Explicit constructions of anti-automorphisms of cyclic and generalized cyclic algebras

Abstract

We present norm criteria for the existence of anti-automorphisms, as well as explicit constructions of anti-automorphisms, both on cyclic and generalized cyclic algebras. Our approach describes anti-automorphisms as polynomial maps and unifies existing approaches. It recovers classical criteria for the existence of involutions as special cases. We obtain norm conditions for the existence of anti-automorphisms of the second kind on the ring of twisted Laurent series $K((t;σ))$ over a field $K$ and the ring of twisted Laurent series $D((t;σ))$ over a division algebra $D$ that is finite-dimensional over its center. Our constructions rely on the isomorphisms between an algebra and its opposite algebra. Along the way, we hence describe monomial isomorphisms between cyclic or generalized cyclic algebras, which ties in with studying the isomorphism problem for central simple algebras. Proper nonassociative cyclic and generalized cyclic algebras, which are canonical generalizations of associative cyclic and generalized cyclic central simple algebras, are included here.