Commutator theory for racks and quandles
| Autor: | Marco Bonatto, David Stanovský |
|---|---|
| Rok vydání: | 2019 |
| Předmět: |
08A30
Pure mathematics General Mathematics quandles and racks Alexander polynomial left distributive quasigroups Group Theory (math.GR) Galois connection 01 natural sciences Mathematics::Quantum Algebra 0103 physical sciences FOS: Mathematics Order (group theory) Universal algebra 0101 mathematics Mathematics Commutator Racks and quandles Group (mathematics) commutator theory 010102 general mathematics nilpotence 20N05 Congruence relation 20N02 Mathematics::Geometric Topology 57M27 010307 mathematical physics Mathematics - Group Theory solvability |
| Zdroj: | J. Math. Soc. Japan 73, no. 1 (2021), 41-75 |
| DOI: | 10.48550/arxiv.1902.08980 |
| Popis: | We adapt the commutator theory of universal algebra to the particular setting of racks and quandles, exploiting a Galois connection between congruences and certain normal subgroups of the displacement group. Congruence properties, such as abelianness and centrality, are reflected by the corresponding relative displacement groups, and the global properties, solvability and nilpotence, are reflected by the properties of the whole displacement group. To show the new tool in action, we present three applications: non-existence theorems for quandles (no connected involutory quandles of order $2^k$, no latin quandles of order $\equiv2\pmod4$), a non-colorability theorem (knots with trivial Alexander polynomial are not colorable by solvable quandles; in particular, by finite latin quandles), and a strengthening of Glauberman's results on Bruck loops of odd order. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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