Lecture 3 - Monday Week 2

Lecture 3 - 19 Jan 2026

Lie Groups

Defined Lie group
Defined morphism of lie groups

A Smooth map: [math]\displaystyle{ f:G \rightarrow H }[/math] between lie groups [math]\displaystyle{ G,H }[/math] which also preserves group homomorphisms.

Examples:

for vector space [math]\displaystyle{ V }[/math], [math]\displaystyle{ (V, + ) }[/math] is a lie group.
[math]\displaystyle{ ({\mathbb{Z}}, + ) }[/math], or even any finite dimensional group, is a [math]\displaystyle{ 0 - }[/math]dimensional lie group. (not really using smoothness because all maps are smooth in discrete setting).
[math]\displaystyle{ \frac{\mathbb{R}}{\mathbb{Z}} }[/math] or [math]\displaystyle{ S^{1} = \left\{ z \in {\mathbb{C}},|z| = 1 \right\} }[/math] is the circle lie group.

Then for more classical examples:

[math]\displaystyle{ \text{GL}_{n}({\mathbb{R}}) }[/math] is the group of [math]\displaystyle{ n \times n }[/math] invertible real-valued matrices.
[math]\displaystyle{ \text{SL}_{n}({\mathbb{R}}) \subset \text{GL}_{n}({\mathbb{R}}) }[/math] is [math]\displaystyle{ \left\{ A \in \text{GL}_{n}({\mathbb{R}}),\det(A) = 1 \right\} }[/math], the group of determinant [math]\displaystyle{ 1 }[/math] [math]\displaystyle{ n \times n }[/math] real-valued matrices.
[math]\displaystyle{ O_{n}({\mathbb{R}}) = \left\{ A \in \text{GL}_{n}({\mathbb{R}}),AA^{T} = \text{id} \right\} }[/math] are the orthogonal matrices.
[math]\displaystyle{ S^{3} = \left\{ \left( x_{1},y_{1},x_{2},y_{2} \right) \in {\mathbb{R}}^{4},||_{||} = 1 \right\} }[/math]

[math]\displaystyle{ S^{3} \subseteq {\mathbb{C}}^{2} }[/math] with group structure [math]\displaystyle{ \left( z_{1},z_{2} \right) \times \left( z_{3},z_{4} \right) = \left( z_{1}z_{3} - z_{2}\underset{¯}{z_{4}},z_{1}z_{4} + z_{2}\underset{¯}{z_{3}} \right) }[/math].

Define [math]\displaystyle{ L_{a},R_{a},C_{a} }[/math] as left mult, right mult, and conj. Let [math]\displaystyle{ G }[/math] be a lie group and [math]\displaystyle{ a \in G }[/math]. Then

[math]\displaystyle{ L_{a}:G \rightarrow G }[/math] given by [math]\displaystyle{ g| \rightarrow ag }[/math]
[math]\displaystyle{ R_{a}:G \rightarrow G }[/math] given by [math]\displaystyle{ g| \rightarrow ga }[/math]
[math]\displaystyle{ C_{a}:G \rightarrow G }[/math] given by [math]\displaystyle{ g| \rightarrow aga^{\left\{ - 1 \right\}} }[/math].

These are all diffeomorphisms of [math]\displaystyle{ G }[/math].

We can use [math]\displaystyle{ L_{a} }[/math] to convert a basis for [math]\displaystyle{ T_{1}G }[/math] to a basis for [math]\displaystyle{ T_{a}G }[/math]. In this way we can show that any lie group is parallelisable.

[math]\displaystyle{ D_{1}L_{a}:T_{1}G \rightarrow T_{a}G }[/math] is an isomporphism of tangent spaces.

The characteristic [math]\displaystyle{ \chi_{i}(a) = D_{1}L_{a}\left( x_{i} \right) }[/math] for [math]\displaystyle{ x_{i} \in X(G) }[/math] where [math]\displaystyle{ X(G) }[/math] is the set of vector spaces on [math]\displaystyle{ G }[/math]. should be curly [math]\displaystyle{ X }[/math].

Lie Algebras

Define Lie Algebra: A lie algebra is a vector space [math]\displaystyle{ \mathfrak{g} }[/math] equipped with a bracket [math]\displaystyle{ \lbrack - , - \rbrack:\mathfrak{g} \times \mathfrak{g} \rightarrow \mathfrak{g} }[/math] which is

(linear)
antisymmetric: [math]\displaystyle{ \lbrack X,Y\rbrack = - \lbrack Y,X\rbrack }[/math]
jacobi identity

Define morphism of lie algebras. A morphism of lie algebras is a linear map which preserves [math]\displaystyle{ \lbrack - , - \rbrack }[/math].

Example of lie algebra:

Given an associative algebra [math]\displaystyle{ A }[/math], define [math]\displaystyle{ \lbrack X,Y\rbrack = XY - YX }[/math]. This is always a lie bracket.

Def: for lie group [math]\displaystyle{ G }[/math], the lie algebra of [math]\displaystyle{ G }[/math] is [math]\displaystyle{ \mathfrak{g} }[/math] = [math]\displaystyle{ \left\{ X \in X(G),\forall a \in G,\left( L_{a} \right)*X = X \right\} }[/math], the space of left-invariant vector fields. The bracket [math]\displaystyle{ \lbrack - , - \rbrack }[/math] is inherited from [math]\displaystyle{ X(G) }[/math].

These are uniquely determined by the value of [math]\displaystyle{ X }[/math] at [math]\displaystyle{ 1 \in G }[/math].

Prop: [math]\displaystyle{ \mathfrak{g} \simeq T_{1}G }[/math]. this isomorphism is given by [math]\displaystyle{ X{| \rightarrow}X(1) }[/math]. proof: Given [math]\displaystyle{ v \in T_{1}G }[/math], [math]\displaystyle{ \chi_{v}(a) = D_{1}L_{a}(v) }[/math].

Prop: let [math]\displaystyle{ G,H }[/math] be lie groups. Then for [math]\displaystyle{ G \rightarrow^{f}H }[/math] lie group homomorphism, [math]\displaystyle{ df:\mathfrak{g} \rightarrow \mathfrak{h} }[/math] is a lie group homomorphism.

Prop: For [math]\displaystyle{ G \subseteq \text{GL}_{n}({\mathbb{R}}) }[/math], we have [math]\displaystyle{ \mathfrak{g} \simeq T_{1}G \subseteq T_{1}\text{GL}_{n}({\mathbb{R}}) = M_{n} }[/math] and [math]\displaystyle{ \lbrack A,B\rbrack = AB - BA }[/math]. proof: Let [math]\displaystyle{ x \in \mathfrak{g} }[/math] and [math]\displaystyle{ x(1) = A \in M_{n} }[/math]. [math]\displaystyle{ x(a) = a \cdot A }[/math]. Flow [math]\displaystyle{ {\Phi^{x}}_{t}(a) = d \cdot e^{tA} }[/math]…

In the matrix case, we have explicit formulas for the flow, and so computing brackets is easy. But in general this is hard.

Examples:

The lie algebra of [math]\displaystyle{ \text{GL}_{n} }[/math] is [math]\displaystyle{ \mathfrak{gl}_{n} = M_{n}({\mathbb{R}}) }[/math], the set of [math]\displaystyle{ n \times n }[/math] real-valued matrices. Then [math]\displaystyle{ \text{SL}_{n} = \det^{-}1(1) }[/math] where [math]\displaystyle{ \det:\text{GL}_{n} \rightarrow {\mathbb{R}} }[/math]. Then we have that the tangent space [math]\displaystyle{ T_{1}\text{SL}_{n} = \ker D_{1}\det = \ker\text{tr} }[/math]

[math]\displaystyle{ \mathfrak{sl}_{n} = \left\{ A \in M_{n},\text{tr}(A) = 0 \right\} }[/math]. For example, [math]\displaystyle{ \mathfrak{sl}_{2} = \lt \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 1 & 0 \\ 0 & - 1 \end{pmatrix} \gt }[/math]