Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

Restricting the Rost invariant to the center


Authors: S. Garibaldi and A. Quéguiner-Mathieu
Original publication: Algebra i Analiz, tom 19 (2007), nomer 2.
Journal: St. Petersburg Math. J. 19 (2008), 197-213
MSC (2000): Primary 12G05
DOI: https://doi.org/10.1090/S1061-0022-08-00993-X
Published electronically: February 1, 2008
MathSciNet review: 2333896
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For simple simply connected algebraic groups of classical type, Merkurjev, Parimala, and Tignol gave a formula for the restriction of the Rost invariant to the torsors induced from the center of the group. This paper completes their results by giving formulas for the exceptional groups. The method is somewhat different and also recovers their formula for classical groups.


References [Enhancements On Off] (What's this?)

  • [Borel] A. Borel, Linear algebraic groups, second ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012 (92d:20001)
  • [Bou] N. Bourbaki, Lie groups and Lie algebras: Chapters 4-6, Springer-Verlag, Berlin, 2002. MR 1890629 (2003a:17001)
  • [BS] A. Borel and T. A. Springer, Rationality properties of linear algebraic groups. II, Tohoku Math. J. (2) 20 (1968), 443-497 (= Borel, Oe., vol. 2, #76). MR 0244259 (39:5576)
  • [BT] A. Borel and J. Tits, Groupes réductifs, Inst. Hautes Études Sci. Publ. Math. 27 (1965), 55-150. MR 0207712 (34:7527)
  • [Brown] Kenneth S. Brown, Cohomology of Groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 0672956 (83k:20002)
  • [Ga01a] R. S. Garibaldi, Groups of type $ {E}_7$ over arbitrary fields, Comm. Algebra 29 (2001), no. 6, 2689-2710. MR 1845137 (2002f:17003)
  • [Ga01b] -, The Rost invariant has trivial kernel for quasi-split groups of low rank, Comment. Math. Helv. 76 (2001), no. 4, 684-711. MR 1881703 (2003g:20079)
  • [Gi] Ph. Gille, Invariants cohomologiques de Rost en caractéristique positive, $ K$-Theory 21 (2000), 57-100. MR 1802626 (2001k:11064)
  • [KMRT] M.-A. Knus, A. S. Merkurjev, M. Rost, and J.-P. Tignol, The book of involutions, Colloquium Publications, vol. 44, Amer. Math. Soc., 1998. MR 1632779 (2000a:16031)
  • [M96] A. S. Merkurjev, The norm principle for algebraic groups, St. Petersburg Math. J. 7 (1996), no. 2, 243-264. MR 1347513 (96k:20088)
  • [M03] -, Rost invariants of simply connected algebraic groups, with a section by S. Garibaldi, in Cohomological invariants in Galois cohomology, University Lecture Series, vol. 28, Amer. Math. Soc., 2003. MR 1999385
  • [MPT] A. S. Merkurjev, R. Parimala, and J.-P. Tignol, Invariants of quasitrivial tori and the Rost invariant, St. Petersburg Math. J. 14 (2003), 791-821. MR 1970336 (2004c:11045)
  • [MT] A. S. Merkurjev and J.-P. Tignol, The multipliers of similitudes and the Brauer group of homogeneous varieties, J. Reine Angew. Math. 461 (1995), 13-47. MR 1324207 (96c:20083)
  • [SS] T. A. Springer and R. Steinberg, Conjugacy classes, Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69), Springer, Berlin, 1970, pp. 167-266. MR 0268192 (42:3091)
  • [St] R. Steinberg, Lectures on Chevalley groups, Yale University, New Haven, Conn., 1968. MR 0466335 (57:6215)
  • [Ti66] J. Tits, Classification of algebraic semisimple groups, Algebraic Groups and Discontinuous Subgroups, Proc. Symp. Pure Math., vol. IX, AMS, 1966, pp. 32-62. MR 0224710 (37:309)
  • [Ti71] -, Représentations linéaires irréductibles d'un groupe réductif sur un corps quelconque, J. Reine Angew. Math. 247 (1971), 196-220. MR 0277536 (43:3269)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 12G05

Retrieve articles in all journals with MSC (2000): 12G05


Additional Information

S. Garibaldi
Affiliation: Department of Mathematics & Computer Science, Emory University, Atlanta, Georgia 30322
Email: skip@member.ams.org

A. Quéguiner-Mathieu
Affiliation: Laboratoire Analyse, Géométrie & Applications, UMR CNRS 7539, Institut Galilée, Université Paris 13, 93430 Villetaneuse, France
Email: queguin@math.univ-paris13.fr

DOI: https://doi.org/10.1090/S1061-0022-08-00993-X
Keywords: Algebraic groups of classical type, exceptional groups, Rost invariant
Received by editor(s): July 27, 2006
Published electronically: February 1, 2008
Article copyright: © Copyright 2008 American Mathematical Society

American Mathematical Society