Loops with involution and the Cayley-Dickson doubling process
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912174545305600 |
|---|---|
| author | Chapman, Adam Levin, Ilan Vishne, Uzi Zaninelli, Marco |
| author_facet | Chapman, Adam Levin, Ilan Vishne, Uzi Zaninelli, Marco |
| contents | 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 $\text{GL}_3(\mathbb{F}_2) \times \{\pm 1\}^{n-3}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_00123 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Loops with involution and the Cayley-Dickson doubling process Chapman, Adam Levin, Ilan Vishne, Uzi Zaninelli, Marco Combinatorics Rings and Algebras Primary 20N05, Secondary 17A36, 20F28 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 $\text{GL}_3(\mathbb{F}_2) \times \{\pm 1\}^{n-3}$. |
| title | Loops with involution and the Cayley-Dickson doubling process |
| topic | Combinatorics Rings and Algebras Primary 20N05, Secondary 17A36, 20F28 |
| url | https://arxiv.org/abs/2501.00123 |