Editing Lecture 3 - Monday Week 2

== Lecture 3 - 19 Jan 2026 ==

=== Lie Groups ===

* Defined Lie group
* Defined morphism of lie groups

A Smooth map: <math display="inline">f:G \rightarrow H</math> between lie groups <math display="inline">G,H</math> which also preserves group homomorphisms.

Examples:

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

Then for more classical examples:

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

<math display="inline">S^{3} \subseteq {\mathbb{C}}^{2}</math> with group structure <math display="inline">\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 display="inline">L_{a},R_{a},C_{a}</math> as left mult, right mult, and conj. Let <math display="inline">G</math> be a lie group and <math display="inline">a \in G</math>. Then

* <math display="inline">L_{a}:G \rightarrow G</math> given by <math display="inline">g| \rightarrow ag</math>
* <math display="inline">R_{a}:G \rightarrow G</math> given by <math display="inline">g| \rightarrow ga</math>
* <math display="inline">C_{a}:G \rightarrow G</math> given by <math display="inline">g| \rightarrow aga^{\left\{ - 1 \right\}}</math>.

These are all diffeomorphisms of <math display="inline">G</math>.

We can use <math display="inline">L_{a}</math> to convert a basis for <math display="inline">T_{1}G</math> to a basis for <math display="inline">T_{a}G</math>. In this way we can show that any lie group is parallelisable.

<math display="inline">D_{1}L_{a}:T_{1}G \rightarrow T_{a}G</math> is an isomporphism of tangent spaces.

The characteristic <math display="inline">\chi_{i}(a) = D_{1}L_{a}\left( x_{i} \right)</math> for <math display="inline">x_{i} \in X(G)</math> where <math display="inline">X(G)</math> is the set of vector spaces on <math display="inline">G</math>. ''should be curly <math display="inline">X</math>''.

=== Lie Algebras ===

Define Lie Algebra: A lie algebra is a vector space <math display="inline">\mathfrak{g}</math> equipped with a bracket <math display="inline">\lbrack - , - \rbrack:\mathfrak{g} \times \mathfrak{g} \rightarrow \mathfrak{g}</math> which is

* (linear)
* antisymmetric: <math display="inline">\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 display="inline">\lbrack - , - \rbrack</math>.

Example of lie algebra:

Given an associative algebra <math display="inline">A</math>, define <math display="inline">\lbrack X,Y\rbrack = XY - YX</math>. This is always a lie bracket.

Def: for lie group <math display="inline">G</math>, the lie algebra of <math display="inline">G</math> is <math display="inline">\mathfrak{g}</math> = <math display="inline">\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 display="inline">\lbrack - , - \rbrack</math> is inherited from <math display="inline">X(G)</math>.

These are uniquely determined by the value of <math display="inline">X</math> at <math display="inline">1 \in G</math>.

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

Prop: let <math display="inline">G,H</math> be lie groups. Then for <math display="inline">G \rightarrow^{f}H</math> lie group homomorphism, <math display="inline">df:\mathfrak{g} \rightarrow \mathfrak{h}</math> is a lie group homomorphism.

Prop: For <math display="inline">G \subseteq \text{GL}_{n}({\mathbb{R}})</math>, we have <math display="inline">\mathfrak{g} \simeq T_{1}G \subseteq T_{1}\text{GL}_{n}({\mathbb{R}}) = M_{n}</math> and <math display="inline">\lbrack A,B\rbrack = AB - BA</math>. proof: Let <math display="inline">x \in \mathfrak{g}</math> and <math display="inline">x(1) = A \in M_{n}</math>. <math display="inline">x(a) = a \cdot A</math>. Flow <math display="inline">{\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 display="inline">\text{GL}_{n}</math> is <math display="inline">\mathfrak{gl}_{n} = M_{n}({\mathbb{R}})</math>, the set of <math display="inline">n \times n</math> real-valued matrices. Then <math display="inline">\text{SL}_{n} = \det^{-}1(1)</math> where <math display="inline">\det:\text{GL}_{n} \rightarrow {\mathbb{R}}</math>. Then we have that the tangent space <math display="inline">T_{1}\text{SL}_{n} = \ker D_{1}\det = \ker\text{tr}</math>

<math display="inline">\mathfrak{sl}_{n} = \left\{ A \in M_{n},\text{tr}(A) = 0 \right\}</math>. For example, <math display="block">\mathfrak{sl}_{2} = < \begin{pmatrix}
0 & 1 \\
0 & 0
\end{pmatrix}\begin{pmatrix}
0 & 0 \\
1 & 0
\end{pmatrix}\begin{pmatrix}
1 & 0 \\
0 & - 1
\end{pmatrix} ></math>