Reversibility and algebraic characterization of the quaternionic Möbius transformations

Abstract

Let $\mathrm {SL}(2, \mathbb{H})$ be the group of $2 \times 2$ quaternionic matrices with quaternionic determinant $1$. The group $\mathrm {SL}(2, \mathbb{H})$ acts on the four-dimensional sphere $\widehat {\mathbb{H}}=\mathbb{H} \cup \{\infty\}$ by the (orientation-preserving) quaternionic Möbius transformations: $$A=\begin{pmatrix} a & b \\ c & d \end{pmatrix}: z \mapsto (az+b)(cz+d)^{-1}.$$ Using this action, the Möbius group may be identified with $\mathrm {PSL}(2, \mathbb{H})$. A quaternionic Möbius transformation $[A]$ is reversible if it is conjugate to its inverse in $\mathrm {PSL}(2, \mathbb{H})$. Given an element $A$ in $\mathrm {SL}(2, \mathbb{H})$, we provide a characterization in terms of the conjugacy invariants of $A$ that recognizes whether or not $[A]$ is reversible. Equivalently, this characterization detects whether $[A]$ is a product of two involutions or not. Further, we revisit the algebraic characterization of the quaternionic Möbius transformations.

Theorems & Definitions (2)

• Theorem 1.1
• Definition 2.1: cf. Let $A \in \mathrm{M}(2,\mathbb {H})$ and write $A = A_1 + A_2 \mathbf {j}$, where $A_1, A_2 \in \mathrm{M}(2,\mathbb {C})$. Consider the embedding $\Phi$ from $\mathrm{M}(2,\mathbb {H})$ to $\mathrm{M}(4,\mathbb {C})$, which is defined as follows: $\Phi(A):= A_1A_2- \overline{A_2}\overline{A_1},$ where $\overline{A_i}$ denotes the complex conjugate of $A_i$ for $i=1,2$. Then the determinant (respectively, characteristic polynomial) of $A \in \mathrm{M}(2,\mathbb {H})$, denoted as ${\rm det_{\mathbb {H}}(A)}$ (respectively, $\chi_{\mathbb {H}}(A)$), is given by the determinant (respectively, characteristic polynomial) of $\Phi(A)$. In other words, ${\rm det_{\mathbb {H}}(A)}:= {\rm det(\Phi(A))}$ and $\chi_{\mathbb {H}}(A):= \chi_{\Phi(A)}$. In view of the Skolem-Noether theorem, the above definition of the quaternionic determinant and characteristic polynomial is independent of the choice of embedding $\Phi$. The groups under consideration in this paper are $\mathrm{GL}(2,\mathbb {H}) := \{ A \in \mathrm{M}(2,\mathbb {H}) \mid {\rm det_{\mathbb {H}}}(g) \neq 0 \}~~ {\rm and }$ $\mathrm{SL}(2,\mathbb {H}) := \{ A \in \mathrm{GL}(2,\mathbb {H}) \mid {\rm det_{\mathbb {H}}}(g) = 1 \}.$ Note that if the characteristic polynomial of $A \in {\rm SL}(2, \mathbb {H})$ is given by $\chi_{\mathbb {H}}(A) := \chi_{\Phi(A)} = x^4-c_3 x^3 + c_2 x^2 -c_1 x+1.$ Then $c_1$, $c_2$ and $c_3$ are real numbers, and they can be expressed as functions of the coefficients of matrix $A$; see Fo. In the following lemma, we recall the well-known conjugacy classification in $\mathrm{SL}(2,\mathbb {H})$, which follows from the Jordan decomposition of matrices over quaternions; see rodman. Every element of $\mathrm{SL}(2,\mathbb {H})$ is conjugate to one of the following matrices: $r e^{\mathbf {i} \theta}00r^{-1} e^{\mathbf {i} \phi}$, where $r \in \mathbb{R}^{+}= \{x\in \mathbb {R} \mid x >0\}$ and $\theta, \phi \in [0,\pi]$, $e^{\mathbf {i} \theta}10e^{\mathbf {i} \theta}$, where $\theta \in [0,\pi]$. In view of the above conjugacy classification in $\mathrm{SL}(2,\mathbb {H})$, we get the following well-defined classification for $\mathrm{PSL}(2,\mathbb {H})$ into three mutually exclusive classes. Let $[A] \in \mathrm{PSL}(2,\mathbb {H})$, where $A$ is a lift of $[A]$ in $\mathrm{SL}(2,\mathbb {H})$. Then $[A]$ is hyperbolic (loxodromic) if $A$ has at least one eigenvalue with a non-unit modulus.$[A]$ is elliptic if $A$ is diagonalizable (semisimple) and each eigenvalue of $A$ has a unit modulus.$[A]$ is parabolic if $A$ is non-diagonalizable and each eigenvalue of $A$ has a unit modulus. The centralizer of an element $A \in \mathrm{SL(2,\mathbb {H})}$ is defined as follows: $\mathcal{Z}_{\mathrm{SL(2,\mathbb {H})}}(A): = \{ g \in \mathrm{SL(2,\mathbb {H})} \mid gAg^{-1} = A\}.$ The reverser set of an element $A$ of $\mathrm{SL(2,\mathbb {H})}$ is defined as follows: $\mathcal{R}_{\mathrm{SL(2,\mathbb {H})}}(A) := \{ g \in \mathrm{SL(2,\mathbb {H})} \mid gAg^{-1} = A^{-1} \}.$ Note that the set $\mathcal{R}_{\mathrm{SL(2,\mathbb {H})}}(A)$ of reversers for a reversible element $A$ is a right coset of the centralizer $\mathcal{Z}_{\mathrm{SL(2,\mathbb {H})}}(A)$. Moreover, for each $A \in \mathrm{SL(2,\mathbb {H})}$, the reversing symmetry group or extended centralizer $\mathcal{E}_{\mathrm{SL(2,\mathbb {H})}}(A) := \mathcal{Z}_{\mathrm{SL(2,\mathbb {H})}}(A) \cup \mathcal{R}_{\mathrm{SL(2,\mathbb {H})}}(A)$ is a subgroup of $\mathrm{SL(2,\mathbb {H})}$ in which $\mathcal{Z}_{\mathrm{SL(2,\mathbb {H})}}(A)$ has an index of at most $2$; see BR, OS for more details. Centralizers of elements of $\mathrm{GL}(2,\mathbb{H})$ are well-studied in the literature; see KG. The following lemma gives the centralizer of each conjugacy class representative in $\mathrm{SL}(2,\mathbb{H})$.