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

Published electronically:
July 7, 2005

MathSciNet review:
2176001

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Abstract: If is a division algebra with its center a number field and with an involution of the second kind, it is unknown if the group , is trivial. We show that, by contrast, if is a function field in one variable over a number field, and if is an algebra with center and with an involution of the second kind, the group can be infinite in general. We give an infinite class of examples.

**1.**Armand Borel,*Linear algebraic groups*, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR**1102012****2.**Patrick J. Morandi and B. A. Sethuraman,*Noncrossed product division algebras with a Baer ordering*, Proc. Amer. Math. Soc.**123**(1995), no. 7, 1995–2003. MR**1246532**, 10.1090/S0002-9939-1995-1246532-6**3.**Richard S. Pierce,*Associative algebras*, Graduate Texts in Mathematics, vol. 88, Springer-Verlag, New York-Berlin, 1982. Studies in the History of Modern Science, 9. MR**674652****4.**Vladimir Platonov and Andrei Rapinchuk,*Algebraic groups and number theory*, Pure and Applied Mathematics, vol. 139, Academic Press, Inc., Boston, MA, 1994. Translated from the 1991 Russian original by Rachel Rowen. MR**1278263****5.**O. F. G. Schilling,*The Theory of Valuations*, Mathematical Surveys, No. 4, American Mathematical Society, New York, N. Y., 1950. MR**0043776****6.**Adrian R. Wadsworth,*Extending valuations to finite-dimensional division algebras*, Proc. Amer. Math. Soc.**98**(1986), no. 1, 20–22. MR**848866**, 10.1090/S0002-9939-1986-0848866-8**7.**Shianghaw Wang,*On the commutator group of a simple algebra*, Amer. J. Math.**72**(1950), 323–334. MR**0034380**

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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