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Quaternionic Kähler reductions of Wolf spaces

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Abstract

The main purpose of the following article is to introduce a Lie theoretical approach to the problem of classifying pseudo quaternionic-Kähler (QK) reductions of the pseudo QK symmetric spaces, otherwise called generalized Wolf spaces.

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References

  1. Alekseevskĭ D.V. (1968). Riemannian spaces with unusual holonomy groups. Funkcional. Anal. i Priložen 2(2): 1–10. MR 37 #6868

    Google Scholar 

  2. Alekseevskĭ D.V. (1975). Classification of quaternionic spaces with transitive solvable group of motions. Izv. Akad. Nauk SSSR Ser. Mat. 39(2): 315–362, 472. MR 53 #6465

    Google Scholar 

  3. Alekseevskĭ D.V. and Cortés V. (1997). Homogeneous quaternionic Kähler manifolds of unimodular group. Boll. Un. Mat. Ital. B (7) 11(2): 217–229. MR 1456262 (98i:53062)

    Google Scholar 

  4. Alekseevskĭ D.V.: Classification of pseudo-Riemannian symmetric spaces of quaternionic Kähler type, Lie groups and invariant theory, Amer. Math. Soc. Transl. Ser. 2, 213, Amer. Math. Soc., Providence, RI, pp. 33–62. MR 2140713 (Review) (2005)

  5. Abraham R., Marsden J.E.: Foundations of Mechanics. Benjamin/Cummings Publishing Co. Inc. Advanced Book Program, Reading, Mass., 1978, Second edition, revised and enlarged, With the assistance of Tudor Raţiu and Richard Cushman. MR 81e:58025

  6. Bates L. (2004). A symmetry completeness criterion for second-order differential equations. Proc. AMS 132(6): 1785–1786. MR 2051142 (2005b:34076)

    Article  MATH  Google Scholar 

  7. Berger M. (1955). Sur les groupes d’holonomie homogène des variétés à connexion affine et des variétés riemanniennes. Bull. Soc. Math. France 83: 279–330. MR 18,149a

    MATH  Google Scholar 

  8. Boyer C.P., Calderbank D.M.J., Galicki K. and Piccinni P. (2005). Toric self-dual Einstein metrics as quotients. Comm. Math. Phys. 253(2): 337–370. MR 2140252

    Article  MATH  Google Scholar 

  9. Boyer C.P., Galicki K. and Mann B.M. (1993). Quaternionic reduction and Einstein manifolds, Comm. Anal. Geom. 1(2): 229–279. MR 95c:53056

    MATH  Google Scholar 

  10. Boyer C.P., Galicki K. and Mann B.M. (1994). The geometry and topology of 3-Sasakian manifolds. J. Reine Angew. Math. 455: 183–220. MR 96e:53057

    MATH  Google Scholar 

  11. Boyer C.P., Galicki K. and Piccinni P. (2002). 3-Sasakian geometry, nilpotent orbits, and exceptional quotients. Ann. Global Anal. Geom. 21(1): 85–110. MR 2003d:53076

    Article  MATH  Google Scholar 

  12. Burgoyne N. and Cushman R. (1977). Conjugacy classes in linear groups. J. Algebra 44(2): 339–362. MR 55 #5761

    Article  MATH  Google Scholar 

  13. Galicki K. (1986). Quaternionic Kähler and hyper-Kähler nonlinear σ-models. Nuclear Phys. B 271(2): 402–416. MR 88m:53087

    Article  Google Scholar 

  14. Galicki K. (1987). A generalization of the momentum mapping construction for quaternionic Kähler manifolds. Comm. Math. Phys. 108(1): 117–138. MR 88f:53088

    Article  MATH  Google Scholar 

  15. Galicki K. and Lawson H.B. (1988). Quaternionic reduction and quaternionic orbifolds. Math. Ann. 282(1): 1–21. MR 89m:53075

    Article  MATH  Google Scholar 

  16. Harvey F.R. (1990). Spinors and calibrations. Perspectives in Mathematics, Val. 9.. Academic Press, Boston

    Google Scholar 

  17. Konishi, M.: On manifolds with Sasakian 3-structure over quaternion Kaehler manifolds. Kōdai Math. Sem. Rep. 26:194–200. MR 51 #13951 (1974/75)

  18. Kobak P.Z. and Swann A. (1993). Quaternionic geometry of a nilpotent variety. Math. Ann. 297(4): 747–764. MR 94j:53057

    Article  MATH  Google Scholar 

  19. Marsden J. and Weinstein A. (1974). Reduction of symplectic manifolds with symmetry. Rep. Mathematical Phys. 5(1): 121–130. MR 0402819 (53 #6633)

    Article  MATH  Google Scholar 

  20. Salamon S.M. (1982). Quaternionic Kähler manifolds. Invent. Math. 67(1): 143–171. MR 83k:53054

    Article  MATH  Google Scholar 

  21. Swann A.  (1991). Hyper-Kähler and quaternionic Kähler geometry. Math. Ann. 289(3): 421–450. MR 92c:53030

    Article  MATH  Google Scholar 

  22. Wolf J.A. (1965). Complex homogeneous contact manifolds and quaternionic symmetric spaces. J. Math. Mech. 14: 1033–1047. MR 32 #3020

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics & Statistics, University of New Mexico, Albuquerque, NM, 87131, USA

    Daniele Grandini

  2. Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, Milano, Italy

    Daniele Grandini

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  1. Daniele Grandini
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Correspondence to Daniele Grandini.

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Grandini, D. Quaternionic Kähler reductions of Wolf spaces. Ann Glob Anal Geom 32, 225–252 (2007). https://doi.org/10.1007/s10455-006-9056-2

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  • DOI: https://doi.org/10.1007/s10455-006-9056-2

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