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Jordan homomorphisms and harmonic mappings

Andrea Blunck, Hans Havlicek

Abstract

We show that each Jordan homomorphism $R\to R'$ of rings gives rise to a harmonic mapping of one connected component of the projective line over $R$ into the projective line over $R'$. If there is more than one connected component then this mapping can be extended in various ways to a harmonic mapping which is defined on the entire projective line over $R$.

Jordan homomorphisms and harmonic mappings

Abstract

We show that each Jordan homomorphism of rings gives rise to a harmonic mapping of one connected component of the projective line over into the projective line over . If there is more than one connected component then this mapping can be extended in various ways to a harmonic mapping which is defined on the entire projective line over .

Paper Structure

This paper contains 5 sections, 11 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. The elementary subgroup ${\mathrm{E}}_2(R)$
  3. Jordan homomorphisms
  4. The projective line over a ring
  5. On homomorphisms of chain geometries

Key Result

Lemma 2.5

If $(x_1,x_2,\ldots,x_{n})\in {\mathcal{S}}({\mathbb Z}\langle X\rangle)$ then

Theorems & Definitions (12)

  • Lemma 2.5
  • Proposition 2.8
  • Proposition 3.3
  • Proposition 3.4
  • Theorem 3.5
  • Theorem 3.6
  • Corollary 3.7
  • Lemma 4.2
  • Theorem 4.4
  • Proposition 4.8
  • ...and 2 more