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June  2019, 11(2): 255-275. doi: 10.3934/jgm.2019014

Dual pairs for matrix groups

Paul Skerritt 1,, and  Cornelia Vizman 2,

1. 

Department of Mathematics, University of Surrey, Guildford GU2 7XH, United Kingdom
2. 

Department of Mathematics, West University of Timişoara, RO–300223 Timişoara, Romania

* Corresponding author

Received  April 2018 Revised  March 2019 Published  May 2019

Fund Project: The first author was partially supported by Leverhulme Trust Research Project Grant 2014-112. The second author was supported by the grant PN-Ⅲ-P4-ID-PCE-2016-0778 of the Romanian Ministry of Research and Innovation CNCS-UEFISCDI, within PNCDI Ⅲ.

In this paper we present two dual pairs that can be seen as the linear analogues of the following two dual pairs related to fluids: the EPDiff dual pair due to Holm and Marsden, and the ideal fluid dual pair due to Marsden and Weinstein.

Keywords: Dual pairs, momentum maps, coadjoint orbits, symplectic reduction.
Mathematics Subject Classification: Primary: 53D20, 53D17; Secondary: 22E60, 17B08.
Citation: Paul Skerritt, Cornelia Vizman. Dual pairs for matrix groups. Journal of Geometric Mechanics, 2019, 11 (2) : 255-275. doi: 10.3934/jgm.2019014
References:
[1]

E. Artin, Geometric Algebra, reprint of the 1957 original, John Wiley & Sons, Inc., New York, 1988. doi: 10.1002/9781118164518.  Google Scholar

[2]

C. Balleier and T. Wurzbacher, On the geometry and quantization of symplectic Howe pairs, Math. Z., 271 (2012), 577-591.  doi: 10.1007/s00209-011-0878-7.  Google Scholar

[3]

A. Blaom, A Geometric Setting for Hamiltonian Perturbation Theory, Memoirs of the American Mathematical Society, vol. 153, AMS, 2001. doi: 10.1090/memo/0727.  Google Scholar

[4]

F. Gay-Balmaz and C. Vizman, Dual pairs in fluid dynamics, Ann. Global Anal. Geom., 41 (2012), 1-24.  doi: 10.1007/s10455-011-9267-z.  Google Scholar

[5]

D. D. Holm and J. E. Marsden, Momentum maps and measure-valued solutions (peakons, filaments and sheets) for the EPDiff equation, in The Breath of Symplectic and Poisson Geometry (eds. J. E. Marsden and T. S. Ratiu), Progress in Mathematics, vol. 232, Birkhäuser, (2005), 203–235. doi: 10.1007/0-8176-4419-9_8.  Google Scholar

[6]

R. Howe, Remarks on classical invariant theory, Trans. Amer. Math. Soc., 313 (1989), 539-570.  doi: 10.1090/S0002-9947-1989-0986027-X.  Google Scholar

[7]

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.  Google Scholar

[8]

S. Kudla, Seesaw dual reductive pairs, in Automorphic Forms of Several Variables (Katata, 1983), (eds. I. Satake and Y. Morita), Progress in Mathematics, 46 (1984), 244–268.  Google Scholar

[9]

J. Lee, Introduction to Smooth Manifolds, 2$^{nd}$ edition, Graduate Texts in Mathematics, vol. 218, Springer, 2013. doi: 10.1007/978-0-387-21752-9.  Google Scholar

[10]

E. Lerman, R. Montgomery and R. Sjamaar, Examples of singular reduction, in Symplectic Geometry (Warwick, 1990) (ed. D. Salamon), London Mathematical Society Lecture Note Series, vol. 192, Cambridge University Press, (1993), 127–155. doi: 10.1017/CBO9780511526343.008.  Google Scholar

[11]

P. Libermann and C.-M. Marle, Symplectic Geometry and Analytical Mechanics, Mathematics and Its Applications, vol. 35, D. Reidel Publishing Company, 1987. doi: 10.1007/978-94-009-3807-6.  Google Scholar

[12]

S. Lie, Theorie der Tranformationsgruppen, (Zweiter Abschnitt, unter mitwirkung von Prof. Dr. Friedrich Engel), Teubner, 1890. Google Scholar

[13]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, 2$^{nd}$ edition, Texts in Applied Mathematics, vol. 17, Springer, 1999. doi: 10.1007/978-0-387-21792-5.  Google Scholar

[14]

J. E. Marsden and A. Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Order in chaos (Los Alamos, N.M., 1982). Phys. D, 7 (1983), 305-323.  doi: 10.1016/0167-2789(83)90134-3.  Google Scholar

[15]

J.-P. Ortega, Singular dual pairs, Differential Geom. Appl., 19 (2003), 61-95.  doi: 10.1016/S0926-2245(03)00015-9.  Google Scholar

[16]

J.-P. Ortega and T. S. Ratiu, Momentum Maps and Hamiltonian Reduction, Progress in Mathematics, vol. 222, Birkhäuser, 2004. doi: 10.1007/978-1-4757-3811-7.  Google Scholar

[17]

P. Skerritt, The frame bundle picture of Gaussian wave packet dynamics in semiclassical mechanics, arXiv: 1802.04362. Google Scholar

[18]

A. Weinstein, The local structure of Poisson manifolds, J. Differential Geom., 18 (1983), 523-557.  doi: 10.4310/jdg/1214437787.  Google Scholar

[19]

H. Xu, An SVD-like matrix decomposition and its applications, Linear Algebra Appl., 368 (2003), 1-24.  doi: 10.1016/S0024-3795(03)00370-7.  Google Scholar

show all references

References:
[1]

E. Artin, Geometric Algebra, reprint of the 1957 original, John Wiley & Sons, Inc., New York, 1988. doi: 10.1002/9781118164518.  Google Scholar

[2]

C. Balleier and T. Wurzbacher, On the geometry and quantization of symplectic Howe pairs, Math. Z., 271 (2012), 577-591.  doi: 10.1007/s00209-011-0878-7.  Google Scholar

[3]

A. Blaom, A Geometric Setting for Hamiltonian Perturbation Theory, Memoirs of the American Mathematical Society, vol. 153, AMS, 2001. doi: 10.1090/memo/0727.  Google Scholar

[4]

F. Gay-Balmaz and C. Vizman, Dual pairs in fluid dynamics, Ann. Global Anal. Geom., 41 (2012), 1-24.  doi: 10.1007/s10455-011-9267-z.  Google Scholar

[5]

D. D. Holm and J. E. Marsden, Momentum maps and measure-valued solutions (peakons, filaments and sheets) for the EPDiff equation, in The Breath of Symplectic and Poisson Geometry (eds. J. E. Marsden and T. S. Ratiu), Progress in Mathematics, vol. 232, Birkhäuser, (2005), 203–235. doi: 10.1007/0-8176-4419-9_8.  Google Scholar

[6]

R. Howe, Remarks on classical invariant theory, Trans. Amer. Math. Soc., 313 (1989), 539-570.  doi: 10.1090/S0002-9947-1989-0986027-X.  Google Scholar

[7]

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.  Google Scholar

[8]

S. Kudla, Seesaw dual reductive pairs, in Automorphic Forms of Several Variables (Katata, 1983), (eds. I. Satake and Y. Morita), Progress in Mathematics, 46 (1984), 244–268.  Google Scholar

[9]

J. Lee, Introduction to Smooth Manifolds, 2$^{nd}$ edition, Graduate Texts in Mathematics, vol. 218, Springer, 2013. doi: 10.1007/978-0-387-21752-9.  Google Scholar

[10]

E. Lerman, R. Montgomery and R. Sjamaar, Examples of singular reduction, in Symplectic Geometry (Warwick, 1990) (ed. D. Salamon), London Mathematical Society Lecture Note Series, vol. 192, Cambridge University Press, (1993), 127–155. doi: 10.1017/CBO9780511526343.008.  Google Scholar

[11]

P. Libermann and C.-M. Marle, Symplectic Geometry and Analytical Mechanics, Mathematics and Its Applications, vol. 35, D. Reidel Publishing Company, 1987. doi: 10.1007/978-94-009-3807-6.  Google Scholar

[12]

S. Lie, Theorie der Tranformationsgruppen, (Zweiter Abschnitt, unter mitwirkung von Prof. Dr. Friedrich Engel), Teubner, 1890. Google Scholar

[13]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, 2$^{nd}$ edition, Texts in Applied Mathematics, vol. 17, Springer, 1999. doi: 10.1007/978-0-387-21792-5.  Google Scholar

[14]

J. E. Marsden and A. Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Order in chaos (Los Alamos, N.M., 1982). Phys. D, 7 (1983), 305-323.  doi: 10.1016/0167-2789(83)90134-3.  Google Scholar

[15]

J.-P. Ortega, Singular dual pairs, Differential Geom. Appl., 19 (2003), 61-95.  doi: 10.1016/S0926-2245(03)00015-9.  Google Scholar

[16]

J.-P. Ortega and T. S. Ratiu, Momentum Maps and Hamiltonian Reduction, Progress in Mathematics, vol. 222, Birkhäuser, 2004. doi: 10.1007/978-1-4757-3811-7.  Google Scholar

[17]

P. Skerritt, The frame bundle picture of Gaussian wave packet dynamics in semiclassical mechanics, arXiv: 1802.04362. Google Scholar

[18]

A. Weinstein, The local structure of Poisson manifolds, J. Differential Geom., 18 (1983), 523-557.  doi: 10.4310/jdg/1214437787.  Google Scholar

[19]

H. Xu, An SVD-like matrix decomposition and its applications, Linear Algebra Appl., 368 (2003), 1-24.  doi: 10.1016/S0024-3795(03)00370-7.  Google Scholar

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