THE EXPLICIT PARAMETRIZATION FORMULAS FOR THE COADJOINT ORBITS OF THE HEISENBERG LIE GROUP
Keywords:
Heisenberg Lie Group, Coadjoint Orbit, Parametrization
Abstract
Downloads
Download data is not yet available.
References
[1] F. Barbaresco and F. Gay-Balmaz, “Lie group cohomology and (Multi)symplectic integrators: New geometric tools for lie group machine learning based on souriau geometric statistical mechanics,” Entropy, vol. 22, no. 5, May 2020, doi: 10.3390/E22050498.
[2] N. T. Varopoulos, “Potential theory and geometry on lie groups,” Potential Theory and Geometry on Lie Groups, pp. 1–596, Oct. 2020, doi: 10.1017/9781139567718.
[3] M. K. Choudhary and B. S, “Basic Concept of Lie Groups,” International Journal of Mathematics Trends and Technology, vol. 67, no. 4, pp. 62–66, Apr. 2021, doi: 10.14445/22315373/IJMTT-V67I4P509.
[4] S. Akter, M. Md. Moheuddin, S. Hossain, and A. Khatun, “Operations and Actions of Lie Groups on Manifolds,” American Journal of Computational Mathematics, vol. 10, no. 03, pp. 460–472, 2020, doi: 10.4236/AJCM.2020.103026.
[5] E. Howard, “Theory of groups and symmetries: Finite groups, Lie groups and Lie algebras,” Contemp Phys, vol. 60, no. 3, pp. 275–275, Jul. 2019, doi: 10.1080/00107514.2019.1663933.
[6] D. Prinz and A. Schmeding, “Lie theory for asymptotic symmetries in general relativity: The NU group,” Class Quantum Gravity, vol. 39, no. 15, Aug. 2022, doi: 10.1088/1361-6382/AC776C.
[7] W. S. Gan, “Lie Group and Lie Algebra,” Time Reversal Acoustics, pp. 7–13, 2021, doi: 10.1007/978-981-16-3235-8_2.
[8] B. C. Hall, “Lie Groups and Lie Algebras,” in Theoretical and Mathematical Physics, 2nd ed., WORLD SCIENTIFIC, 2018, pp. 516–554. doi: 10.1142/9789813275386_0022.
[9] D. Prinz and A. Schmeding, “Lie theory for asymptotic symmetries in general relativity: The BMS group,” Class Quantum Gravity, vol. 39, no. 6, p. 065004, Feb. 2022, doi: 10.1088/1361-6382/AC4AE2.
[10] M. Torrisi and R. Traciná, “Lie symmetries and solutions of reaction diffusion systems arising in biomathematics,” Symmetry (Basel), vol. 13, no. 8, Aug. 2021, doi: 10.3390/sym13081530.
[11] I. Hernández, C. Mateos, J. Núñez, and Á. F. Tenorio, “Lie Theory: Applications to Problems in Mathematical Finance and Economics,” 2009.
[12] V. R. Chintala, “On Suslin matrices and their connection to spin groups,” Documenta Mathematica, vol. 20, no. 2015, pp. 531–550, Jan. 2015, doi: 10.4171/DM/498.
[13] “Algebra & Number Theory Vol. 16, No. 6, 2022.” Accessed: Sep. 14, 2024. [Online]. Available: https://msp.org/ant/2022/16-6/p06.xhtml
[14] A. Shapovalov and A. Breev, “Harmonic Oscillator Coherent States from the Standpoint of Orbit Theory,” Symmetry 2023, Vol. 15, Page 282, vol. 15, no. 2, p. 282, Jan. 2023, doi: 10.3390/SYM15020282.
[15] A. V. Podobryaev, “Coadjoint Orbits of Three-Step Free Nilpotent Lie Groups and Time-Optimal Control Problem,” Doklady Mathematics, vol. 102, no. 1, pp. 293–295, Jul. 2020, doi: 10.1134/S1064562420040158/METRICS.
[16] J. Figueroa-O’Farrill, R. Grassie, and S. Prohazka, “Lifshitz symmetry: Lie algebras, spacetimes and particles,” SciPost Physics, vol. 14, no. 3, p. 035, Mar. 2023, doi: 10.21468/SCIPOSTPHYS.14.3.035/PDF.
[17] A. A. Kirillov, “Lectures on Orbit Method,” vol. 64, pp. 71–72, 2004.
[18] O. L. Kurnyavko and I. V. Shirokov, “Construction of invariants of the coadjoint representation of Lie groups using linear algebra methods,” Theoretical and Mathematical Physics(Russian Federation), vol. 188, no. 1, pp. 965–979, Jul. 2016, doi: 10.1134/S0040577916070011.
[19] D. A. Vogan, “The Orbit Method and Unitary Representations for Reductive Lie Groups,” 1994.
[20] D. A. Vogan, “The method of coadjoint orbits for real reductive groups,” 1998.
[21] L. J. Corwin and F. P. Greenleaf, “Representations of Nilpotent Lie Groups and their Applications_ Part 1, Basic Theory and Examples-Cambridge,” 1990.
[22] A. Mcinerney, “First Steps in Differential Geometry,” 2013. [Online]. Available: http://www.springer.com/series/666
[2] N. T. Varopoulos, “Potential theory and geometry on lie groups,” Potential Theory and Geometry on Lie Groups, pp. 1–596, Oct. 2020, doi: 10.1017/9781139567718.
[3] M. K. Choudhary and B. S, “Basic Concept of Lie Groups,” International Journal of Mathematics Trends and Technology, vol. 67, no. 4, pp. 62–66, Apr. 2021, doi: 10.14445/22315373/IJMTT-V67I4P509.
[4] S. Akter, M. Md. Moheuddin, S. Hossain, and A. Khatun, “Operations and Actions of Lie Groups on Manifolds,” American Journal of Computational Mathematics, vol. 10, no. 03, pp. 460–472, 2020, doi: 10.4236/AJCM.2020.103026.
[5] E. Howard, “Theory of groups and symmetries: Finite groups, Lie groups and Lie algebras,” Contemp Phys, vol. 60, no. 3, pp. 275–275, Jul. 2019, doi: 10.1080/00107514.2019.1663933.
[6] D. Prinz and A. Schmeding, “Lie theory for asymptotic symmetries in general relativity: The NU group,” Class Quantum Gravity, vol. 39, no. 15, Aug. 2022, doi: 10.1088/1361-6382/AC776C.
[7] W. S. Gan, “Lie Group and Lie Algebra,” Time Reversal Acoustics, pp. 7–13, 2021, doi: 10.1007/978-981-16-3235-8_2.
[8] B. C. Hall, “Lie Groups and Lie Algebras,” in Theoretical and Mathematical Physics, 2nd ed., WORLD SCIENTIFIC, 2018, pp. 516–554. doi: 10.1142/9789813275386_0022.
[9] D. Prinz and A. Schmeding, “Lie theory for asymptotic symmetries in general relativity: The BMS group,” Class Quantum Gravity, vol. 39, no. 6, p. 065004, Feb. 2022, doi: 10.1088/1361-6382/AC4AE2.
[10] M. Torrisi and R. Traciná, “Lie symmetries and solutions of reaction diffusion systems arising in biomathematics,” Symmetry (Basel), vol. 13, no. 8, Aug. 2021, doi: 10.3390/sym13081530.
[11] I. Hernández, C. Mateos, J. Núñez, and Á. F. Tenorio, “Lie Theory: Applications to Problems in Mathematical Finance and Economics,” 2009.
[12] V. R. Chintala, “On Suslin matrices and their connection to spin groups,” Documenta Mathematica, vol. 20, no. 2015, pp. 531–550, Jan. 2015, doi: 10.4171/DM/498.
[13] “Algebra & Number Theory Vol. 16, No. 6, 2022.” Accessed: Sep. 14, 2024. [Online]. Available: https://msp.org/ant/2022/16-6/p06.xhtml
[14] A. Shapovalov and A. Breev, “Harmonic Oscillator Coherent States from the Standpoint of Orbit Theory,” Symmetry 2023, Vol. 15, Page 282, vol. 15, no. 2, p. 282, Jan. 2023, doi: 10.3390/SYM15020282.
[15] A. V. Podobryaev, “Coadjoint Orbits of Three-Step Free Nilpotent Lie Groups and Time-Optimal Control Problem,” Doklady Mathematics, vol. 102, no. 1, pp. 293–295, Jul. 2020, doi: 10.1134/S1064562420040158/METRICS.
[16] J. Figueroa-O’Farrill, R. Grassie, and S. Prohazka, “Lifshitz symmetry: Lie algebras, spacetimes and particles,” SciPost Physics, vol. 14, no. 3, p. 035, Mar. 2023, doi: 10.21468/SCIPOSTPHYS.14.3.035/PDF.
[17] A. A. Kirillov, “Lectures on Orbit Method,” vol. 64, pp. 71–72, 2004.
[18] O. L. Kurnyavko and I. V. Shirokov, “Construction of invariants of the coadjoint representation of Lie groups using linear algebra methods,” Theoretical and Mathematical Physics(Russian Federation), vol. 188, no. 1, pp. 965–979, Jul. 2016, doi: 10.1134/S0040577916070011.
[19] D. A. Vogan, “The Orbit Method and Unitary Representations for Reductive Lie Groups,” 1994.
[20] D. A. Vogan, “The method of coadjoint orbits for real reductive groups,” 1998.
[21] L. J. Corwin and F. P. Greenleaf, “Representations of Nilpotent Lie Groups and their Applications_ Part 1, Basic Theory and Examples-Cambridge,” 1990.
[22] A. Mcinerney, “First Steps in Differential Geometry,” 2013. [Online]. Available: http://www.springer.com/series/666
Published
2026-04-08
How to Cite
[1]
M. Z. Zachary, E. Kurniadi, and S. Sylviani, “THE EXPLICIT PARAMETRIZATION FORMULAS FOR THE COADJOINT ORBITS OF THE HEISENBERG LIE GROUP”, BAREKENG: J. Math. & App., vol. 20, no. 3, pp. 2063-2074, Apr. 2026.
Section
Articles
Copyright (c) 2026 Muhammad Zaky Zachary, Edi Kurniadi, Sisilia Sylviani
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
Authors who publish with this Journal agree to the following terms:
- Author retain copyright and grant the journal right of first publication with the work simultaneously licensed under a creative commons attribution license that allow others to share the work within an acknowledgement of the work’s authorship and initial publication of this journal.
- Authors are able to enter into separate, additional contractual arrangement for the non-exclusive distribution of the journal’s published version of the work (e.g. acknowledgement of its initial publication in this journal).
- Authors are permitted and encouraged to post their work online (e.g. in institutional repositories or on their websites) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published works.
1.gif)
