Topologically conjugate classifications of the translation actions on low-dimensional compact connected Lie groups

  • Xiaotian Pan
  • Bingzhe HouEmail author


In this article, we focus on the left translation actions on noncommutative compact connected Lie groups with topological dimension 3 or 4, consisting of SU(2), U(2), SO(3), SO(3) × S1 and Spin(3). We define the rotation vectors (numbers) of the left actions induced by the elements in the maximal tori of these groups, and utilize rotation vectors (numbers) to give the topologically conjugate classification of the left actions. Algebraic conjugacy and smooth conjugacy are also considered. As a by-product, we show that for any homeomorphism f : L(p, −1) × S1L(p, −1) × S1, the induced isomorphism (τfi)* maps each element in the fundamental group of L(p, −1) to itself or its inverse, where i : L(p, −1) → L(p, −1) × S1 is the natural inclusion and τ : L(p, −1) × S1L(p, −1) is the projection.


topological conjugacy rotation vectors translation actions C*-dynamical system classification 


37C15 55S37 22C05 


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© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of MathematicsJilin UniversityChangchunChina

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