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arXiv · 2306.12052

Some invariants of $U(1,1;\mathbb{H})$ and diagonalization

Cailing Yao·Bingzhe Hou·Xiaoqi Feng
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Abstract

Denote by $\mathbb{H}$ the set of all quaternions. We are interested in the group $U(1,1;\mathbb{H})$, which is a subgroup of $2\times 2$ quaternionic matrix group and is sometimes called $Sp(1,1)$. As well known, $U(1,1;\mathbb{H})$ corresponds to the quaternionic M\"{o}bius transformations on the unit ball in $\mathbb{H}$. In this article, some similar invariants on $U(1,1;\mathbb{H})$ are discussed. Our main result shows that each matrix $T\in U(1,1;\mathbb{H})$, which corresponds to an elliptic quaternionic M\"{o}bius transformation $g_T(z)$, could be $U(1,1;\mathbb{H})$-similar to a diagonal matrix.

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Cailing Yao, Bingzhe Hou, Xiaoqi Feng. 2023-06-21. Some invariants of $U(1,1;\mathbb{H})$ and diagonalization. https://arxiv.org/abs/2306.12052

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