Reduced Whitehead groups and the conjugacy problem for special unitary groups of anisotropic Hermitian forms
Let \(K/k\) be a separable field extension of degree 2, \(D\) be a finite-dimensional central division algebra over \(K\) with a \(K/k\)-involution \(\tau,\) \(h\) be an Hermitian anisotropic form on a right \(D\)-vector space with respect to \(\tau\), and let \(\mathrm U(h)\) be the unitary group of \(h\). Then the reduced Whitehead group of its special linear subgroup is defined as follows: \(\mathrm{SU}K_1^{\text{an}} (h)=\mathrm{SU}(h)/[\mathrm U(h),\mathrm U(h)]\), where \([\mathrm U(h),\mathrm U(h)] \) is the commutator subgroup of \(\mathrm U(h)\). The first main result establishes a link between the above group and its analog \(\mathrm{SU}K_ 1 (h) \)for the case of isotropic \(h\) (with respect to the same \(\tau\)). Theorem. There exists a surjective homomorphism \(\mathrm{SU}K_1^{\text{an}} (h)) \to\mathrm{SU}K _1 (h)\). Furthermore, a solution of the conjugacy problem for special unitary subgroups of anisotropic Hermitian forms over quaternion division algebras as subgroups of their multiplicative groups is given.
scientific article; zbMATH DE number 6347693
scientific article