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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 ${\mathfrak so}_n$, 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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  • Christine Bessenrodt and Lieven Le Bruyn, Stable rationality of certain ${\rm PGL}_n$-quotients, Invent. Math. 104 (1991), no. 1, 179–199. MR 1094051, DOI
  • Grégory Berhuy, Marina Monsurrò, and Jean-Pierre Tignol, Cohomological invariants and $R$-triviality of adjoint classical groups, Math. Z. 248 (2004), no. 2, 313–323. MR 2088930, DOI
  • Armand Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012
  • N. Bourbaki, Éléments de mathématique. Fasc. XXXVII. Groupes et algèbres de Lie. Chapitre II: Algèbres de Lie libres. Chapitre III: Groupes de Lie, Hermann, Paris, 1972. Actualités Scientifiques et Industrielles, No. 1349. MR 0573068
  • N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238
  • N. Bourbaki, Éléments de mathématique. Fasc. XXXVIII: Groupes et algèbres de Lie. Chapitre VII: Sous-algèbres de Cartan, éléments réguliers. Chapitre VIII: Algèbres de Lie semi-simples déployées, Actualités Scientifiques et Industrielles, No. 1364. Hermann, Paris, 1975 (French). MR 0453824
  • Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 672956
  • Jon F. Carlson, Modules and group algebras, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1996. Notes by Ruedi Suter. MR 1393196
  • [Ca]cayley A. Cayley, Sur Quelques propriétés des déterminants gauches, J. Rein. Angew. Math. (Crelle) 32 (1846), 119–123. Reprinted in: The Coll. Math. Papers of Arthur Cayley, Vol. I, No. 52, Cambridge University Press, 1889, 332–336.
  • C. Chevalley, On algebraic group varieties, J. Math. Soc. Japan 6 (1954), 303–324. MR 67122, DOI
  • Claude Chevalley, The algebraic theory of spinors and Clifford algebras, Springer-Verlag, Berlin, 1997. Collected works. Vol. 2; Edited and with a foreword by Pierre Cartier and Catherine Chevalley; With a postface by J.-P. Bourguignon. MR 1636473
  • Gerald Cliff and Alfred Weiss, Summands of permutation lattices for finite groups, Proc. Amer. Math. Soc. 110 (1990), no. 1, 17–20. MR 1027091, DOI
  • Jean-Louis Colliot-Thélène and Jean-Jacques Sansuc, La $R$-équivalence sur les tores, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 2, 175–229 (French). MR 450280
  • Jean-Louis Colliot-Thélène and Jean-Jacques Sansuc, Principal homogeneous spaces under flasque tori: applications, J. Algebra 106 (1987), no. 1, 148–205. MR 878473, DOI
  • [C–T]colliot J.–L. Colliot–Thélène (with the collaboration of J.–J. Sansuc), The rationality problem for fields under linear algebraic groups (with special regards to the Brauer group), IX Escuela Latinoamericana de Mathemáticas, Santiago de Chile, July 1988.
  • Anne Cortella and Boris Kunyavskiĭ, Rationality problem for generic tori in simple groups, J. Algebra 225 (2000), no. 2, 771–793. MR 1741561, DOI
  • V. Chernousov and A. Merkurjev, $R$-equivalence and special unitary groups, J. Algebra 209 (1998), no. 1, 175–198. MR 1652122, DOI
  • Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1990. With applications to finite groups and orders; Reprint of the 1981 original; A Wiley-Interscience Publication. MR 1038525
  • Jean A. Dieudonné, La géométrie des groupes classiques, Springer-Verlag, Berlin-New York, 1971 (French). Troisième édition; Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 5. MR 0310083
  • Igor V. Dolgachev, Rationality of fields of invariants, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 3–16. MR 927970
  • Mowaffaq Hajja and Ming Chang Kang, Some actions of symmetric groups, J. Algebra 177 (1995), no. 2, 511–535. MR 1355213, DOI
  • Otto Hölder, Die Gruppen der Ordnungen $p^3$, $pq^2$, $pqr$, $p^4$, Math. Ann. 43 (1893), no. 2-3, 301–412 (German). MR 1510814, DOI
  • V. A. Iskovskih, Minimal models of rational surfaces over arbitrary fields, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 1, 19–43, 237 (Russian). MR 525940
  • [Isk2]iskovskih-enc V. A. Iskovskikh, Cremona group, in: Math. Encyclopaedia, Vol. 3, Sov. Encycl., Moscow, 1982, (in Russian), p. 95.
  • V. A. Iskovskikh, Factorization of birational mappings of rational surfaces from the point of view of Mori theory, Uspekhi Mat. Nauk 51 (1996), no. 4(310), 3–72 (Russian); English transl., Russian Math. Surveys 51 (1996), no. 4, 585–652. MR 1422227, DOI
  • V. A. Iskovskikh, Two nonconjugate embeddings of the group $S_3\times Z_2$ into the Cremona group, Tr. Mat. Inst. Steklova 241 (2003), no. Teor. Chisel, Algebra i Algebr. Geom., 105–109 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 2(241) (2003), 93–97. MR 2024046
  • [Isk$_5$]iskovskih5 V. A. Iskovskikh, Two non-conjugate embeddings of $\ {\operatorname {S}}_3 \times {\mathbb Z}_2$ into the Cremona group, II, arXiv:math.AG/0508484.
  • Bertram Kostant and Peter W. Michor, The generalized Cayley map from an algebraic group to its Lie algebra, The orbit method in geometry and physics (Marseille, 2000) Progr. Math., vol. 213, Birkhäuser Boston, Boston, MA, 2003, pp. 259–296. MR 1995382
  • Max-Albert Knus, Alexander Merkurjev, Markus Rost, and Jean-Pierre Tignol, The book of involutions, American Mathematical Society Colloquium Publications, vol. 44, American Mathematical Society, Providence, RI, 1998. With a preface in French by J. Tits. MR 1632779
  • M. Kneser, Lectures on Galois cohomology of classical groups, Tata Institute of Fundamental Research, Bombay, 1969. With an appendix by T. A. Springer; Notes by P. Jothilingam; Tata Institute of Fundamental Research Lectures on Mathematics, No. 47. MR 0340440
  • Lieven Le Bruyn, Centers of generic division algebras, the rationality problem 1965–1990, Israel J. Math. 76 (1991), no. 1-2, 97–111. MR 1177334, DOI
  • Lieven Le Bruyn, Generic norm one tori, Nieuw Arch. Wisk. (4) 13 (1995), no. 3, 401–407. MR 1378805
  • Nicole Lemire and Martin Lorenz, On certain lattices associated with generic division algebras, J. Group Theory 3 (2000), no. 4, 385–405. MR 1790337, DOI
  • [LPR]lpr N. Lemire, V. L. Popov, Z. Reichstein, Cayley groups, Forschungsinstitut für Mathematik ETH Zürich preprint, August 2004. [Lun1]lu1 D. Luna, Sur la linéarité au voisinage d’un point fixe, Unpublished manuscript, May 1972.
  • Domingo Luna, Slices étales, Sur les groupes algébriques, Soc. Math. France, Paris, 1973, pp. 81–105. Bull. Soc. Math. France, Paris, Mémoire 33 (French). MR 0342523, DOI
  • [Lun3]lu2 D. Luna, Letter to V. L. Popov, January 18, 1975.
  • Ju. I. Manin, Rational surfaces over perfect fields. II, Mat. Sb. (N.S.) 72 (114) (1967), 161–192 (Russian). MR 0225781
  • Yu. I. Manin and M. Hazewinkel, Cubic forms: algebra, geometry, arithmetic, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., New York, 1974. Translated from the Russian by M. Hazewinkel; North-Holland Mathematical Library, Vol. 4. MR 0460349
  • A. S. Merkurjev, $R$-equivalence and rationality problem for semisimple adjoint classical algebraic groups, Inst. Hautes Études Sci. Publ. Math. 84 (1996), 189–213 (1997). MR 1441008
  • D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR 1304906
  • V. L. Popov, Picard groups of homogeneous spaces of linear algebraic groups and one-dimensional homogeneous vector fiberings, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 294–322 (Russian). MR 0357399
  • [Pop2]popov-luna V. L. Popov, Letter to D. Luna, March 12, 1975.
  • Vladimir Popov, Sections in invariant theory, The Sophus Lie Memorial Conference (Oslo, 1992) Scand. Univ. Press, Oslo, 1994, pp. 315–361. MR 1456471
  • È. B. Vinberg and V. L. Popov, Invariant theory, Algebraic geometry, 4 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989, pp. 137–314, 315 (Russian). MR 1100485
  • M. Postnikov, Lie groups and Lie algebras, “Mir”, Moscow, 1986. Lectures in geometry. Semester V; Translated from the Russian by Vladimir Shokurov. MR 905471
  • David J. Saltman, Invariant fields of linear groups and division algebras, Perspectives in ring theory (Antwerp, 1987) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 233, Kluwer Acad. Publ., Dordrecht, 1988, pp. 279–297. MR 1048416
  • [Se]serre J.–P. Serre, Espaces fibrés algébriques, in: Séminaire C. Chevalley E.N.S., 1958, Anneaux de Chow et applications, Exposé no. 1, Secr. math. 11 rue Pierre Curie, Paris 5e, 1958, 1-01– 1-37. Reprinted in: Exposés de Séminaires 1950–1999, Doc. Math. 1, Soc. Math. France, 2001, 107–139.
  • Igor R. Shafarevich, Basic algebraic geometry. 1, 2nd ed., Springer-Verlag, Berlin, 1994. Varieties in projective space; Translated from the 1988 Russian edition and with notes by Miles Reid. MR 1328833
  • A. Speiser, Zahlentheoretische Sätze aus der Gruppentheorie, Math. Z. 5 (1919), no. 1-2, 1–6 (German). MR 1544369, DOI
  • T. A. Springer, Linear algebraic groups, 2nd ed., Progress in Mathematics, vol. 9, Birkhäuser Boston, Inc., Boston, MA, 1998. MR 1642713
  • Richard G. Swan, Invariant rational functions and a problem of Steenrod, Invent. Math. 7 (1969), 148–158. MR 244215, DOI
  • V. E. Voskresenskiĭ, Algebraic groups and their birational invariants, Translations of Mathematical Monographs, vol. 179, American Mathematical Society, Providence, RI, 1998. Translated from the Russian manuscript by Boris Kunyavski [Boris È. Kunyavskiĭ]. MR 1634406
  • V. E. Voskresenskiĭ and A. A. Klyachko, Toric Fano varieties and systems of roots, Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984), no. 2, 237–263 (Russian). MR 740791
  • Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR 1269324
  • André Weil, Algebras with involutions and the classical groups, J. Indian Math. Soc. (N.S.) 24 (1960), 589–623 (1961). MR 136682
  • Hermann Weyl, The Classical Groups. Their Invariants and Representations, Princeton University Press, Princeton, N.J., 1939. MR 0000255

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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
MR Author ID: 268803

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 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.