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

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 $\T^n = (\S^1)^n$.
• Important subgroups of $GL(n, \R)$ and $GL(n, \C)$
  • special orthogonal group $SO(n, \R)$,
  • Lorentz group $SO(1,3)$ ,
  • Symplectic group $Sp(2n, \R) \subset GL(2n, \R)$.
  • Unitary group $SU(n) \subset GL(n, \C)$

Let $G, H$ be Lie group, we say $\varphi: G \to H$ is a Lie group homomorphism, if it is a smooth map and also a group homomorphism.

• $(\R, +) \to (\R_+, *)$, $t \mapsto e^t$.
• $S^1 \into \C$
• Let $V, W$ be vector spaces, viewed as Lie group by translation, then any linear map $V \to W$ is a group homomorphism.
• $\det: GL(n, \R) \to \R^*$, since $\det(AB) = \det(A)\det(B)$.
• Let $g \in G$, the conjugation action $Ad_g: G \to G$, $h \mapsto g h g^{-1}$ is a group homomoprhism.

Thm : Group homomorphisms are constant rank maps.

Proof: Let $\varphi: G \to H$ be a Lie group homomorphism. We use left translation to move all the maps on the tangent space $T_g G \to T_{\varphi(g)} H$ back to identity $T_e G \to T_e H$.

A Lie subgroup of $G$ is a subgroup of $G$ endowed with a topology and smooth structure making it into a Lie group and an immersed submanifold of $G$.

Prop 7.11 (Lee) : Let $G$ be a Lie group and $H \subset G$ a subgroup, which is also an embedded submanifold, then $H$ is a Lie subgroup.

Proof: