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 Zap. Nauchn. Sem. POMI, 2012, Volume 400, Pages 222–245 (Mi znsl5621)

Reduced Whitehead groups and conjugacy problem for special unitary groups of anisotropic hermitian forms

V. I. Yanchevskii

Institute of Mathematics of the National Academy of Sciences of Belarus

Abstract: Let $K/k$ be a separable field extension of degree 2, $D$ be a finite-dimensional central division algebra over $K$ with $K/k$-involution $\tau$, $h$ be an hermitian anisotropic form on a right $D$-vector space with respect to $\tau$ and let $U(h)$ be the unitary group of $h$. Then the reduced Whitehead group of its special linear subgroup is defined as follows: $\mathrm{SUK_1^{an}}(h)=\mathrm{SU}(h)/[U(h),U(h)]$, where $[U(h),U(h)]$ is the commutator subgroup of $U(h)$. The first main result establishes a link between the above group and its analog $\mathrm{SUK}_1(h)$ for the case of isotropic $h$ (with respect to the same $\tau$).
Theorem. There exists a surjective homomorphism from $\mathrm{SUK_1^{an}}(h)$ to $\mathrm{SUK}_1(h)$.
Furthermore, we give also a solution of conjugacy problem for special unitary subgroups of anisotropic hermitian forms over quaternion division algebras as subgroups of their multiplicative groups.

Key words and phrases: anisotropic and isotropic algebraic groups, reduced Whitehead groups, hermitian forms, special unitary groups.

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English version:
Journal of Mathematical Sciences (New York), 2013, 192:2, 250–262

Bibliographic databases:

Document Type: Article
UDC: 514.142

Citation: V. I. Yanchevskii, “Reduced Whitehead groups and conjugacy problem for special unitary groups of anisotropic hermitian forms”, Problems in the theory of representations of algebras and groups. Part 23, Zap. Nauchn. Sem. POMI, 400, POMI, St. Petersburg, 2012, 222–245; J. Math. Sci. (N. Y.), 192:2 (2013), 250–262

Citation in format AMSBIB
\Bibitem{Yan12} \by V.~I.~Yanchevskii \paper Reduced Whitehead groups and conjugacy problem for special unitary groups of anisotropic hermitian forms \inbook Problems in the theory of representations of algebras and groups. Part~23 \serial Zap. Nauchn. Sem. POMI \yr 2012 \vol 400 \pages 222--245 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl5621} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3029575} \transl \jour J. Math. Sci. (N. Y.) \yr 2013 \vol 192 \issue 2 \pages 250--262 \crossref{https://doi.org/10.1007/s10958-013-1391-9} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84884989607}