Year: 2024
Type: preprint
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 $\text{GL}_3(\mathbb{F}_2) \times \{\pm 1\}^{n-3}$.
Source: arXiv (Cornell University)
Authors: Chapman, Adam, Levin, Ilan, Vishne, Uzi, Zaninelli, Marco
Institutions: Bar-Ilan University, University of Pennsylvania
Language: English
Topic: Mathematics and Applications
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4406052129