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,

$$(\gamma_1 \gamma_2)^\prime(0) = \gamma_1(0) \gamma_2^\prime(0) + \gamma_1^\prime(0) \gamma_2(0)$$

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 $$R_{g_2(0)}:\,g\mapsto gg_2(0).$$