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

V. I. Yanchevskii

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 23, Tome 400 (2012), pp. 222-245

Résumé

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.