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Cayley groups

Authors: Nicole Lemire, Vladimir L. Popov and Zinovy Reichstein
Journal: J. Amer. Math. Soc. 19 (2006), 921-967
MSC (2000): Primary 14L35, 14L40, 14L30, 17B45, 20C10
Published electronically: February 6, 2006
MathSciNet review: 2219306
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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{s}o}_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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Additional Information

Nicole Lemire
Affiliation: Department of Mathematics, University of Western Ontario, London, Ontario N6A 5B7, Canada

Vladimir L. Popov
Affiliation: Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina 8, Moscow 119991, Russia

Zinovy Reichstein
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada

Keywords: Algebraic group, Lie algebra, reductive group, algebraic torus, Weyl group, root system, birational isomorphism, Cayley map, rationality, cohomology, permutation lattice
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, Zürich, Switzerland, Russian grants \Russian{RFFI 05–01–00455, NSH–123.2003.01}, and a (granting) program of the Mathematics Branch of the Russian Academy of Sciences.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.