Lie Groups and Lie Algebras

A group is a pair $(G,\cdot)$ consisting of a set $G$ together with a binary operation \[ \cdot : G \times G \to G \] such that: • Associativity: For all $a,b,c \in G$, $(a \cdot b)\cdot c = a \cdot (b \cdot c)…

A group is a group object in the category $\mathbf{Set}$. That is, a set $G$ equipped with morphisms: \[ m : G \times G \to G, \quad e : 1 \to G, \quad i : G \to G \] such that the following identities hold: • As…

Let $(G, \cdot)$ and $(H, \ast)$ be groups in the sense of Definition~. A map $\phi : G \to H$ is a group homomorphism if \[ \phi(a \cdot b) = \phi(a) \ast \phi(b) \quad for all a, b \in G. \] If $\phi$ is also a bij…

For a homomorphism $\phi : G \to H$ (Definition~), the kernel and image are \[ \ker \phi := \{ g \in G \mid \phi(g) = e_H \}, \qquad \operatorname{im} \phi := \{ \phi(g) \mid g \in G \}. \]

Let $G$ be a group (Definition~). A subset $H \subseteq G$ is a subgroup, written $H \leq G$, if it is closed under multiplication and inverses and contains the identity. It is normal, written $H \trianglelefteq G$, if …

For any homomorphism $\phi : G \to H$ (Definition~), $\ker \phi \trianglelefteq G$ and $\operatorname{im} \phi \leq H$ in the sense of Definition~, where $\ker\phi$ and $\operatorname{im}\phi$ are as in Definition~.

Let $N \trianglelefteq G$ be a normal subgroup (Definition~). The quotient group $G/N$ is the set of left cosets $\{ gN \mid g \in G \}$ equipped with the operation $(aN)(bN) := (ab)N$.

Let $\phi : G \to H$ be a group homomorphism (Definition~). Then the quotient group (Definition~) by the kernel (Definition~) satisfies \[ G / \ker\phi \;\cong\; \operatorname{im}\phi. \]

A left action of a group $G$ (Definition~) on a set $X$ is a map $G \times X \to X$, written $(g, x) \mapsto g \cdot x$, satisfying \[ e \cdot x = x \quad and \quad g \cdot (h \cdot x) = (gh) \cdot x \quad for all g,…

Let $G$ act on $X$ (Definition~). For $x \in X$, the orbit and stabilizer of $x$ are \[ G \cdot x := \{ g \cdot x \mid g \in G \}, \qquad G_x := \{ g \in G \mid g \cdot x = x \}. \] Note that $G_x \leq G$ is a subgr…

For any $x \in X$ with orbit and stabilizer as in Definition~, \[ |G \cdot x| = [G : G_x], \] where $[G : G_x]$ is the index of the subgroup $G_x \leq G$ (Definition~), equivalently the cardinality of the quotient $G/…

The direct product $G \times H$ of two groups (Definition~) is the Cartesian product with componentwise operations. More generally, given a normal subgroup $N \trianglelefteq G$ and a subgroup $H \leq G$ (Definition~) w…

A topological space is a pair $(X, \mathcal{T})$ where $\mathcal{T}$ is a collection of subsets of $X$ (the open sets) satisfying: $\emptyset, X \in \mathcal{T}$; arbitrary unions of open sets are open; finite intersect…

Let $(X, \mathcal{T}_X)$ and $(Y, \mathcal{T}_Y)$ be topological spaces (Definition~). A map $f : X \to Y$ is continuous if $f^{-1}(U) \in \mathcal{T}_X$ for every $U \in \mathcal{T}_Y$. If $f$ is a continuous bijection…

A topological space $X$ (Definition~) is connected if it cannot be written as a disjoint union of two nonempty open sets. It is path-connected if for every $x, y \in X$ there exists a continuous path $\gamma : [0,1] \to…

Path-connectedness implies connectedness, but not conversely. For Lie groups the two notions coincide: every connected Lie group is path-connected.

A path-connected space $X$ (Definition~) is simply connected if every loop $\gamma : [0,1] \to X$ with $\gamma(0) = \gamma(1)$ can be continuously contracted to a point, i.e., the fundamental group $\pi_1(X) = 0$.

A topological space $X$ (Definition~) is compact if every open cover of $X$ has a finite subcover. A subset $A \subseteq \mathbb{R}^n$ is compact if and only if it is closed and bounded (Heine--Borel).

A topological $n$-manifold is a Hausdorff, second-countable topological space $M$ (Definition~) such that every point $p \in M$ has a neighbourhood homeomorphic (Definition~) to an open subset of $\mathbb{R}^n$.

A topological $n$-manifold $M$ (Definition~) is a smooth manifold if it is equipped with a maximal smooth atlas: a collection of charts $(U_\alpha, \varphi_\alpha)$ covering $M$ such that all transition maps $\varphi_\b…

For $p \in M$ a smooth manifold (Definition~), the tangent space $T_pM$ is the vector space of derivations on smooth functions near $p$. Concretely in a chart, it is spanned by $\partial/\partial x^i|_p$. The tangent bu…

A map $f : M \to N$ between smooth manifolds (Definition~) is smooth if its coordinate representations are smooth. It is a diffeomorphism if it is a smooth bijection with smooth inverse; in particular a diffeomorphism i…

A smooth map $f : M \to N$ (Definition~) is an immersion if the differential $df_p : T_pM \to T_{f(p)}N$ (Definition~) is injective for all $p$. It is an embedding if it is additionally a homeomorphism (Definition~) ont…

This distinction matters for Lie subgroups: closed subgroups are always embedded (Definition~), but immersed subgroups (e.g.\ a dense winding on a torus) need not be.

A continuous map $p : \widetilde{X} \to X$ (Definition~) between topological spaces (Definition~) is a covering map if every point $x \in X$ has an open neighbourhood $U$ such that $p^{-1}(U)$ is a disjoint union of ope…

If $G$ is a connected (Definition~) Lie group, its universal cover $\widetilde{G}$ (Definition~) carries a unique Lie group structure such that the covering map $p : \widetilde{G} \to G$ is a Lie group homomorphism (Def…

A (linear) representation of a group $G$ (Definition~) on a vector space $V$ over a field $k$ is a group homomorphism (Definition~) \[ \rho : G \to \mathrm{GL}(V). \] We say $(V, \rho)$ is a $G$-representation, or sim…

Given a $G$-representation $(V, \rho)$ (Definition~), a subspace $W \subseteq V$ is $G$-invariant (a subrepresentation) if $\rho(g)w \in W$ for all $g \in G$, $w \in W$. The restriction $\rho|_W$ then defines a represen…

A representation $(V, \rho)$ (Definition~) is irreducible (or simple) if $V \neq 0$ and its only subrepresentations (Definition~) are $\{0\}$ and $V$ itself.

A linear map $T : V \to W$ between $G$-representations (Definition~) is a $G$-equivariant map (or intertwiner) if \[ T \circ \rho_V(g) = \rho_W(g) \circ T \quad for all g \in G. \] An invertible intertwiner is an iso…

Let $(V, \rho)$ and $(W, \sigma)$ be irreducible $G$-representations (Definition~) over an algebraically closed field, and let $T : V \to W$ be an intertwiner (Definition~). • Either $T = 0$ or $T$ is an isomorphism.…

Given $G$-representations $(V, \rho)$ and $(W, \sigma)$ (Definition~), their direct sum is $(V \oplus W, \rho \oplus \sigma)$ where $(\rho \oplus \sigma)(g)(v, w) = (\rho(g)v, \sigma(g)w)$.

A representation $V$ (Definition~) is completely reducible (semisimple) if it decomposes as a direct sum (Definition~) of irreducible subrepresentations (Definition~, Definition~).

Every finite-dimensional continuous representation (Definition~) of a compact Lie group (Definition~) over $\mathbb{R}$ or $\mathbb{C}$ is completely reducible (Definition~).

The character of a finite-dimensional representation $(V, \rho)$ (Definition~) is the function $\chi_V : G \to k$ defined by $\chi_V(g) = \mathrm{tr}(\rho(g))$. Characters are class functions: $\chi_V(hgh^{-1}) = \chi_V…

Isomorphic representations (Definition~) have equal characters (Definition~). For compact groups (Definition~), the converse holds: two representations are isomorphic if and only if their characters are equal.

Let $\rho_1 : G \to \mathrm{GL}(V_1)$ and $\rho_2 : G \to \mathrm{V_2}$ be two linear representations of $G$ (Definition ), and let $\chi_1$ and $\chi_2$ be their characters (Definition ). Then: • The character …

The inner product of two characters (Definition ) of representations of a finite group $G$ (Definition is \[ \langle \chi_1, \chi_2 \rangle = \frac{1}{|G|}\sum_{g \in G} \chi_1 (g) \overline{\chi_2(g)} \] …

A Lie Group is a group (Definition ) that is also a finite dimensional smooth differentiable manifold (Definition ), with the added condition that the group operations of multiplication and inversion are smooth maps…

The unit circle in $\mathbb{C}$ denoted $$S^1 = \{e^{i\theta} \colon \theta \in [0,2\pi)\} = \{z \in \mathbb{C}\colon |z|= 1\}$$ endowed with group multiplication as \[ e^{i\alpha} \cdot e^{i\beta} = e^{i(\alpha+…

$S^1$ (Example ) is a Lie group and is isomorphic to $SO(2)$ the rotation group.

A representation of a Lie group (Definition ) is the exact same as a representation of a finite group (Definition ), however with the added condition that the representation must be a continuous map.