The Topologically Conjugate Classifications Of The Translation Actions On The Low-dimensional Compact Connected Lie Groups

The main content of this paper is related to two important branches of topologically dynamical systems-rotation theory and topological conjugacy. We apply some results about the maximal torus of Lie groups in this paper and combine the definition of the rotation vectors of the rotations of the n-dimension torus Tn to define the rotation vectors of the translation actions induced by the elements in the maximal torus on some compact connected Lie groups, and we may utilize the rotation vec-tors defined in the present paper to describe the topologically conjugate classifications of these translation actions, and then describe the topo-logically conjugate classifications of all the translation actions on these compact connected Lie groups completely.In this paper, we will focus on those non-commutative compact con-nected Lie groups whose topological dimensions are 3 or 4, consisting of SU(2), U(2), SO(3), SO(3) x S1 Spinc(3), and the left actions on these Lie groups (the relevant results of the right actions on these Lie groups are the same as the left actions). We define the rotation vectors of the left actions induced by the elements in the maximal torus on these five non-commutative compact connected Lie groups, and utilize the rotation vectors defined to describe the topologically conjugate classifications of these left actions, and then describe the topologically conjugate classifications of all the left actions on these compact connected Lie groups completely.First of all, let G be a compact connected Lie groups and TG be a maximal torus of G. Then for any g ∈G, there always exists t ∈TG such that g and t are conjugate, and then the left actions Lg and Lt are topologically conjugate.Next, we will show the main conclusions of this paper.(1) For the Lie group SU(2), choose one maximal torus of SU(2). Obviously, and choose one isomorphism Φ:TSU(2)�'S1 defined by Then, for any Lg∈MTSU(2), the left action Lg acting on the maximal torus TSU(2) is equivalent to a rotation of S1, so we can define the rotation number of the left action Lg as the same as the definition of the rotation f of S1. Therefore, for any Lg ∈ MTSU(2), we define And then, for left actions Lg, Lg’∈MTSU(2), Lg and Lg’ are topologically conjugate if and only ifAccording to the equivalence relation above, we describe topologi-cally conjugate classification of all the left actions on SU(2). Further more, we also prove that the topologically conjugate classification of the left actions on the Lie groups SU(2) are equivalent to the algebraically conjugate classification and the smooth conjugate classification.(2) For the Lie group U(2), choose one maximal torus of U(2). Obviously, and choose one isomorphism Φ:TU(2)�'T2 defined by Then, for any Lg ∈MTU(2), the left action Lg acting on the maximal torus Tu(2) is equivalent to a rotation of T2, so we can define the rotation vector of the left action Lg as the same as the definition of the rotation f of T2. Therefor, for any Lg∈MTU(2), we define And then, for the left actions Lg, Lg’∈MTU(2), set Lg and Lg’ are topologically conjugate if and only ifAccording to the equivalence relation above, we describe topolog- ically conjugate classification of all the left actions on U(2). Further more, we also prove that the topologically conjugate classification of the left actions on the Lie groups U(2) is not equivalent to the algebraically conjugate classification, but it is equivalent to the smooth conjugate clas-sification.(3) For the Lie group SO(3), since SU(2) is the covering space of SO(3), then according to the maximal torus TSU(2) in part (1), choose one maximal torus TSO(3) of SO(3) such that TSO(3) is just the image of TSU(2) under the covering mapping. Then, according to the properties of the covering map π:SU(2)�'SO (3), we know that the elements in TSU(2) could be denoted by the form Obviously, TSO(3)(?)T1=S1. and choose one isomorphism Φ:TSO(3)�'S1 defined by Then, for any Lg mtso(3), the left action Lg acting on the maximal torus TSO(3) is equivalent to a rotation f:z(?)e2πiθ,z∈S1 of S1, so we can define the rotation number of the left action Lg as the same as the definition of the rotation f of S1. Therefore, for any Lg ∈ MTSO(3), we define ρ(Lg)=ρ(f)=θ∈[0,1). And then, for the left actions Lg, Lg’∈MTSO(3), Lg and Lg’ are topologi-cally conjugate if and only if ρ(Lg’)=±ρ(Lg) (mod Z).According to the equivalence relation above, we describe topologi-cally conjugate classification of all the left actions on SO(3). Further more, we also prove that the topologically conjugate classification of the left actions on the Lie groups SO(3) are equivalent to the algebraically conjugate classification and the smooth conjugate classification.(4) For the Lie group SO(3)×S1, since U(2) is the covering space of SO(3)×S1, then according to the maximal torus TU(2) in part (2), choose one maximal torus TSO(3)×S1 of SO(3)×S1 such that TSO(3)×S1 is just the image of TU(2) under the covering mapping. Since TSO(3)×S1≌TSO(3)×S1, and according to the properties of the covering mapping π:U(2)�' SO(3)×S1, we know that the elements in TSO(3)×S1 could be denoted by the form (u, λ) with Obviously, SO(3)×S1(?)T2. and choose one isomorphism Φ:TSO(3)×s1 �'T2 defined by Then, for any Lg∈MTSO(3)×S1, the left action Lg acting on the maximal torus TTSO(3)×S1 is equivalent to a rotation of T2, thus we can define the rotation vector of the left action Lg as the same as the definition of the rotation f of T2. Therefor, for any Lg∈MTSO(3)×S1, we define And then, for the left actions Lg, Lg’∈MTSO(3)×S1, set Lg and Lg’ are topologically conjugate if and only ifAccording to the equivalence relation above, we describe topologi-cally conjugate classification of all the left actions on SO(3) x S1. Further more, we also prove that the topologically conjugate classification of the left actions on the Lie groups SO(3)×S1 is equivalent to the smooth conjugate classification.(5) For the Lie group SpinC(3), since U(2) is the covering space of SpinC(3), then according to the maximal torus TU(2) in part (2), choose one maximal torus TSpinc(3) of Spin (3) such that TSpinc(3) is just the image of TU(2) under the covering mapping. Set Φ:TU(2)�'T2 is the isomorphism defined in part (2),πt:U(2)�'SpinC(3) is the covering mapping, and π’ is a mapping defined byObviously, π and π’ are group homomorphisms, andThen, there exists an isomorphism Φ’:TSpinC(3)�'T2 induced by the homomorphisms Φ,7r and π’such that Now we let (TSpinc(3),Φ’) be the representation of the maximal torus Tspinc(3).Obviously, Then, for any Lg∈MTSpinC(3), the left action Lg acting on the maximal torus TSpinC(3) is equivalent to a rotation of T2, so we can define the rotation vector of the left action Lg as the same as the definition of the rotation f of T2. Therefor, for any Lg∈MTSpinC(3), we define And then, for the left actions Lg, Lg’∈MTSinC(3),set Lg and Lg’ are topologically conjugate if and only ifAccording to the equivalence relation above, we describe topologi-cally conjugate classification of all the left actions on Spinc(3). Further more, we also prove that the topologically conjugate classification of the left actions on the Lie groups Spinc(3) is equivalent to the smooth con-jugate classification.Since all the commutative compact connected Lie groups are isomor-phic to Tn, then every left action on these Lie groups is a rotation of Tn, and the topologically conjugate classifications of the rotations of Tn have been described by the rotation vectors. Therefore, together with the results in this paper, the problem of the topologically conjugate classifi-cations of the translation actions on the compact connected Lie groups which topological dimensions are equal or less than to 4 has been solved completely.