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

Susanne Pumpluen

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 $K((t;σ))$ 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 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.

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

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 over a field and the ring of twisted Laurent series over a division algebra 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.

Paper Structure

This paper contains 18 sections, 38 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Nonassociative rings and algebras
  4. Anti-automorphisms
  5. Cyclic algebras
  6. Monomial anti-automorphisms of degree one on associative cyclic algebras
  7. Monomial anti-automorphisms of associative cyclic algebras of degree greater than one
  8. Ring isomorphisms between cyclic algebras
  9. Anti-automorphisms of cyclic algebras
  10. Examples of anti-automorphisms of the second kind and of infinite order on algebras without involutions
  11. Anti-automorphisms of infinite order on cyclic algebras
  12. Anti-automorphisms on the ring twisted Laurent series
  13. Monomial anti-automorphisms of degree one on generalized cyclic algebras
  14. Nonassociative generalized cyclic algebras
  15. Monomial anti-automorphisms of degree one
  16. ...and 3 more sections

Key Result

Lemma 2.2

(i) The elements of the group ${\rm Aut}(K)_\sigma$ are automorphisms of $K$ that preserve the subfield $F$ (that is, that restrict to automorphisms of $F$). (ii) For every automorphism $\tau$ of $K$ that restricts to an automorphism of $F$ there exists a positive integer $k$, $\gcd(k,n)=1$, such th

Theorems & Definitions (63)

  • Remark 2.1
  • Lemma 2.2
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • proof
  • Lemma 3.4
  • ...and 53 more