Reversibility in the groups of PL^+(S^1) and PL(S^1)

Reversibility and Banach Algebras

An element $g$ of a group is called {\em reversible} if it is conjugate in the group to its inverse. In this paper we review some results about the structure of groups involving the reversible elements and we pose some questions about groups associated to a Banach algebra.

Reversibility and Real Adjoint Orbits of Linear Maps

We extend classical results on the classification of reversible elements of the group $\mathrm{GL}(n, \mathbb{C})$ (and $\mathrm{GL}(n, \mathbb{R})$) to $\mathrm{GL}(n, \mathbb{H})$ using an infinitesimal version of the classical reversibility, namely adjoint reality in the Lie algebra set-up. We also provide a new proof of such a classification for the general linear groups over $\mathbb{R}$ and $\mathbb{C}$. Further, we classify the real adjoint orbits in the Lie algebra $\mathfrak{gl}(n, \mathbb{D})$ for $ \mathbb{D}=\mathbb{R}, \mathbb{C}$ or $\mathbb{H} $.

仿射变换的可逆性

Reversibility of Affine Transformations

本研究围绕仿射变换的可逆性展开，明确群中可逆元与强可逆元的定义，针对仿射群$\mathrm{GL}(n, \mathbb{D}) \ltimes \mathbb{D}^n $中的可逆元与强可逆元进行分类研究，为仿射变换相关群论研究提供分类依据。

On groups generated by bi-reversible automata: the two-state case over a changing alphabet

The notion of an automaton over a changing alphabet $X=(X_i)_{i\geq 1}$ is used to define and study automorphism groups of the tree $X^*$ of finite words over $X$. The concept of bi-reversibility for Mealy-type automata is extended to automata over a changing alphabet. It is proved that a non-abelian free group can be generated by a two-state bi-reversible automaton over a changing alphabet $X=(X_i)_{i\geq 1}$ if and only if $X$ is unbounded. The characterization of groups generated by a two-state bi-reversible automaton over the sequence of binary alphabets is established.

On the reversibility and the closed image property of linear cellular automata

When $G$ is an arbitrary group and $V$ is a finite-dimensional vector space, it is known that every bijective linear cellular automaton $τ\colon V^G \to V^G$ is reversible and that the image of every linear cellular automaton $τ\colon V^G \to V^G$ is closed in $V^G$ for the prodiscrete topology. In this paper, we present a new proof of these two results which is based on the Mittag-Leffler lemma for projective sequences of sets. We also show that if $G$ is a non-periodic group and $V$ is an infinite-dimensional vector space, then there exist a linear cellular automaton $τ_1 \colon V^G \to V^G$…