doi: 10.3934/dcdss.2020066

On Lie algebra actions

Department of Mathematics and Statistics, University of Calgary, Calgary, AB, T2N 1N4, Canada

* Corresponding author: R. H. Cushman

Received  October 2017 Revised  July 2018 Published  April 2019

In this paper we define an action of a Lie algebra on a smooth manifold. We get nearly the same results as those for group actions, when the flows of the symmetry vector fields are complete. We show that the orbit space of a Lie algebra action is a differential space. We discuss differential spaces occuring in the reduction of symmetries in integrable Hamiltonian systems.

Citation: Richard H. Cushman, Jędrzej Śniatycki. On Lie algebra actions. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020066
References:
[1]

N. Aronszajn, Subcartesian and subRiemannian spaces, Notices American Mathematical Society, 14 (1967), 111-111. Google Scholar

[2]

R. Cushman and J. Śniatycki, Differential structure of orbit spaces, Canad. Math. J., 54 (2001), 715-755. doi: 10.4153/CJM-2001-029-1. Google Scholar

[3]

R. H. Cushman and L. M. Bates, Global Aspects of Classical Integrable Systems, second edition, Birkhäuser, Basel, 2015. doi: 10.1007/978-3-0348-0918-4. Google Scholar

[4]

R. S. Palais, A Global Formulation of the Lie Theory of Transformation Groups, Memoir 22, American Mathematical Society, Providence, R.I. 1957. Google Scholar

[5]

T. Ratiu, C. Wacheux and N. T. Zung, Convexity of singular affine structures and toric-focus integrable Hamiltonian systems, arXiv: 1706.01093v1.Google Scholar

[6] J. Śniatycki, Differential Geometry of Singular Spaces and Reduction of Symmetry, Cambridge University Press, Cambridge, UK, 2013. doi: 10.1017/CBO9781139136990.
[7]

H. Sussmann, Orbits of families of vector fields and foliations with singularities, Trans. Amer. Math. Soc., 180 (1973), 171-188. doi: 10.1090/S0002-9947-1973-0321133-2. Google Scholar

show all references

References:
[1]

N. Aronszajn, Subcartesian and subRiemannian spaces, Notices American Mathematical Society, 14 (1967), 111-111. Google Scholar

[2]

R. Cushman and J. Śniatycki, Differential structure of orbit spaces, Canad. Math. J., 54 (2001), 715-755. doi: 10.4153/CJM-2001-029-1. Google Scholar

[3]

R. H. Cushman and L. M. Bates, Global Aspects of Classical Integrable Systems, second edition, Birkhäuser, Basel, 2015. doi: 10.1007/978-3-0348-0918-4. Google Scholar

[4]

R. S. Palais, A Global Formulation of the Lie Theory of Transformation Groups, Memoir 22, American Mathematical Society, Providence, R.I. 1957. Google Scholar

[5]

T. Ratiu, C. Wacheux and N. T. Zung, Convexity of singular affine structures and toric-focus integrable Hamiltonian systems, arXiv: 1706.01093v1.Google Scholar

[6] J. Śniatycki, Differential Geometry of Singular Spaces and Reduction of Symmetry, Cambridge University Press, Cambridge, UK, 2013. doi: 10.1017/CBO9781139136990.
[7]

H. Sussmann, Orbits of families of vector fields and foliations with singularities, Trans. Amer. Math. Soc., 180 (1973), 171-188. doi: 10.1090/S0002-9947-1973-0321133-2. Google Scholar

[1]

Feimin Zhong, Jinxing Xie, Jing Jiao. Solutions for bargaining games with incomplete information: General type space and action space. Journal of Industrial & Management Optimization, 2018, 14 (3) : 953-966. doi: 10.3934/jimo.2017084

[2]

Evgeny L. Korotyaev. Estimates for solutions of KDV on the phase space of periodic distributions in terms of action variables. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 219-225. doi: 10.3934/dcds.2011.30.219

[3]

Brandon Seward. Every action of a nonamenable group is the factor of a small action. Journal of Modern Dynamics, 2014, 8 (2) : 251-270. doi: 10.3934/jmd.2014.8.251

[4]

Michael Hutchings. Mean action and the Calabi invariant. Journal of Modern Dynamics, 2016, 10: 511-539. doi: 10.3934/jmd.2016.10.511

[5]

Paolo Perfetti. Fixed point theorems in the Arnol'd model about instability of the action-variables in phase-space. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 379-391. doi: 10.3934/dcds.1998.4.379

[6]

Helmut Kröger. From quantum action to quantum chaos. Conference Publications, 2003, 2003 (Special) : 492-500. doi: 10.3934/proc.2003.2003.492

[7]

Franz W. Kamber and Peter W. Michor. Completing Lie algebra actions to Lie group actions. Electronic Research Announcements, 2004, 10: 1-10.

[8]

Yu Tian, John R. Graef, Lingju Kong, Min Wang. Existence of solutions to a multi-point boundary value problem for a second order differential system via the dual least action principle. Conference Publications, 2013, 2013 (special) : 759-769. doi: 10.3934/proc.2013.2013.759

[9]

S. A. Krat. On pairs of metrics invariant under a cocompact action of a group. Electronic Research Announcements, 2001, 7: 79-86.

[10]

Alexandre Rocha, Mário Jorge Dias Carneiro. A dynamical condition for differentiability of Mather's average action. Journal of Geometric Mechanics, 2014, 6 (4) : 549-566. doi: 10.3934/jgm.2014.6.549

[11]

Răzvan M. Tudoran. On the control of stability of periodic orbits of completely integrable systems. Journal of Geometric Mechanics, 2015, 7 (1) : 109-124. doi: 10.3934/jgm.2015.7.109

[12]

Rainer Steinwandt, Adriana Suárez Corona. Cryptanalysis of a 2-party key establishment based on a semigroup action problem. Advances in Mathematics of Communications, 2011, 5 (1) : 87-92. doi: 10.3934/amc.2011.5.87

[13]

Roy Malka, Vered Rom-Kedar. Bacteria--phagocyte dynamics, axiomatic modelling and mass-action kinetics. Mathematical Biosciences & Engineering, 2011, 8 (2) : 475-502. doi: 10.3934/mbe.2011.8.475

[14]

Kaizhi Wang. Action minimizing stochastic invariant measures for a class of Lagrangian systems. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1211-1223. doi: 10.3934/cpaa.2008.7.1211

[15]

Jungsoo Kang. Survival of infinitely many critical points for the Rabinowitz action functional. Journal of Modern Dynamics, 2010, 4 (4) : 733-739. doi: 10.3934/jmd.2010.4.733

[16]

J. I. Díaz, J. F. Padial. On a free-boundary problem modeling the action of a limiter on a plasma. Conference Publications, 2007, 2007 (Special) : 313-322. doi: 10.3934/proc.2007.2007.313

[17]

Alessandro Ferriero. Action functionals that attain regular minima in presence of energy gaps. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 675-690. doi: 10.3934/dcds.2007.19.675

[18]

Rostyslav Kravchenko. The action of finite-state tree automorphisms on Bernoulli measures. Journal of Modern Dynamics, 2010, 4 (3) : 443-451. doi: 10.3934/jmd.2010.4.443

[19]

Alessandro Fonda, Antonio J. Ureña. Periodic, subharmonic, and quasi-periodic oscillations under the action of a central force. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 169-192. doi: 10.3934/dcds.2011.29.169

[20]

Carlos Matheus, Jean-Christophe Yoccoz. The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis. Journal of Modern Dynamics, 2010, 4 (3) : 453-486. doi: 10.3934/jmd.2010.4.453

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (10)
  • HTML views (228)
  • Cited by (0)

Other articles
by authors

[Back to Top]