Reduced anisotropic unitary Whitehead groups of henselian division algebras with special residue fields of their centers
https://doi.org/10.29235/1561-2430-2021-57-1-7-13
Abstract
The reduced anisotropic unitary Whitehead groups of henselian division algebras with unitary involutions are computed in the cases where the centers of residue algebras are of special types.
About the Author
V. I. Yanchevskii
Institute of Mathematics of the National Academy of Sciences of Belarus
Russian Federation
Russian Federation
Vyacheslav I. Yanchevskii – Academician of the National Academy of Sciences of Belarus, Dr. Sc. (Physics and Mathematics), Professor, Head of the Department of Algebra
11, Surganov Str., 220072, Minsk
References
1. Platonov V. P., Yanchevskii V. I. On the Kneser - Tits conjecture for unitary groups. Doklady Akademiinauk SSSR [Proceedings of the Academy of Sciences of the USSR], 1975, vol. 225, no. 1, pp. 48–51 (in Russian).
2. Platonov V. P. The Tannaka – Artin problem and reduced K-theory. Mathematics of the USSR – Izvestiya, 1976, vol. 10, no. 2, pp. 211–243. https://doi.org/10.1070/IM1976v010n02ABEH001686
3. Yanchevskii V. I. Reduced unitary K-theory and division rings over discretely valued hensel fields. Mathematics of the USSR – Izvestiya, 1979, vol. 13, no. 1, pp. 175–213. https://doi.org/10.1070/IM1979v013n01ABEH002018
4. Yanchevskii V. I. A converse problem in reduced unitary K-theory. Mathematical notes of the Academy of Sciences of the USSR, 1979, vol. 26, pp.728–731. https://doi.org/10.1007/BF01138683
5. Yanchevskii V. I. Reduced unitary K-theory.Aplications to algebraic groups. Mathematics of the USSR-Sbornik, 1981, vol. 38, no. 4, pp. 533–548. https://doi.org/10.1070/SM1981v038n04ABEH001460
6. Draxl P. SK1 von Algebren über vollständig diskret bewerteten Körpern und Galoiskohomologie abelscher Körpererweiterungen. Journal für die reine und angewandteMathematik (Crelles Journal), 1977, vol. 293–294, pp. 116–142 (in Gwerman). https://doi.org/10.1515/crll.1977.293-294.116
7. Draxl P. Ostrowski’s theorem for Henselian valued skew fields. Journal für die reine und angewandte Mathematik (Crelles Journal), 1984, vol. 1984, no. 354, pp. 213–218 (in German). https://doi.org/10.1515/crll.1984.354.213
8. Gille P. Le probléme de Kneser-Tits. Séminaire Boubaki. Astérisque, 2009. x+409 p.
9. Hazrat R., Wadsworth A. R. SK1 of graded division algebras. Israel Journal of Mathematics, 2011, vol. 183, no. 1, pp. 117–163. https://doi.org/10.1007/s11856-011-0045-1
10. Hazrat R., Wadsworth A. R. Unitary SK1 of graded and valued division algebras. Proceedings of the London Mathematical Society, 2011, vol. 103, no. 3, pp. 508–534. https://doi.org/10.1112/plms/pdr010
11. Suslin A. A. SK1 of division algebras and Galois cohomology revisited. Proceedings of the St. Petersburg Mathematical Society, 2006, vol. 12, pp. 125–147. https://doi.org/10.1090/trans2/219/04
12. Wadsworth A. R. Unitary SK1 of semiramified graded and valued division algebras. Manuscripta Mathematica, 2012, vol. 139, no. 3–4, pp. 343–389. https://doi.org/10.1007/s00229-011-0519-9
13. Sethuraman B. A., Sury B. A note on the special unitary group of a division algebra. Proceedings of the American Mathematical Society, 2005, vol. 134, no. 02, pp. 351–354. https://doi.org/10.1090/s0002-9939-05-07985-2
14. Sury B. On SU(1,D)/[U(1,D),U(1,D)] for a quaternion division algebra D. Archiv der Mathematik, 2008, vol. 90, no. 6, pp. 493–500. https://doi.org/10.1007/s00013-008-2438-x
15. Yanchevskii V. I. Reduced Whitehead groups and the conjugacy problem for special unitary groups of anisotropic Hermitian forms. Journal of Mathematical Sciences, 2013, vol. 192, no. 2, pp. 250–262. https://doi.org/10.1007/s10958-0131391-9
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