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Reversibility of affine transformations

Part of: Automorphic functions Complex spaces with a group of automorphisms Structure and classification of infinite or finite groups Classical differential geometry Special matrices

Published online by Cambridge University Press:  08 November 2023

Krishnendu Gongopadhyay Open the ORCID record for Krishnendu Gongopadhyay [Opens in a new window] ,
Tejbir Lohan Open the ORCID record for Tejbir Lohan [Opens in a new window]  and
Chandan Maity Open the ORCID record for Chandan Maity [Opens in a new window]
Krishnendu Gongopadhyay
Affiliation:
Department of Mathematical Sciences, Indian Institute of Science Education and Research (IISER) Mohali, Knowledge City, Punjab, India (krishnendu@iisermohali.ac.in; tejbirlohan70@gmail.com; maity.chandan1@gmail.com)
Tejbir Lohan
Affiliation:
Department of Mathematical Sciences, Indian Institute of Science Education and Research (IISER) Mohali, Knowledge City, Punjab, India (krishnendu@iisermohali.ac.in; tejbirlohan70@gmail.com; maity.chandan1@gmail.com)
Chandan Maity
Affiliation:
Department of Mathematical Sciences, Indian Institute of Science Education and Research (IISER) Mohali, Knowledge City, Punjab, India (krishnendu@iisermohali.ac.in; tejbirlohan70@gmail.com; maity.chandan1@gmail.com)
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Abstract

An element g in a group G is called reversible if g is conjugate to g−1 in G. An element g in G is strongly reversible if g is conjugate to g−1 by an involution in G. The group of affine transformations of $\mathbb D^n$ may be identified with the semi-direct product $\mathrm{GL}(n, \mathbb D) \ltimes \mathbb D^n $, where $\mathbb D:=\mathbb R, \mathbb C$ or $ \mathbb H $. This paper classifies reversible and strongly reversible elements in the affine group $\mathrm{GL}(n, \mathbb D) \ltimes \mathbb D^n $.

Keywords

affine groupreversible elementsstrongly reversible elementsreal elementsstrongly real elementsadjoint reality

MSC classification

Primary: 53A15: Affine differential geometry
Secondary: 20E45: Conjugacy classes 32M17: Automorphism groups of $^n$ and affine manifolds 15B33: Matrices over special rings (quaternions, finite fields, etc.)

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 66 , Issue 4 , November 2023 , pp. 1217 - 1228
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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