Unitary groups over semilocal domains. I

AbstractLet O be a semilocal domain that contains 2 as a unit. Let ∗ be an involution on O with the property that there exists a unit θ in O such that θ∗ = −θ. In this paper, we study the structure of a lattice L with canonical splitting (see Section 2) over O and the generation theorems for the unitary group U(L) of the lattice and its commutator subgroup Ω(L). First, we prove the Witt theorem, i.e., every isometry from an orthogonal component of L into L can be lifted up to L, listed as Theorem 3.7. Then, we determine all the invariant sublattices of L (see Theorem 3.9). Let E(L) be the subgroup of U(L) generated by pure unitary transvections and pure quasi-transvections. We show that Ω(L) = E(L) and U(L) is generated by E(L) and the semideletions PH(α) (Listed as Theorems 4.4 and 4.5)