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

TL;DR

This paper develops explicit criteria and constructions for anti-automorphisms in cyclic and generalized cyclic algebras, unifying and extending classical involution results.

Contribution

It introduces norm-based criteria and polynomial map constructions for anti-automorphisms, covering cases of infinite order and nonassociative algebras.

Findings

Norm criteria for anti-automorphisms of second kind

Explicit constructions for cyclic and generalized cyclic algebras

Unified approach recovering classical involution conditions

Abstract

While involutions on central simple algebras have been studied extensively and are well understood, much less is known about general anti-automorphisms. We present norm criteria for the existence of anti-automorphisms as well as explicit constructions of anti-automorphisms, 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 and of infinite order on the ring of twisted Laurent series $K((t;\sigma))$ over a field $K$ and the ring of twisted Laurent series $D((t;\sigma))$ over a division algebra $D$ that is finite-dimensional over its center. Our construction produces all possible anti-automorphisms of proper nonassociative cyclic or generalized cyclic algebras, i.e. for special classes of Petit algebras which can be viewed as canonical generalizations of associative cyclic and generalized cyclic central simple algebras.