, Mir, M., 1989

[5] Fels M., Olver P. J., “Moving Coframes: I. A Practical Algorithm”, Acta Appl. Math., 51:2 (1998), 161–213 | DOI | MR

[6] Fels M., Olver P. J., “Moving coframes: II. Regularization and theoretical foundations”, Acta Appl. Math., 55:2 (1999), 127–208 | DOI | MR | Zbl

[7] Chupakhin A. P., “Differential invariants: theorem of commutativity”, Commun. in Nonlinear Sci. and Numerical Simulation, 9:1 (2004), 25–33 | DOI | MR | Zbl

[8] Gazizov R. K., Gainetdinova A. A., “Operator invariantnogo differentsirovaniya i ego primenenie dlya integrirovaniya sistem obyknovennykh differentsialnykh uravnenii”, Ufimsk. matem. zhurn., 9:4 (2017), 12–21 | MR | Zbl

[9] Shirokov I. V., “Differentsialnye invarianty gruppy preobrazovanii odnorodnogo prostranstva”, Sib. matem. zhurn., 48:6 (2007), 1405–1421 | MR | Zbl

[10] Magazev A. A., Mikheyev V. V., Shirokov I. V., “Computation of Composition Functions and Invariant Vector Fields in Terms of Structure Constants of Associated Lie Algebras”, Symmetry, Integrability and Geom.: Methods and Appl., 11 (2015), 066, 17 pp. | MR | Zbl