Structure preserving transformations in hyperkähler Euclidean spaces

Thumbnail Image
Full text at PDC
Publication Date
Advisors (or tutors)
Journal Title
Journal ISSN
Volume Title
Google Scholar
Research Projects
Organizational Units
Journal Issue
The definition and structure of hyperkähler structure preserving transformations (invariance group) for quaternionic structures have been recently studied and some preliminary results on the Euclidean case discussed. In this work we present the whole structure of the invariance Lie algebra in the Euclidean case for any dimension.
© 2015 Elsevier B.V. MAR was supported by the Spanish Ministry of Science and Innovation under project FIS2011-22566. GG is supported by the Italian MIUR-PRIN program under project 2010-JJ4KPA. This article was started in the course of visits of GG at Universidad Complutense of Madrid and of MAR at Università degli Studi di Milano.
Unesco subjects
[1] D.V. Alekseevsky and S. Marchiafava, “Quaternionic structures on a manifold and subordinated structures”, Ann. Mat.Pura Appl. 171 (1996), 205-273. [2] V.I. Arnold, Mathematical Methods of Classical Mechanics (Springer, Berlin, 1989). [3] M.F. Atiyah, Geometry of Yang-Mills fields (lezioni fermiane) (SNS, Pisa, 1979). [4] M.F. Atiyah, “Hyper-Kahler manifolds”, in Complex geometry and analysis, Lect. Notes Math. 1422, ed. V. Villani (Springer, Berlin, 1990). [5] M.F. Atiyah and N.J. Hitchin, The geometry and dynamics of magnetic monopoles, (Princeton University Press, Princeton, 1988). [6] M. Berger, “Sur les groupes d’holonomie homog`enes de variétés `a connexion affine et des variétés riemanniennes”, Bull. Soc. Math. France 83 (1955), 279-330. [7] A. Baker, Matrix Groups: An Introduction to Lie Group Theory (Springer, London 2002). [8] E. Calabi, “Métriques kahleriennes et fibrés holomorphes”, Ann. Sci. ENS 12 (1979), 269-294. [9] E. Calabi, “Isometric families of Kahler structures”, in The Chern symposium 1979 ed. W. Y. Hsiang et al. (Springer, New York, 1980). [10] A.S. Dancer, “Nahm’s equations and hyperkhler geometry”, Comm. Math. Phys. 158 (1993), 545-568. [11] M. Dunajski, Solitons, instantons and twistors (Oxford University Press, Oxford, 2010). [12] E.B. Dynkin, “The maximal subgroups of the classical groups”, Trudy Moskov Mat. Obsc. 1, 39 (1952) (AMS Transl. (Series 2) 245 (1957)). [13] G. Gaeta, “Quaternionic integrability”, J. Nonlin. Math. Phys. 18 (2011), 461-474. [14] G. Gaeta and P. Morando, “Hyper-Hamiltonian dynamics”, J. Phys. A: Math. Gen. 35 (2002), 3925-3943. [15] G. Gaeta and P. Morando, “Quaternionic integrable systems”, in Symmetry and Perturbation Theory – SPT2002 ed. S.. Abenda, G. Gaeta and S. Walcher (World Scientific, Singapore, 2003). [16] G. Gaeta and P. Morando, “A variational principle for volume-preserving dynamics”, J. Nonlin. Math. Phys. 10 (2003), 539-554. [17] G. Gaeta and M. A. Rodríguez, “On the physical applications of hyper-Hamiltonian dynamics”, J. Phys. A: Math. Theor. 41 (2008), 175203. [18] G. Gaeta and M. A. Rodríguez, “Hyperkahler structure of the Taub-NUT metric”, J. Nonlin. Math. Phys. 19 (2012), 1250014. [19] G. Gaeta and M.A. Rodríguez “Canonical transformations for hyperkahler structures and hyperhamiltonian dynamics”, J. Math. Phys. 55 (2014), 052901. [20] Gaeta G and Rodríguez, “Symmetry and quaternionic integrable systems”, J. Geom. Phys. 87 (2015), 134-148. [21] G. Gentili, S. Marchiafava and M. Pontecorvo (eds.) Proceedings of the Meeting on Quaternionic Structures in Mathematics and Physics (SISSA, Trieste, Italy, September 5-9, 1994) 270 pp. Available at [22] N.J. Hitchin, A. Karlhede, U. Lindstrom and M. Rocek, “Hyperkahler metrics and Supersymmetry”, Commun. Math. Phys. 108 (1987), 535-589. [23] D. D. Joyce, Compact manifolds with special holonomy (Oxford Sci. Pub., 2000). [24] A. A. Kirillov, Elements of the Theory of Representations (Springer, Berlin, 1976). [25] L.D. Landau and E.M. Lifshitz, Quantum Mechanics (Pergamon Press, London, 1958). [26] S. Marchiafava, P. Piccinni and M. Pontecorvo, Quaternionic Structures in Mathematics and Physics (World Scientific, 2001). [27] P. Morando and M. Tarallo “Quaternionic Hamilton equations”, Mod. Phys. Lett. A 18 (2003), 1841-1847. [28] A. Newlander and L. Nirenberg, “Complex analytic coordinates in almost complex manifolds.” Ann. Math. 65 (1957), 391–40.