Unitary SK_1 of semiramified graded and valued division algebras

We prove formulas for the unitary SK_1 of a semiramified graded division algebra (or valued division algebra over a Henselian field) with a unitary involution. These formulas generalize earlier formulas of Yanchevskii, (and Platonov and Ershov for the nonunitary SK_1).

💡 Research Summary

The paper investigates the reduced Whitehead group SK₁ of division algebras equipped with a unitary involution in the special setting of semiramified graded division algebras, and by extension, valued division algebras over Henselian fields. A semiramified algebra is characterized by a value group extension of index two and a residue division algebra that is a cyclic extension of its center, also of degree two. In this situation the algebra contains a maximal cyclic subfield L such that L/K is a non‑Galois quadratic extension. The authors consider a unitary involution τ on the algebra D, which restricts to the non‑trivial automorphism σ of the center K. τ induces an involution on the associated graded algebra gr(D) and on the residue algebra (\overline{D}); the latter is denoted (\overline{τ}).

The main achievement is an explicit description of the unitary SK₁ group, denoted SK₁(D,τ), in two equivalent forms: one expressed through the residue algebra and the other through the value group. For the graded algebra E = gr(D) with the induced involution τ̂ the authors prove the isomorphisms

\