This shows you the differences between two versions of the page.
Next revision | Previous revision | ||
math214:03-04 [2020/03/04 08:44] pzhou created |
math214:03-04 [2020/03/04 11:01] (current) pzhou |
||
---|---|---|---|
Line 1: | Line 1: | ||
====== 2020-03-04, Wednesday ====== | ====== 2020-03-04, Wednesday ====== | ||
$$\gdef\T{\mathbb T}$$ | $$\gdef\T{\mathbb T}$$ | ||
+ | |||
+ | ===== Definitions ===== | ||
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. | 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. | ||
Line 7: | Line 9: | ||
$$ L_g(h) = gh, \quad R_g(h) = hg $$ | $$ L_g(h) = gh, \quad R_g(h) = hg $$ | ||
- | ==== Example | + | ==== Examples |
* $GL(n, \R)$, 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 $GL_+(n, \R)$. | * $GL(n, \R)$, 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. | * $GL(n, \C)$, as a complex manifold. | ||
Line 13: | Line 15: | ||
* Translation group $\R^n$ acting. | * Translation group $\R^n$ acting. | ||
* The circle group $S^1 \subset \C^*$. | * The circle group $S^1 \subset \C^*$. | ||
- | * The n-dimensional torus $\T^n = \S^1 \times \cdots \times | + | * The n-dimensional torus $\T^n = (\S^1)^n$. |
+ | * Important subgroups of $GL(n, \R)$ and $GL(n, \C)$ (later when we know how to produce subgroups) | ||
+ | * special orthogonal group $SO(n, \R)$, | ||
+ | * Lorentz group $SO(1,3)$ , | ||
+ | * Symplectic group $Sp(2n, \R) \subset GL(2n, \R)$. | ||
+ | * Unitary group $SU(n) \subset GL(n, \C)$ | ||
+ | |||
+ | ===== Group Homomorphism ===== | ||
+ | Let $G, H$ be Lie group, we say $\varphi: G \to H$ is a Lie 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, | ||
+ | * $\det: GL(n, \R) \to \R^*$, 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 $T_e G \to T_e H$. | ||
+ | |||
+ | ===== Lie Subgroup ===== | ||
+ | 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, | ||
+ | |||
+ | //Proof//: This uses Corollary 5.30, which says, if $F: M \to N$ is a smooth map, $S \subset N$ an embedded submanifold, | ||
+ | |||
+ | ===== Lie group Action ===== | ||
+ | First, we consider just group action. Let $G$ be a group, $M$ be a set. A left group action is a map | ||
+ | $$ \rho: G \times | ||
+ | such that for all $p \in M$, $g_1, g_2 \in G$, | ||
+ | $$ (g_1 g_2) \cdot p) = g_1 \cdot (g_2 \cdot p) $$ | ||
+ | and | ||
+ | $$ e \cdot p = p. $$ | ||
+ | |||
+ | A continuous action, or a smooth action is defined the same way, just imposing the corresponding conditions on $G, M$ and the map $G \times M \to M$. | ||
+ | |||
+ | Right action. And how to translate a right action into a left action. | ||
+ | |||
+ | Lie group usually arises as ' | ||
+ | |||
+ | Some terminologies: | ||
+ | * $\theta_g: M \to M$ is the map $p \mapsto \theta(g, | ||
+ | * for any $p \in M$, the **orbit** $G \cdot p$ is the set $\{ g \cdot p \mid g \in G\}$. | ||
+ | * the **isotropy group** or the **stabilizer** of $p$ is the subgroup $\{g \in G \mid g \cdot p = p \}$, denoted as $G_p$. | ||
+ | * The action is **transitive**, | ||
+ | * The action is free, if the only element of $G$ that fixes some element in $M$ is the identity element, namely if $g \cdot p = p$ for some $p \in M$ then $g = e$. Equivalently, | ||
+ | |||
+ | Examples: | ||
+ | * Lie group acts by conjugation on itself. | ||
+ | * Lie group acts by left translation on itself. | ||
+ | * $GL(n, \R)$ acts on $\R^n$, it is transitive. What is the isotropy group of $(1,0,\cdots, 0)$? | ||
+ | |||
+ | ==== $G$-equivariant maps ==== | ||
+ | Suppose $M$ and $N$ are two manifolds where the Lie group $G$ acts on the left. We say a smooth map $F: M \to N$ is $G$-equivariant, | ||
+ | $$ F(g \cdot p) = g \cdot F(p). $$ | ||
+ | |||
+ | ** Thm (constant rank theorem) (7.25) **: If $M, N$ are smooth manifold with left $G$-action. Suppose $G$ acts on $M$ transitively. Then any equivariant map $F: M \to N$ is constant rank. In particular, if $F$ is a surjection, then it is a submersion, $F$ is an injectiion then it is a an immersion; finally if $F$ is a bijection, then it is a diffeomorphism. ' | ||
+ | |||
+ | This follows immediately from the global rank theorem, which says, if $F: M \to N$ is constant rank, then $F$ is a surjection implies $F$ is a smooth submersion; $F$ is an injection implies $F$ is a smooth immersion; $F$ is a bijection implies $F$ is a diffeomorphism. Morally, it allows one to upgrade a set-wise statement to a smooth manifold statement with control on the differential of $F$. | ||
+ | |||
+ | |||
+ | |||
+ | |||
+ | |||
+ | |||
+ | |||
+ | |||
+ | |||
+ |