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I'm having a bit of trouble with the last bit of Problem 3.2 in Kirillov Jr.'s Introduction to Lie Groups and Lie Algebras.

(3.2) Let $f: \mathfrak{g} \rightarrow G$ be any smooth map such that $f(0)=1_G$ and $f_*(0) = \text{id}_{\mathfrak{g}}.$ Show that the group law in this coordinate system near the identity has the form $f(x) f(y) = f(x+y+B_f(x,y)+ \cdots )$ for some bilinear map $B: \mathfrak{g} \otimes \mathfrak{g} \rightarrow \mathfrak{g}$ and that $B_f(x,y)-B_f(y,x)= [x,y]$.

The inverse function theorem lets us write $f(x)\cdot f(y) = f(\mu_f(x,y))$ for some smooth map $\mu_f: \tilde{U} \rightarrow \mathfrak{g}$ defined on some open set $\tilde{U} \subset \mathfrak{g} \times \mathfrak{g}$. Restricting to $x=0$ then $y=0$ in the Taylor expansion of $\mu_f$ around $0 \times 0$ reveals that $\mu_f(x,y) = x+y+ B_f(x,y) + \cdots$ for some bilinear map $B_f : \mathfrak{g} \otimes \mathfrak{g} \rightarrow \mathfrak{g}$ which is defined everywhere even though $\mu_f$ might not be since we can rescale. We know that in the case where $f= \text{exp}: \mathfrak{g} \rightarrow G$ we have $B_{\text{exp}} (x,y)= \frac{1}{2} [x,y]$, which means it's enough to show that for any other $\tilde{f}$ satisfying the conditions of the problem statement we have $$B_f(x,y)-B_f(y,x)=B_{\tilde{f}}(x,y)-B_{\tilde{f}}(y,x)$$ for all $x,y \in \mathfrak{g}$. This is all I've got so far - I can't seem to find the right view to take in order to finish the problem.

Note that unlike the exponential map, in general we don't expect there to be a relationship between $f(x)$ and $f( \lambda x)$ for $\lambda \in \mathbb{R}$ since all we know is that $f$ is smooth.

Any help would be greatly appreciated!

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do I have to beg? :P –  StackQs Dec 14 '11 at 19:58

1 Answer 1

I also got stuck on this problem. Here is what I came up with:

Let $G$ be a lie group and let $x_1, \dots, x_n$ be coordinates around the identity such that the identity is $(0,\dots,0)$. If we express the group multiplication in coordinates, we get a Formal Group Law. Write $a$ and $b$ for coordinate vectors. Then we have $$ a b = a + b + B(a,b) + \text{degree $\geq 3 $}$$ $$ aba^{-1}b^{-1} = B(a,b) - B(b,a) + \text{degree $\geq$ 3} $$ Therefore $B(a,b) - B(b,a)$ is the lowest degree term of the commutator in the coordinate system x_i. The lie bracket $[-,-]$ is the lowest degree term of the commutator in log coordinates...

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