# Thread: Tangent Space in Lie Group

1. ## Tangent Space in Lie Group

Hello,

i try to understand how one can compute the tangent space on a Lie Group G at a given point, say $\displaystyle e \ in G$ (That is to compute the Lie Algebra of G).

Is there a general method or trick?

My second question is about the special tangent space of SO(3) at e. Is there any simple way to compute the tangent space at e? I think it is more easy than in the general case, but i don't know how to do it.

Thanks in advance.

2. I'll confine G as a matrix lie group.

Think of tangent vectors as "velocity vectors" of points moving smoothly through the point 1.

If $\displaystyle M \in O(n)$, then $\displaystyle MM^T=1$. Let M=M(t) be a smooth path originating at 1. Apply d/dt to $\displaystyle M(t)M(t)^T=1$. Then, we have

$\displaystyle M^\prime(t)M(t)^T + M(t)M^\prime(t)^T=0$.

Since $\displaystyle M(0)=M(0)^T=1$, if we let $\displaystyle X=M^\prime(0)$, then we have

$\displaystyle X+X^T=0$ for t=0. Further, for any vector X with $\displaystyle X + X^T=0$, the matrix $\displaystyle e^X$ is in $\displaystyle SO(n)$ (verify this). Thus, the tangent space of SO(n) consists of $\displaystyle n \times n$ real vectors X such that $\displaystyle X+X^T=0$.

You can apply this method for U(n), SP(n) as well.

3. ## Re: Tangent Space in Lie Group

Hello!
Many thanks for your answer to Sogan. Could you also please show how to construct the tangent space to SO(3) at a generic point $g \in SO(3)$ ?

thanks,
roth