- #1
mnb96
- 715
- 5
Hello,
I have a doubt on the definition of Lie groups that I would like to clarify.
Let's have the set of functions [tex]G=\{ f:R^2 \rightarrow R^2 \; | \; < f(x),f(y)>=<x,y> \: \forall x,y \in R^2 \}[/tex], that is the set of all linear functions ℝ2→ℝ2 that preserve the inner product. Let's associate the operation of composition to the elements of G and we obtain a group [itex](G,\circ)[/itex].
Now in order to say that G is indeed a Lie group we must prove that G is also a smooth manifold. How can I do this if we don't specify a parametrization (e.g. a matrix representation) for the group G ?
And also in that case, wouldn't the definition of G being a smooth manifold depend on the parametrization?
I have a doubt on the definition of Lie groups that I would like to clarify.
Let's have the set of functions [tex]G=\{ f:R^2 \rightarrow R^2 \; | \; < f(x),f(y)>=<x,y> \: \forall x,y \in R^2 \}[/tex], that is the set of all linear functions ℝ2→ℝ2 that preserve the inner product. Let's associate the operation of composition to the elements of G and we obtain a group [itex](G,\circ)[/itex].
Now in order to say that G is indeed a Lie group we must prove that G is also a smooth manifold. How can I do this if we don't specify a parametrization (e.g. a matrix representation) for the group G ?
And also in that case, wouldn't the definition of G being a smooth manifold depend on the parametrization?