$$\gdef\T{\mathbb T}$$

A Lie group $G$ is a smooth manifold that is also a group in the algebraic sense, such that the multiplication map $m: G \times G \to G$ and the inverse $i: G \to G$ are all smooth maps.

Let $g \in G$, we define the left translation $L_g$ and right translation $R_g$ as maps $G \to G$ by $$ L_g(h) = gh, \quad R_g(h) = hg $$

• $GL(n, \R)$, the invertible $n \times n$ matrices with real entries. It's dimension is $n^2$. We can check multiplication is smooth by writing down the formula $C = AB$, then $c_{ij} = \sum_k a_{ik}b_{kj}$. And the inverse is smooth, since we can write $G^{-1} = (\det G)^{-1} G_{adj}$. An open subgroup of $GL(n, \R)$ is $GL_+(n, \R)$.
• $GL(n, \C)$, as a complex manifold.
• If $V$ is a real or complex vector space, we can talk about $GL(V)$, the group of invertible linear maps from $V$ to $V$.
• Translation group $\R^n$ acting.
• The circle group $S^1 \subset \C^*$.
• The n-dimensional torus $\T^n = \S^1 \times \cdots \times S^1$.