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Journal of the American Mathematical Society

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ISSN 1088-6834 (online) ISSN 0894-0347 (print)

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Cayley groups
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by Nicole Lemire, Vladimir L. Popov and Zinovy Reichstein
J. Amer. Math. Soc. 19 (2006), 921-967
DOI: https://doi.org/10.1090/S0894-0347-06-00522-4
Published electronically: February 6, 2006
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Abstract:

The classical Cayley map, $X \mapsto (I_n-X)(I_n+X)^{-1}$, is a birational isomorphism between the special orthogonal group SO$_n$ and its Lie algebra ${\mathfrak so}_n$, which is SO$_n$-equivariant with respect to the conjugating and adjoint actions, respectively. We ask whether or not maps with these properties can be constructed for other algebraic groups. We show that the answer is usually “no", with a few exceptions. In particular, we show that a Cayley map for the group SL$_n$ exists if and only if $n \leqslant 3$, answering an old question of Luna.
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Bibliographic Information
  • Nicole Lemire
  • Affiliation: Department of Mathematics, University of Western Ontario, London, Ontario N6A 5B7, Canada
  • Email: nlemire@uwo.ca
  • Vladimir L. Popov
  • Affiliation: Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina 8, Moscow 119991, Russia
  • Email: popovvl@orc.ru
  • Zinovy Reichstein
  • Affiliation: Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada
  • MR Author ID: 268803
  • Email: reichst@math.ubc.ca
  • Received by editor(s): January 14, 2005
  • Published electronically: February 6, 2006
  • Additional Notes: The first and last authors were supported in part by NSERC research grants. The second author was supported in part by ETH, ZĂŒrich, Switzerland, Russian grants RFFI 05–01–00455, NSH–123.2003.01, and a (granting) program of the Mathematics Branch of the Russian Academy of Sciences.
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 19 (2006), 921-967
  • MSC (2000): Primary 14L35, 14L40, 14L30, 17B45, 20C10
  • DOI: https://doi.org/10.1090/S0894-0347-06-00522-4
  • MathSciNet review: 2219306