Linear phase space deformations with angular momentum symmetry

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March 2019, 11(1): 45-58. doi: 10.3934/jgm.2019003

Linear phase space deformations with angular momentum symmetry

Claudio Meneses ^,

Mathematisches Seminar, Christian-Albrechts Universität zu Kiel, Ludewig-Meyn-Str. 4, 24118 Kiel, Germany

* Corresponding author: Claudio Meneses

Received March 2018 Revised November 2018 Published January 2019

Fund Project: The author is supported by the DFG SPP 2026 priority programme "Geometry at infinity".

Motivated by the work of Leznov-Mostovoy [17], we classify the linear deformations of standard $ 2n $-dimensional phase space that preserve the obvious symplectic $ \mathfrak{o}(n) $-symmetry. As a consequence, we describe standard phase space, as well as $ T^{*}S^{n} $ and $ T^{*}\mathbb{H}^{n} $ with their standard symplectic forms, as degenerations of a 3-dimensional family of coadjoint orbits, which in a generic regime are identified with the Grassmannian of oriented 2-planes in $ {\mathbb{R}}^{n+2} $.

Keywords: Coadjoint orbits, Lie algebra deformations, momentum maps.

Mathematics Subject Classification: Primary: 17B08, 17B56, 17B80; Secondary: 53D20, 14H70.

Citation: Claudio Meneses. Linear phase space deformations with angular momentum symmetry. Journal of Geometric Mechanics, 2019, 11 (1) : 45-58. doi: 10.3934/jgm.2019003

References:

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[2]	O. M. Boyarskyi and T. V. Skrypnik, Degenerate orbits of adjoint representation of orthogonal and unitary groups regarded as algebraic submanifolds, Ukrainian Math. J., 49 (1997), 1003-1015. doi: 10.1007/BF02528745.
[3]	C. Chevalley and S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc., 63 (1948), 85-124. doi: 10.2307/1990637.
[4]	A. Fialowski, Deformations of Lie algebras, Mat. Sb. (N. S.), 127 (1985), 476-482.
[5]	D. M. Fradkin, Three-dimensional isotropic harmonic oscillator and $SU_3$, Am. J. Phys., 33 (1965), 207-211. doi: 10.1119/1.1971373.
[6]	M. Gerstenhaber, On the deformation of rings and algebras, Ann. Math., 79 (1964), 59-103. doi: 10.2307/1970484.
[7]	V. Guillemin and S. Sternberg, Symplectic Techniques in Physics, Cambridge University Press, 1984.
[8]	_____, Variations on a Theme by Kepler, Colloquium Publications, Vol. 42, American Mathematical Soc., 2006.
[9]	P. W. Higgs, Dynamical symmetries in a spherical geometry I, J. Phys. A, 12 (1979), 309-323.
[10]	G. Hochschild and J.-P. Serre, Cohomology of Lie algebras, Ann. Math., 57 (1953), 591-603. doi: 10.2307/1969740.
[11]	R. Howe, Remarks on classical invariant theory, Trans. Amer. Math. Soc., 313 (1989), 539-570. doi: 10.2307/2001418.
[12]	E. Inonu and E. P. Wigner, On the contraction of groups and their representations, Proc. Nat. Acad. Sci. U. S. A., 39 (1953), 510-524. doi: 10.1073/pnas.39.6.510.
[13]	D. Kazhdan, B. Kostant and S. Sternberg, Hamiltonian group actions and dynamical systems of Calogero type, Comm. Pure Appl. Math., 31 (1978), 481-507. doi: 10.1002/cpa.3160310405.
[14]	A. A. Kirillov, Lectures on the Orbit Method, Graduate Studies in Mathematics, Vol. 64, American Mathematical Soc., 2004. doi: 10.1090/gsm/064.
[15]	W. Lenz, Über den Bewegungsverlauf und die Quantenzustände der gestörten Keplerbewegung, Z. Phys., 24 (1924), 197-207.
[16]	M. Levy-Nahas, Deformation and contraction of Lie algebras, J. Math. Phys., 8 (1967), 1211-1222. doi: 10.1063/1.1705338.
[17]	A. Leznov and J. Mostovoy, Classical dynamics in deformed spaces, J. Phys. A, 36 (2003), 1439-1449. doi: 10.1088/0305-4470/36/5/317.
[18]	S. P. Novikov, The Hamiltonian formalism and a multivalued analogue of Morse theory, Uspekhi Mat. Nauk, 37 (1982), 3-49.
[19]	A. Perelomov, Integrable Systems of Classical Mechanics and Lie Algebras, Birkhäuser, 1990. doi: 10.1007/978-3-0348-9257-5.
[20]	A. Reyman and M. A. Semenov-Tian-Shansky, Group-theoretical methods in the theory of finite-dimensional integrable systems, Dynamical Systems VII, Springer Berlin Heidelberg, 16 (1994), 116-225. doi: 10.1007/978-3-662-06796-3_7.
[21]	C. A. Weibel, An Introduction to Homological Algebra, Studies in Advanced Mathematics, Vol. 38, Cambridge University Press, Cambridge, 1994. doi: 10.1017/CBO9781139644136.
[22]	J. Wolf, Representations associated to minimal co-adjoint orbits, Differential Geometrical Methods in Mathematical Physics II., Springer Berlin Heidelberg, (1978), 329-349.

show all references

References:

[1]	A. Borel, Kählerian coset spaces of semisimple Lie groups, Proc. Nat. Acad. Sci. U. S. A., 40 (1954), 1147-1151. doi: 10.1073/pnas.40.12.1147.
[2]	O. M. Boyarskyi and T. V. Skrypnik, Degenerate orbits of adjoint representation of orthogonal and unitary groups regarded as algebraic submanifolds, Ukrainian Math. J., 49 (1997), 1003-1015. doi: 10.1007/BF02528745.
[3]	C. Chevalley and S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc., 63 (1948), 85-124. doi: 10.2307/1990637.
[4]	A. Fialowski, Deformations of Lie algebras, Mat. Sb. (N. S.), 127 (1985), 476-482.
[5]	D. M. Fradkin, Three-dimensional isotropic harmonic oscillator and $SU_3$, Am. J. Phys., 33 (1965), 207-211. doi: 10.1119/1.1971373.
[6]	M. Gerstenhaber, On the deformation of rings and algebras, Ann. Math., 79 (1964), 59-103. doi: 10.2307/1970484.
[7]	V. Guillemin and S. Sternberg, Symplectic Techniques in Physics, Cambridge University Press, 1984.
[8]	_____, Variations on a Theme by Kepler, Colloquium Publications, Vol. 42, American Mathematical Soc., 2006.
[9]	P. W. Higgs, Dynamical symmetries in a spherical geometry I, J. Phys. A, 12 (1979), 309-323.
[10]	G. Hochschild and J.-P. Serre, Cohomology of Lie algebras, Ann. Math., 57 (1953), 591-603. doi: 10.2307/1969740.
[11]	R. Howe, Remarks on classical invariant theory, Trans. Amer. Math. Soc., 313 (1989), 539-570. doi: 10.2307/2001418.
[12]	E. Inonu and E. P. Wigner, On the contraction of groups and their representations, Proc. Nat. Acad. Sci. U. S. A., 39 (1953), 510-524. doi: 10.1073/pnas.39.6.510.
[13]	D. Kazhdan, B. Kostant and S. Sternberg, Hamiltonian group actions and dynamical systems of Calogero type, Comm. Pure Appl. Math., 31 (1978), 481-507. doi: 10.1002/cpa.3160310405.
[14]	A. A. Kirillov, Lectures on the Orbit Method, Graduate Studies in Mathematics, Vol. 64, American Mathematical Soc., 2004. doi: 10.1090/gsm/064.
[15]	W. Lenz, Über den Bewegungsverlauf und die Quantenzustände der gestörten Keplerbewegung, Z. Phys., 24 (1924), 197-207.
[16]	M. Levy-Nahas, Deformation and contraction of Lie algebras, J. Math. Phys., 8 (1967), 1211-1222. doi: 10.1063/1.1705338.
[17]	A. Leznov and J. Mostovoy, Classical dynamics in deformed spaces, J. Phys. A, 36 (2003), 1439-1449. doi: 10.1088/0305-4470/36/5/317.
[18]	S. P. Novikov, The Hamiltonian formalism and a multivalued analogue of Morse theory, Uspekhi Mat. Nauk, 37 (1982), 3-49.
[19]	A. Perelomov, Integrable Systems of Classical Mechanics and Lie Algebras, Birkhäuser, 1990. doi: 10.1007/978-3-0348-9257-5.
[20]	A. Reyman and M. A. Semenov-Tian-Shansky, Group-theoretical methods in the theory of finite-dimensional integrable systems, Dynamical Systems VII, Springer Berlin Heidelberg, 16 (1994), 116-225. doi: 10.1007/978-3-662-06796-3_7.
[21]	C. A. Weibel, An Introduction to Homological Algebra, Studies in Advanced Mathematics, Vol. 38, Cambridge University Press, Cambridge, 1994. doi: 10.1017/CBO9781139644136.
[22]	J. Wolf, Representations associated to minimal co-adjoint orbits, Differential Geometrical Methods in Mathematical Physics II., Springer Berlin Heidelberg, (1978), 329-349.

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