I need help understanding notation from group theory articles and literature

Question

I need help understanding the notation of this question. I have a general understanding of groups but I don't know what a left-translation map is.

We will study the left-translation maps and left-invariant vector fields for two Lie group examples: Rⁿ (with vector addition as the group operation) and GL(n).

Given v ∈ Rⁿ, what is the corresponding left-translation map L_v : Rⁿ → Rⁿ?

I understand that Rⁿ is a Lie group. But what is v? Is it a transformation or group action? I'm not entirely sure what an element of a group actually is.

And could someone explain to me what L_v : Rⁿ → Rⁿ is conveying? I see it all over literature and need a solid understanding of what it means to continue my reading. I also constantly see μ: G × G → G. Does this mean any operation on an element in G will map back to another element in G?

Thanks!

Answer 1

$v$ is a group element, and $L_v$ denotes left multiplication by $v$. For the abelian group $\Bbb R^n$, we have $L_v(x) = v+x \ (=x+v)$. For the group $GL(n)$, $v$ is now an invertible matrix, and $L_v(A)=vA$.