- #1
mnb96
- 715
- 5
Hello,
I consider the groups of rotations [itex]R=SO(2)[/itex] and the group [itex]T[/itex] of translations on the 2D Cartesian plane.
Let's define Ω as the group Ω=RT.
Thus Ω is essentially SE(2), the special Euclidean group.
It is known that R and T are respectively 1-dimensional and 2-dimensional Lie groups diffeomorphic to the unit circle and the 2D plane.
My question is:
If I consider now [itex]\left\langle R \right\rangle ^\Omega[/itex], the conjugate closure of R with respect to Ω, what is the "structure" of such a group? Is it still a Lie group? if so, what is the manifold associated with it?
I consider the groups of rotations [itex]R=SO(2)[/itex] and the group [itex]T[/itex] of translations on the 2D Cartesian plane.
Let's define Ω as the group Ω=RT.
Thus Ω is essentially SE(2), the special Euclidean group.
It is known that R and T are respectively 1-dimensional and 2-dimensional Lie groups diffeomorphic to the unit circle and the 2D plane.
My question is:
If I consider now [itex]\left\langle R \right\rangle ^\Omega[/itex], the conjugate closure of R with respect to Ω, what is the "structure" of such a group? Is it still a Lie group? if so, what is the manifold associated with it?