Finding a lie group structure on $\mathbb R^n\setminus\{0\}$

Question

I want to find all maps $g: \mathbb R^n\setminus \{0\} \rightarrow GL_n(\mathbb R)$ which satisfy the properties

• $g$ is differentiable and injective
• $g(g(a)b) = g(a)g(b)$ for all $a,b\in\mathbb R^n\setminus \{0\}$
• $g(e_1)=I_n$ whereby $e_1$ is the vector $(1,0,\ldots,0)^T$ and $I_n$ is the identity matrix of $GL_n(\mathbb R)$

Do you have any tip, how I can solve my problem? Can you recommend a book or article I shall read?

My Attempt:

If I define $\circ: \mathbb R^n\setminus \{0\} \times \mathbb R^n\setminus \{0\} \rightarrow \mathbb R^n\setminus \{0\}$ via $a\circ b = g(a)b$, then $(\mathbb R^n\setminus \{0\}, \circ)$ should be a lie group. $g$ should be a representation of $(\mathbb R^n\setminus \{0\}, \circ)$.

So I thought the lie group theory and lie group representation theory should help me. Unfortunately after scanning some books about those theories I still have no idea how to solve my problem. I have the feeling, that in the theory of lie groups and their representations the lie group structure is always known but in my case it is not. Did I miss something? Can I use those theories to find all maps $g$?

Answer 1

Comment by Qiaochu Yuan: "$\mathbb R^n\setminus\{0\}$ can't have a Lie group structure for $n$ odd and greater than 1 because it's homotopy equivalent to $S^{n−1}$ and in particular has Euler characteristic 2. But a connected Lie group of positive dimension has Euler characteristic either 0 or 1."

Answer 2

Here is my brief answer for the question

"Finding a lie group structure on $R^n\setminus\{0\}$"

For $n$ odd - see above.

Now let $n$ be even.

$n=2$ this space is diffeomorphic to the Lie group $\mathbb{R} \times S^1$

$n=4$ this space is diffeomorphic to the Lie group $\mathbb{R} \times S^3 =\mathbb{R}\times SU(2)$

$n=2k>4$ - there is no structure of a Lie group, since the space of three-dimensional cohomology for non-Abelian compact Lie groups is always non-trivial (due to E.Cartan), while for the space under consideration (considering up to homotopy equivalence!) it is trivial for $n>4$.