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Semi-simple generalized Nijenhuis operators
Invariant forms and automorphisms of locally homogeneous multisymplectic manifolds
1. | Departamento de Matemática Aplicada IV. Universitat Politècnica de Catalunya-BarcelonaTech., Campus Norte, Ed. C-3. C/ Jordi Girona 1, E-08034 Barcelona, Spain |
2. | Department of Mathematics, University of California at Berkeley, Berkeley, CA 94720, USA, and Departamento de Matemáticas, Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911 Leganés, Madrid, Spain |
3. | Departamento de Matemática Aplicada IV, Universitat Politècnica de Catalunya-BarcelonaTech., Edificio C-3, Campus Norte UPC. C/ Jordi Girona 1, E-08034 Barcelona, Spain |
4. | Departamento de Matemática Aplicada IV. Universitat Politècnica de Catalunya-BarcelonaTech., Edificio C-3, Campus Norte UPC, C/ Jordi Girona 1. 08034 Barcelona |
References:
[1] |
C. J. Atkin and J. Grabowsk, Homomorphisms of the Lie algebras associated with a symplectic manifold, Comp. Math., 76 (1990), 315-349. |
[2] |
A. Banyaga, Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique, Comment. Math. Helv., 53 (1978), 174-227.
doi: 10.1007/BF02566074. |
[3] |
A. Banyaga, On isomorphic classical diffeomorphism groups. I, Proc. Am. Math. Soc., 98 (1986), 113-118.
doi: 10.2307/2045779. |
[4] |
A. Banyaga, On isomorphic classical diffeomorphism groups. II, J. Diff. Geom., 28 (1988), 23-35. |
[5] |
A. Banyaga, The structure of classical diffeomorphism groups, in "Mathematics and Its Applications," 400, Kluwer Acad. Pub. Group., Dordrecht, (1997), 113-118. |
[6] |
A. Banyaga and A. McInerney, On isomorphic classical diffeomorphism groups. III, Ann. Global Anal. Geom., 13 (1995), 117-127.
doi: 10.1007/BF01120327. |
[7] |
W. M. Boothby, Transitivity of the automorphisms of certain geometric structures, Amer. Math. Soc., 137 (1969), 93-100. |
[8] |
R. L. Bryant, Metrics with exceptional holonomy, Ann. Math. (2), 126 (1987), 525-576.
doi: 10.2307/1971360. |
[9] |
F. Cantrijn, A. Ibort and M. de León, Hamiltonian structures on multisymplectic manifolds, Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996), 225-236. |
[10] |
F. Cantrijn, A. Ibort and M. de León, On the geometry of multisymplectic manifolds, J. Austral. Math. Soc. Ser., 66 (1999), 303-330. |
[11] |
J. F. Cariñena, M. Crampin and L. A. Ibort, On the multisymplectic formalism for first order field theories, Diff. Geom. Appl., 1 (1991), 345-374.
doi: 10.1016/0926-2245(91)90013-Y. |
[12] |
J. F. Cariñena, J. Gomis, L. A. Ibort and N. Román-Roy, Canonical transformation theory for presymplectic systems, J. Math. Phys., 26 (1985), 1961-1969.
doi: 10.1063/1.526864. |
[13] |
A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Multivector fields and connections. Setting Lagrangian equations in field theories, J. Math. Phys., 39 (1998), 4578-4603.
doi: 10.1063/1.532525. |
[14] |
A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Multivector field formulation of Hamiltonian field theories: Equations and symmetries, J. Phys. A: Math. Gen., 32 (1999), 8461-8484.
doi: 10.1088/0305-4470/32/48/309. |
[15] |
A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Geometry of multisymplectic Hamiltonian first-order field theories, J. Math. Phys., 41 (2000), 7402-7444.
doi: 10.1063/1.1308075. |
[16] |
A. Echeverría-Enríquez, M. de León, M. C. Muñoz-Lecanda and N. Román-Roy, Extended Hamiltonian systems in multisymplectic field theories, J. Math. Phys., 48 (2007), 112901 (30 pp).
doi: 10.1063/1.2801875. |
[17] |
G. Giachetta, L. Mangiarotti and G. Sardanashvily, "New Lagrangian and Hamiltonian Methods in Field Theory," World Sci. Pub. Co., Singapore, 1997. |
[18] |
J. Gomis, J. Llosa and N. Román-Roy, Lee Hwa Chung theorem for presymplectic manifolds. Canonical transformations for constrained systems, J. Math. Phys., 25 (1984), 1348-1355.
doi: 10.1063/1.526303. |
[19] |
M. J. Gotay, A multisymplectic framework for classical field theory and the calculus of variations I: Covariant Hamiltonian formalism, in "Mechanics, Analysis and Geometry: 200 Years after Lagrange" (ed. M. Francaviglia), Elsevier Science Pub., (1991), 203-235. |
[20] |
J. Grabowski, "Isomorphisms of Poisson and Jacobi Brackets," Banach Center Publ., 51, Polish Acad. Sci., Warsaw, (2000), 79-85. |
[21] |
F. Helein and J. Kouneiher, Finite dimensional Hamiltonian formalism for gauge and quantum field theories, J. Math. Phys., 43 (2002), 2306-2347.
doi: 10.1063/1.1467710. |
[22] |
L. Hwa Chung, The universal integral invariants of Hamiltonian systems and applications to the theory of canonical transformations, Proc. Roy. Soc., LXIIA (1947), 237-246. |
[23] |
L. A. Ibort, Multisymplectic geometry: Generic and exceptional, in "IX Fall Workshop on Geometry and Physics, Vilanova i la Geltrú, Spain (2000)" (eds. X. Gràcia, J. Marín-Solano, M. C. Muñoz-Lecanda and N. Román-Roy), UPC Eds., (2001), 79-88. |
[24] |
I. V. Kanatchikov, Canonical structure of classical field theory in the polymomentum phase space, Rep. Math. Phys., 41 (1998), 49-90.
doi: 10.1016/S0034-4877(98)80182-1. |
[25] |
J. Kijowski and W. M. Tulckzyjew, "A Symplectic Framework for Field Theories," Lecture Notes in Physics 107, Springer-Verlag, New York, 1979. |
[26] |
M. de León D. Martín de Diego and A. Santamaría-Merino, Tulczyjew triples and Lagrangian submanifolds in classical field theories, in "Applied Differential Geometry and Mechanics" (eds. W. Sarlet and F. Cantrijn), Univ. of Gent, Gent, Academia Press, (2003), 21-47. |
[27] |
M. de León, D. Martín de Diego and A. Santamaría-Merino, Symmetries in classical field theories, Int. J. Geom. Meth. Mod. Phys., 1 (2004), 651-710.
doi: 10.1142/S0219887804000290. |
[28] |
J. Llosa and N. Román-Roy, Invariant forms and Hamiltonian systems: A geometrical setting, Int. J. Theor. Phys., 27 (1988), 1533-1543.
doi: 10.1007/BF00669290. |
[29] |
C. M. Marle, The Schouten-Nijenhuis bracket and interior products, J. Geom. Phys., 23 (1997), 350-359.
doi: 10.1016/S0393-0440(97)80009-5. |
[30] |
J. E. Marsden and S. Shkoller, Multisymplectic geometry, covariant Hamiltonians and water waves, Math. Proc. Camb. Phil. Soc., 125 (1999), 553-575.
doi: 10.1017/S0305004198002953. |
[31] |
J. Martinet, Sur les singularités des formes différentielles, Ann. Inst. Fourier, Grenoble, 20 (1970), 95-178. |
[32] |
H. Omori, "Infinite Dimensional Lie Transformation Groups," Lect. Notes in Maths., 427, Springer-Verlag, Berlin and New York , 1974. |
[33] |
C. Paufler and H. Romer, Geometry of Hamiltonian $n$-vector fields in multisymplectic field theory, J. Geom. Phys., 44 (2002), 52-69.
doi: 10.1016/S0393-0440(02)00031-1. |
[34] |
L. E. Pursell and M. E. Shanks, The Lie algebra of smooth manifolds, Proc. Am. Math. Soc., 5 (1954), 468-472. |
[35] |
N. Román-Roy, A. M. Rey, M. Salgado and S. Vilariño, On the $k$-symplectic, $k$-cosymplectic and multisymplectic formalisms of classical field theories, J. Geom. Mech., 3 (2011), 113-137. |
[36] |
G. Sardanashvily, "Generalized Hamiltonian Formalism for Field Theory," World Scientific Pub., Singapore, 1995.
doi: 10.1142/9789812831484. |
[37] |
D. J. Saunders, "The Geometry of Jet Bundles," London Math. Soc. Lect. Notes Ser. 142, Cambridge, Univ. Press, 1989.
doi: 10.1017/CBO9780511526411. |
[38] |
M. Shafiee, On Hamiltonian group of multisymplectic manifolds, Int. J. Geom. Meth. Mod. Phys., 8 (2011), 929-935.
doi: 10.1142/S0219887811005506. |
[39] |
F. Takens, Characterization of a differentiable structure by its group of diffeomorphisms, Bol Soc. Brasil. Mat., 10 (1979), 17-25.
doi: 10.1007/BF02588337. |
[40] |
W. M. Tulczyjew, Les sous-variétés Lagrangiennes et la dynamique Hamiltoniènne, C.R. Acad Sci. Paris (Sér. A), 283 (1976), 15-18. |
[41] |
W. M. Tulczyjew, Les sous-variétés Lagrangiennes et la dynamique Lagrangiènne, C.R. Acad Sci. Paris (Sér. A), 283 (1976), 675-678. |
[42] |
M. Wechsler, Homeomorphism groups of certain topological spaces, Ann. Math., 62 (1954), 360-373. |
show all references
References:
[1] |
C. J. Atkin and J. Grabowsk, Homomorphisms of the Lie algebras associated with a symplectic manifold, Comp. Math., 76 (1990), 315-349. |
[2] |
A. Banyaga, Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique, Comment. Math. Helv., 53 (1978), 174-227.
doi: 10.1007/BF02566074. |
[3] |
A. Banyaga, On isomorphic classical diffeomorphism groups. I, Proc. Am. Math. Soc., 98 (1986), 113-118.
doi: 10.2307/2045779. |
[4] |
A. Banyaga, On isomorphic classical diffeomorphism groups. II, J. Diff. Geom., 28 (1988), 23-35. |
[5] |
A. Banyaga, The structure of classical diffeomorphism groups, in "Mathematics and Its Applications," 400, Kluwer Acad. Pub. Group., Dordrecht, (1997), 113-118. |
[6] |
A. Banyaga and A. McInerney, On isomorphic classical diffeomorphism groups. III, Ann. Global Anal. Geom., 13 (1995), 117-127.
doi: 10.1007/BF01120327. |
[7] |
W. M. Boothby, Transitivity of the automorphisms of certain geometric structures, Amer. Math. Soc., 137 (1969), 93-100. |
[8] |
R. L. Bryant, Metrics with exceptional holonomy, Ann. Math. (2), 126 (1987), 525-576.
doi: 10.2307/1971360. |
[9] |
F. Cantrijn, A. Ibort and M. de León, Hamiltonian structures on multisymplectic manifolds, Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996), 225-236. |
[10] |
F. Cantrijn, A. Ibort and M. de León, On the geometry of multisymplectic manifolds, J. Austral. Math. Soc. Ser., 66 (1999), 303-330. |
[11] |
J. F. Cariñena, M. Crampin and L. A. Ibort, On the multisymplectic formalism for first order field theories, Diff. Geom. Appl., 1 (1991), 345-374.
doi: 10.1016/0926-2245(91)90013-Y. |
[12] |
J. F. Cariñena, J. Gomis, L. A. Ibort and N. Román-Roy, Canonical transformation theory for presymplectic systems, J. Math. Phys., 26 (1985), 1961-1969.
doi: 10.1063/1.526864. |
[13] |
A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Multivector fields and connections. Setting Lagrangian equations in field theories, J. Math. Phys., 39 (1998), 4578-4603.
doi: 10.1063/1.532525. |
[14] |
A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Multivector field formulation of Hamiltonian field theories: Equations and symmetries, J. Phys. A: Math. Gen., 32 (1999), 8461-8484.
doi: 10.1088/0305-4470/32/48/309. |
[15] |
A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Geometry of multisymplectic Hamiltonian first-order field theories, J. Math. Phys., 41 (2000), 7402-7444.
doi: 10.1063/1.1308075. |
[16] |
A. Echeverría-Enríquez, M. de León, M. C. Muñoz-Lecanda and N. Román-Roy, Extended Hamiltonian systems in multisymplectic field theories, J. Math. Phys., 48 (2007), 112901 (30 pp).
doi: 10.1063/1.2801875. |
[17] |
G. Giachetta, L. Mangiarotti and G. Sardanashvily, "New Lagrangian and Hamiltonian Methods in Field Theory," World Sci. Pub. Co., Singapore, 1997. |
[18] |
J. Gomis, J. Llosa and N. Román-Roy, Lee Hwa Chung theorem for presymplectic manifolds. Canonical transformations for constrained systems, J. Math. Phys., 25 (1984), 1348-1355.
doi: 10.1063/1.526303. |
[19] |
M. J. Gotay, A multisymplectic framework for classical field theory and the calculus of variations I: Covariant Hamiltonian formalism, in "Mechanics, Analysis and Geometry: 200 Years after Lagrange" (ed. M. Francaviglia), Elsevier Science Pub., (1991), 203-235. |
[20] |
J. Grabowski, "Isomorphisms of Poisson and Jacobi Brackets," Banach Center Publ., 51, Polish Acad. Sci., Warsaw, (2000), 79-85. |
[21] |
F. Helein and J. Kouneiher, Finite dimensional Hamiltonian formalism for gauge and quantum field theories, J. Math. Phys., 43 (2002), 2306-2347.
doi: 10.1063/1.1467710. |
[22] |
L. Hwa Chung, The universal integral invariants of Hamiltonian systems and applications to the theory of canonical transformations, Proc. Roy. Soc., LXIIA (1947), 237-246. |
[23] |
L. A. Ibort, Multisymplectic geometry: Generic and exceptional, in "IX Fall Workshop on Geometry and Physics, Vilanova i la Geltrú, Spain (2000)" (eds. X. Gràcia, J. Marín-Solano, M. C. Muñoz-Lecanda and N. Román-Roy), UPC Eds., (2001), 79-88. |
[24] |
I. V. Kanatchikov, Canonical structure of classical field theory in the polymomentum phase space, Rep. Math. Phys., 41 (1998), 49-90.
doi: 10.1016/S0034-4877(98)80182-1. |
[25] |
J. Kijowski and W. M. Tulckzyjew, "A Symplectic Framework for Field Theories," Lecture Notes in Physics 107, Springer-Verlag, New York, 1979. |
[26] |
M. de León D. Martín de Diego and A. Santamaría-Merino, Tulczyjew triples and Lagrangian submanifolds in classical field theories, in "Applied Differential Geometry and Mechanics" (eds. W. Sarlet and F. Cantrijn), Univ. of Gent, Gent, Academia Press, (2003), 21-47. |
[27] |
M. de León, D. Martín de Diego and A. Santamaría-Merino, Symmetries in classical field theories, Int. J. Geom. Meth. Mod. Phys., 1 (2004), 651-710.
doi: 10.1142/S0219887804000290. |
[28] |
J. Llosa and N. Román-Roy, Invariant forms and Hamiltonian systems: A geometrical setting, Int. J. Theor. Phys., 27 (1988), 1533-1543.
doi: 10.1007/BF00669290. |
[29] |
C. M. Marle, The Schouten-Nijenhuis bracket and interior products, J. Geom. Phys., 23 (1997), 350-359.
doi: 10.1016/S0393-0440(97)80009-5. |
[30] |
J. E. Marsden and S. Shkoller, Multisymplectic geometry, covariant Hamiltonians and water waves, Math. Proc. Camb. Phil. Soc., 125 (1999), 553-575.
doi: 10.1017/S0305004198002953. |
[31] |
J. Martinet, Sur les singularités des formes différentielles, Ann. Inst. Fourier, Grenoble, 20 (1970), 95-178. |
[32] |
H. Omori, "Infinite Dimensional Lie Transformation Groups," Lect. Notes in Maths., 427, Springer-Verlag, Berlin and New York , 1974. |
[33] |
C. Paufler and H. Romer, Geometry of Hamiltonian $n$-vector fields in multisymplectic field theory, J. Geom. Phys., 44 (2002), 52-69.
doi: 10.1016/S0393-0440(02)00031-1. |
[34] |
L. E. Pursell and M. E. Shanks, The Lie algebra of smooth manifolds, Proc. Am. Math. Soc., 5 (1954), 468-472. |
[35] |
N. Román-Roy, A. M. Rey, M. Salgado and S. Vilariño, On the $k$-symplectic, $k$-cosymplectic and multisymplectic formalisms of classical field theories, J. Geom. Mech., 3 (2011), 113-137. |
[36] |
G. Sardanashvily, "Generalized Hamiltonian Formalism for Field Theory," World Scientific Pub., Singapore, 1995.
doi: 10.1142/9789812831484. |
[37] |
D. J. Saunders, "The Geometry of Jet Bundles," London Math. Soc. Lect. Notes Ser. 142, Cambridge, Univ. Press, 1989.
doi: 10.1017/CBO9780511526411. |
[38] |
M. Shafiee, On Hamiltonian group of multisymplectic manifolds, Int. J. Geom. Meth. Mod. Phys., 8 (2011), 929-935.
doi: 10.1142/S0219887811005506. |
[39] |
F. Takens, Characterization of a differentiable structure by its group of diffeomorphisms, Bol Soc. Brasil. Mat., 10 (1979), 17-25.
doi: 10.1007/BF02588337. |
[40] |
W. M. Tulczyjew, Les sous-variétés Lagrangiennes et la dynamique Hamiltoniènne, C.R. Acad Sci. Paris (Sér. A), 283 (1976), 15-18. |
[41] |
W. M. Tulczyjew, Les sous-variétés Lagrangiennes et la dynamique Lagrangiènne, C.R. Acad Sci. Paris (Sér. A), 283 (1976), 675-678. |
[42] |
M. Wechsler, Homeomorphism groups of certain topological spaces, Ann. Math., 62 (1954), 360-373. |
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