Article
| Title: | Algebraic characterization of reversibility in the quaternionic Möbius group (English) |
| Author: | Gongopadhyay, Krishnendu |
| Author: | Lohan, Tejbir |
| Author: | Mukherjee, Abhishek |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 76 |
| Issue: | 3 |
| Year: | 2026 |
| Pages: | 1029-1043 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | An element of a group is called reversible if it is conjugate to its inverse. While reversibility in the quaternionic Möbius group ${\rm PSL}(2,\mathbb {H})$ has traditionally been studied using geometric and dynamical methods, we develop a purely algebraic approach. We obtain an explicit, computable criterion for the reversibility of a quaternionic Möbius transformation, expressed solely in terms of the entries of a matrix representative. More precisely, we prove that \[ [A]\in {\rm PSL}(2,\mathbb {H}) \ \text {is reversible} \Longleftrightarrow \beta _A^{2}=\delta _A^{2}, \] where $\beta _A$ and $\delta _A$ are real conjugacy invariants associated with a lift $A\in {\rm SL}(2,\mathbb {H})$. Furthermore, we give a complete characterization of reversing symmetries of reversible elements in ${\rm SL}(2,\mathbb {H})$ and ${\rm PSL}(2,\mathbb {H})$. (English) |
| Keyword: | reversibility |
| Keyword: | reversible element |
| Keyword: | reversing symmetry |
| Keyword: | quaternionic Möbius transformation |
| Keyword: | algebraic characterization |
| Keyword: | conjugacy invariant |
| MSC: | 15A21 |
| MSC: | 15B33 |
| MSC: | 20E45 |
| MSC: | 51B10 |
| DOI: | 10.21136/CMJ.2026.0020-26 |
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| Date available: | 2026-09-11T13:17:39Z |
| Last updated: | 2026-09-14 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153741 |
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