Understanding the Product Rule in Lie Groups: How Does it Differ from Calculus?

[Monocles]

Out of curiosity, how does the product rule work in Lie groups? I ended up needing it because I approached a problem incorrectly and then saw that the product rule was unnecessary, but it seems to create a strange scenario. For example:

Consider a Lie group G and two smooth curves \gamma_1, \gamma_2: [-1, 1] \rightarrow G such that \gamma_1(0) = \gamma_2(0) = e. Let's say we wish to compute the tangent vector of the curve \gamma_1 \gamma_2 at e. Then,

<br /> (\gamma_1 \gamma_2)^\prime(0) = \gamma_1(0) \gamma_2^\prime(0) + \gamma_1^\prime(0) \gamma_2(0)<br />

But, now we are multiplying a group element by a vector. This would work for, say, GL_n (\mathbb{R}), but not for Lie groups in general. So, I am guessing that the product rule works differently than it does in calculus.

There is even more confusion when you consider the tangent vector at some point other than e!

EDIT: I realized that the product of two curves might not be injective, so let's assume that it is.

 

[arkajad]

The product rule works provided you understand (\gamma_1 \gamma_2)^\prime(0) = \gamma_1(0) \gamma_2^\prime(0)

as acting on the tangent vector by the left translation. The map

L_{g_1(0)}:\, g\mapsto g_1(0)g is a diffemorphism. Its derivative dL_{g_1(0)} maps, in particular, the tangent space at g_2(0) to the tangent space at g_1(0)g_2(0). Similarly for the right translation R_{g_2(0)}:\,g\mapsto gg_2(0).