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Geometry of Routh reduction
Linear phase space deformations with angular momentum symmetry
Mathematisches Seminar, Christian-Albrechts Universität zu Kiel, Ludewig-Meyn-Str. 4, 24118 Kiel, Germany |
Motivated by the work of Leznov-Mostovoy [
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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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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