Lie Groups and Lie Algebra

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.title[
# Lie Groups and Lie Algebra
]
.author[
### Ashan De Silva
]
.institute[
### Department of Mathematics<br>University of Manitoba
]
.date[
### 2026-02-08
]

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# Table of Contents

.large[
1. **Introduction to Lie Groups**  
   *Definition and basic properties*

2. **Examples of Lie Groups**  
   *Classical groups and matrix groups*

3. **Lie Algebras**  
   *The tangent space perspective*

4. **The Exponential Map**  
   *Connecting groups and algebras*

5. **Applications**  
   *Physics and geometry*
]

---
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# What Are Lie Groups?
### Combining Geometry and Algebra

.footnote[Image Credit: Wikeipedia]
---
# Definition: Lie Group

.definition-box[
**DEFINITION**

A **Lie group** is a smooth manifold `$G$` which is also a group, such that:

1. The group multiplication map is smooth:
`$\mu: G \times G \to G, \quad (g,h) \mapsto gh$`

2. The inverse map is smooth:
`$\iota: G \to G, \quad g \mapsto g^{-1}$`
]

.insight-box[
💡 **Key Insight**

Lie groups blend continuous symmetry (manifold structure) with algebraic operations (group structure).
]

---

# Intuitive Understanding

.two-column[
.column-left[
### As a Manifold
- Smooth, continuous space
- Has tangent vectors
- Locally looks like `$\mathbb{R}^n$`
]

.column-right[
### As a Group
- Multiplication is smooth
- Has identity and inverses
- Algebraic structure
]
]

<div class="venn-area"> 
  <div class="v-circle v-left" style="border: 2px solid #586e75;">Groups</div> 
  <div class="v-circle v-right" style="border: 2px solid #586e75;">Manifolds</div> 
  <div class="v-label">Lie<br>Groups</div> 
</div>

---

# Example 1: The Circle Group `$S^1$`

.example-box[
**EXAMPLE 1** The unit circle in the complex plane:
`$S^1 = \{z \in \mathbb{C} : |z| = 1\}=\{e^{i\theta}:0\leq \theta <2\pi\}$`
- **Group operation:**
`$z_1 \cdot z_2 = e^{i\theta_1} \cdot e^{i\theta_2} = e^{i(\theta_1+\theta_2)}$`
- **Inverse:** `$(e^{i\theta})^{-1} = e^{-i\theta}$`
]

.insight-box[
**Geometric View:** Multiplication = Rotation

Rotating by angle `$\theta_1$` then by angle `$\theta_2$` equals rotating by angle `$\theta_1 + \theta_2$`

`$S^1$` models rotational symmetry!
]

---

# Example 2: General Linear Group `$GL(n,\mathbb{R})$`

.example-box[
**EXAMPLE 2**

The group of invertible `$n \times n$` real matrices:

`$$GL(n,\mathbb{R}) = \{A \in \text{Mat}(n \times n, \mathbb{R}) : \det(A) \neq 0\}$$`
]

**Properties:**

- **Manifold structure:** Open subset of `$\mathbb{R}^{n^2}$`  
  (condition `$\det(A) \neq 0$` defines open set)
- **Group operation:** Matrix multiplication
- **Identity:** `$I_n$` (identity matrix)
- **Inverse:** `$A^{-1}$` (matrix inverse)
- **Dimension:** `$n^2$` as a manifold

---
<!---
# Classical Lie Groups

| Symbol | Name | Definition | Dimension |
|--------|------|------------|-----------|
| `$SL(n,\mathbb{R})$` | Special Linear Group | Matrices with `$\det(A) = 1$` | `$n^2 - 1$` |
| `$O(n)$` | Orthogonal Group | `$A^T A = I$` (preserves inner product) | `$\frac{n(n-1)}{2}$` |
| `$SO(n)$` | Special Orthogonal Group | `$O(n)$` with `$\det(A) = 1$` (rotations) | `$\frac{n(n-1)}{2}$` |
| `$U(n)$` | Unitary Group | `$A^* A = I$` in `$\mathbb{C}^{n \times n}$` | `$n^2$` |
| `$Sp(2n,\mathbb{R})$` | Symplectic Group | Preserves symplectic form | `$n(2n+1)$` |

<br>

.small-text[
These groups appear throughout mathematics and physics as fundamental symmetry groups.
]

--->

# Intutive Idea Cont ...
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#Intutive Idean Cont ...
<img src="pic/Picture7.png" width="50%">

--
`$$Z= X + Y + \frac{1}{2}[X,Y]+ \frac{1}{12}[X,[X,Y]]- \frac{1}{12}[Y,[X,Y]]- \frac{1}{24}[Y,[X,[X,Y]]]+ \cdots$$`

--
This formula is called **Baker–Campbell–Hausdorff Formula**

---
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# Lie Algebras

### The Infinitesimal Structure

---
## Intuitive Idea
--

|Curved Space|Flat Space|
|:-:|:-:|
|<img src="https://imgs.search.brave.com/LYPN50ihJGwGZAJBFhcKNHLmgFnCjVQ02hhm99t64to/rs:fit:860:0:0:0/g:ce/aHR0cHM6Ly90aHVt/YnMuZHJlYW1zdGlt/ZS5jb20vYi9nbG9i/ZS03MzQ2MTc2Lmpw/Zw" width="200px" />|<img src="https://imgs.search.brave.com/H28MCU5JiRpf5zL7AshdyinUh6PesIuL0cWnZBWm3lc/rs:fit:860:0:0:0/g:ce/aHR0cHM6Ly93b3Js/ZG1hcGJsYW5rLmNv/bS93cC1jb250ZW50/L3VwbG9hZHMvMjAy/My8wNy9Mb25naXR1/ZGUtYW5kLUxhdGl0/dWRlLU1hcC1NaWxs/ZXItUHJvamVjdGlv/bi53ZWJw" width="200px" />|
|Lie Group|Lie Algebra|
---

# Definition: Lie Algebra

.definition-box[
**DEFINITION**

A **Lie algebra** is a vector space `$\mathfrak{g}$` equipped with a bilinear operation

`$$[\cdot, \cdot]: \mathfrak{g} \times \mathfrak{g} \to \mathfrak{g}$$`

called the **Lie bracket**, satisfying:

1. **Antisymmetry:** `$[X, Y] = -[Y, X]$`

2. **Jacobi identity:** `$[X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0$`
]

.insight-box[
**Connection to Lie Groups**

For a Lie group `$G$`, its Lie algebra `$\mathfrak{g} = T_e G$` is the tangent space at the identity.

The Lie bracket captures how tangent vectors "fail to commute" in the group.
]

---

# Example: Lie Algebra of `$GL(n,\mathbb{R})$`

.example-box[
The Lie algebra of `$GL(n,\mathbb{R})$` is:

`$$\mathfrak{gl}(n,\mathbb{R}) = \text{Mat}(n \times n, \mathbb{R})$$`

(all `$n \times n$` real matrices)
]

**Lie Bracket:**

The Lie bracket is the **matrix commutator**:$[A, B] = AB - BA$

**Verification of Properties:**

- **Antisymmetry:** `$[A, B] = AB - BA = -(BA - AB) = -[B, A]$`
 - **Jacobi identity:** Can be verified by direct computation

---

# The Exponential Map

.theorem-box[
**THEOREM**

The exponential map `$\exp: \mathfrak{g} \to G$` is defined by:

`$$\exp(X) = e^X = I + X + \frac{X^2}{2!} + \frac{X^3}{3!} + \cdots$$`
]

.two-column[
.column-left[
### Key Properties

1. `$\exp(0) = e$` (identity)
2. `$\exp(tX)$` is a 1-parameter subgroup for each `$X \in \mathfrak{g}$`
3. `$\frac{d}{dt}\exp(tX)\big|_{t=0} = X$`
4. `$\exp$` is a local diffeomorphism near `$0 \in \mathfrak{g}$`
]

.column-right[
### Geometric Meaning

The exponential map takes:

**Tangent vectors at identity**  
(infinitesimal symmetries)

↓

**Group elements**  
(actual symmetries)
]
]

---

# Example: Exponential of Matrices

**Example 1: Diagonal Matrix**

If `$X = \text{diag}(\lambda_1, \lambda_2, \ldots, \lambda_n)$`, then:
`$$\exp(X) = \text{diag}(e^{\lambda_1}, e^{\lambda_2}, \ldots, e^{\lambda_n})$$`

<br>

**Example 2: Rotation Matrix**

For the `$2 \times 2$` skew-symmetric matrix:

`$$X = \begin{pmatrix} 0 & \theta \\ -\theta & 0 \end{pmatrix}, \quad \exp(X) = \begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix}$$`

This gives a rotation by angle `$\theta$`!

.insight-box[
The exponential map converts infinitesimal rotations into actual rotations!
]

---

# Applications of Lie Theory

.example-box[
### Physics

- Quantum mechanics: Angular momentum, spin
- Particle physics: Gauge theories, Standard Model
- General relativity: Lorentz group, spacetime symmetries
]

.example-box[
### Differential Geometry

- Isometry groups of Riemannian manifolds
- Holonomy groups and parallel transport
- Symmetric spaces and homogeneous spaces
]

.example-box[
### Control Theory

- Robotics: Configuration spaces
- Nonlinear control systems
- Optimal control on manifolds
]

---

# Summary

### Key Takeaways

1. **Lie groups** combine smooth manifold structure with group structure

2. **Classical examples:** `$GL(n,\mathbb{R})$`, `$SO(n)$`, `$SU(n)$` — fundamental in mathematics and physics

3. **Lie algebras** are the "infinitesimal" version — tangent space at identity

4. **The exponential map** connects Lie algebras to Lie groups

5. **Applications** span physics, geometry, control theory, and beyond

<br>

.insight-box[
📚 **Further Study:** Representation theory, Homogeneous spaces, Lie group cohomology
]

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# Thank You!

### Questions?

**Ashan De Silva**  
Department of Mathematics  
University of Manitoba

---

# References and Further Reading

.small-text[
**Recommended Textbooks:**

- Stillwell, J. (2008). *Naive Lie Theory*. Springer.
- Hall, B. (2015). *Lie Groups, Lie Algebras, and Representations*. Springer.
- Knapp, A. (2002). *Lie Groups Beyond an Introduction*. Birkhäuser.

**Online Resources:**

- nLab: Lie group (https://ncatlab.org/nlab/show/Lie+group)
- MIT OpenCourseWare: Topics in Lie Theory
- University of Chicago: Lie Groups Notes

# Visualizing  and its Lie Algebra

.pull-left[

* This is the Lie Group  (the unit circle).

* We pick the identity element .

* The Lie Algebra  is the tangent space .
]

.pull-right[

]