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Loops with involution and the Cayley-Dickson doubling process

Adam Chapman, Ilan Levin, Uzi Vishne, Marco Zaninelli

Abstract

We develop a theory of loops with involution. On this basis we define a Cayley-Dickson doubling on loops, and use it to investigate the lattice of varieties of loops with involution, focusing on properties that remain valid in the Cayley-Dickson double. Specializing to central-by-abelian loops with elementary abelian $2$-group quotients, we find conditions under which one can characterize the automorphism groups of iterated Cayley-Dickson doubles. A key result is a corrected proof that for $n>3$, the automorphism group of the Cayley-Dickson loop $Q_n$ is $\text{GL}_3(\mathbb{F}_2) \times \{\pm 1\}^{n-3}$.

Loops with involution and the Cayley-Dickson doubling process

Abstract

We develop a theory of loops with involution. On this basis we define a Cayley-Dickson doubling on loops, and use it to investigate the lattice of varieties of loops with involution, focusing on properties that remain valid in the Cayley-Dickson double. Specializing to central-by-abelian loops with elementary abelian -group quotients, we find conditions under which one can characterize the automorphism groups of iterated Cayley-Dickson doubles. A key result is a corrected proof that for , the automorphism group of the Cayley-Dickson loop is .
Paper Structure (32 sections, 67 theorems, 84 equations, 1 figure)

This paper contains 32 sections, 67 theorems, 84 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Loops
  4. Commutators and associators
  5. Loops as central extensions
  6. Involutions on loops
  7. Central involutions
  8. Super-central involutions
  9. Normal involutions
  10. Central-by-abelian loops with involution
  11. Cayley-Dickson doubling
  12. The doubling process
  13. Extending the involution
  14. Some automorphisms of the double
  15. The center of the double
  16. ...and 17 more sections

Key Result

Proposition 2.1

In any loop $L$, the intersection of ${\operatorname{K}}(L)$ with any two of the one-sided nuclei is the center.

Figures (1)

  • Figure 1: Products of $a,b,c$ in a central-by-abelian loop.

Theorems & Definitions (156)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Corollary 2.4
  • Example 3.1
  • Remark 3.2
  • Example 3.3: A central-by-abelian loop with involution which does not have a central involution
  • Proposition 3.4
  • ...and 146 more