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On the number of abelian left symmetric algebras


Authors: Karel Dekimpe and Veerle Ongenae
Journal: Proc. Amer. Math. Soc. 128 (2000), 3191-3200
MSC (2000): Primary 17A30, 17B30; Secondary 57M60, 53B05
DOI: https://doi.org/10.1090/S0002-9939-00-05484-8
Published electronically: May 11, 2000
MathSciNet review: 1695151
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Abstract:

In this paper we prove that there are infinitely many abelian left symmetric algebras in dimensions $\geq 6$. Equivalently this means that there are, up to affine conjugation, infinitely many simply transitive affine actions of $\mathbb R^k$, for $k\geq 6$. This is a result which is usually credited to A.T. Vasquez, but for which there is no proof in the literature.


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  • 1. Auslander, L.
    Simply Transitive Groups of Affine Motions.
    Amer. J. Math., 1977, 99 (4), pp. 809-826. MR 56:5782
  • 2. Benoist, Y.
    Une nilvariété non affine.
    J. Differential Geom., 1995, 41 pp. 21-52. MR 96c:53077
  • 3. Burde, D.
    Affine structures on nilmanifolds.
    Internat. J. Math, 1996, 7 5, pp. 599 - 616. MR 97i:53056
  • 4. Burde, D. and Grunewald, F.
    Modules for certain Lie algebras of maximal class.
    J. Pure Appl. Algebra, 1995, 99 pp. 239-254. MR 96d:17007
  • 5. Dekimpe, K., Igodt, P., and Ongenae, V.
    The five-dimensional complete left symmetric algebra structures compatible with an abelian Lie algebra structure.
    Linear Algebra and its Applications, 1997, 263, pp. 349-375. MR 98g:17001
  • 6. Dekimpe, K. and Malfait, W.
    Affine structures on a class of virtually nilpotent groups.
    Topol. and its Applications, 1996, 73 (2), pp. 97-119. MR 97j:57060
  • 7. Fried, D., Goldman, W., and Hirsch, M.
    Affine manifolds with nilpotent holonomy.
    Comment. Math. Helv., 1981, 56 pp. 487-523.
  • 8. Fried, D. and Goldman, W. M.
    Three-Dimensional Affine Crystallographic Groups.
    Adv. in Math., 1983, 47 1, pp. 1-49. MR 84d:20047
  • 9. Helmstetter, J.
    Radical d'une algèbre symérique a gauche.
    Ann. Inst. Fourier Grenoble, 1979, 29 (4), pp. 17-35. MR 81j:17002
  • 10. Kim, H.
    Complete left-invariant affine structures on nilpotent Lie groups.
    J. Differential Geom., 1986, 24, pp. 373-394. MR 88c:53030
  • 11. Matsushima, Y.
    Affine structures on complex manifolds.
    Osaka J. Math, 1968, 5, pp. 215-222. MR 39:2086
  • 12. Milnor, J.
    On fundamental groups of complete affinely flat manifolds.
    Adv. Math., 1977, 25 pp. 178-187. MR 56:13130
  • 13. Segal, D.
    The structure of complete left-symmetric algebras.
    Math. Ann., 1992, 293 (3), pp. 569-578. MR 93i:17026

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Additional Information

Karel Dekimpe
Affiliation: Katholieke Universiteit Leuven Campus Kortrijk, Universitaire Campus, B-8500 Kortrijk, Belgium
Email: Karel.Dekimpe@kulak.ac.be

Veerle Ongenae
Affiliation: Katholieke Universiteit Leuven Campus Kortrijk, Universitaire Campus, B-8500 Kortrijk, Belgium
Address at time of publication: Department of Pure Mathematics and Computer Algebra, University of Ghent, Galglaan 2, B-9000 Gent, Belgium
Email: vo@cage.rug.ac.be

DOI: https://doi.org/10.1090/S0002-9939-00-05484-8
Keywords: Left symmetric algebra, simply transitive affine action
Received by editor(s): January 11, 1999
Published electronically: May 11, 2000
Additional Notes: The first author is a Research Fellow of the Fund for Scientific Research – Flanders (Belgium) (F.W.O.)
Communicated by: Christopher Croke
Article copyright: © Copyright 2000 American Mathematical Society

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