Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A note on the special unitary group of a division algebra


Authors: B. A. Sethuraman and B. Sury
Journal: Proc. Amer. Math. Soc. 134 (2006), 351-354
MSC (2000): Primary 16K20, 12E15
DOI: https://doi.org/10.1090/S0002-9939-05-07985-2
Published electronically: July 7, 2005
MathSciNet review: 2176001
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: If $D$ is a division algebra with its center a number field $K$ and with an involution of the second kind, it is unknown if the group $SU(1,D)/[U(1,d)$, $U(1,D)]$ is trivial. We show that, by contrast, if $K$is a function field in one variable over a number field, and if $D$ is an algebra with center $K$ and with an involution of the second kind, the group $SU(1,D)/[U(1,d),U(1,D)]$ can be infinite in general. We give an infinite class of examples.


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

  • 1. A. Borel, Linear algebraic groups, Springer-Verlag, 2nd edition, 1991. MR 1102012 (92d:20001)
  • 2. Patrick J. Morandi and B.A. Sethuraman, Noncrossed product division algebras with a Baer ordering, Proc. Amer. Math. Soc., 123 1995, 1995-2003. MR 1246532 (95i:16019)
  • 3. Richard S. Pierce, Associative Alegbras, Graduate Texts in Mathematics, 88, Springer-Verlag, 1982.MR 0674652 (84c:16001)
  • 4. V.P. Platonov and A.S. Rapinchuk, Algebraic groups and number theory, Academic Press, 1994. MR 1278263 (95b:11039)
  • 5. O.F.G. Schilling, The Theory of Valuations, Math Surveys, No. 4., Amer. Math. Soc., Providence, R.I., 1950. MR 0043776 (13:315b)
  • 6. Adrian R. Wadsworth, Extending valuations to finite dimensional division algebras, Proc. Amer. Math. Soc., 98 1986, 20-22. MR 0848866 (87i:16025)
  • 7. S.Wang, On the commutator group of a simple algebra, Amer. J. Math., 72 1950, 323-334.MR 0034380 (11:577d)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 16K20, 12E15

Retrieve articles in all journals with MSC (2000): 16K20, 12E15


Additional Information

B. A. Sethuraman
Affiliation: Department of Mathematics, California State University Northridge, Northridge, California 91330
Email: al.sethuraman@csun.edu

B. Sury
Affiliation: Stat-Math Unit, Indian Statistical Institute, 8th Mile Mysore Road, Bangalore 560 059, India
Email: sury@isibang.ac.in

DOI: https://doi.org/10.1090/S0002-9939-05-07985-2
Received by editor(s): April 19, 2004
Received by editor(s) in revised form: September 21, 2004
Published electronically: July 7, 2005
Additional Notes: This work was done when the first-named author visited the Indian Statistical Institute, Bangalore. He thanks the Institute for the wonderful hospitality it showed during his stay there.
Communicated by: Martin Lorenz
Article copyright: © Copyright 2005 American Mathematical Society

American Mathematical Society