Motivation:

Historically, Lie Groups have played an important role in physics as they have allowed us to analyse the symmetries of physical systems. They have allowed physicists to model the continuous symmetries of differential equations in a manner analogous to the way finite groups are used in Galois theory to model the discrete symmetries of algebraic equations.

From a bird’s eye view, Lie Groups have also provided physicists with a powerful model for smooth manifolds.

Several ways to think about Lie Groups:

Definition via smooth maps:

A real Lie Group is a finite-dimensional real smooth manifold where the group operations of multiplication and inversion are smooth maps. Smoothness of the group operation:

\begin{equation} \mu: G \times G \rightarrow G \end{equation}

means that \(\mu\) is a smooth mapping of the product manifold \(G \times G\) into \(G\).

These two requirements are reducible to the requirement that the mapping:

\begin{equation} (x,y) \mapsto x^{-1} \cdot y \end{equation}

is a smooth mapping of \(G \times G\) into \(G\).

Topological definition:

Topological Groups:

A topological group \(G\) is a Hausdorff space with a group operation \(\cdot\) such that:

\(1.\) \((x,y) \mapsto x \cdot y\) is continuous.

\(2.\) \(x \mapsto x^{-1}\) is continuous.

Lie Groups as Topological Groups:

A Topological Group \(G\) is a real Lie Group if:

\(1.\) \(G\) is a differentiable manifold.

\(2.\) The group operations \((x,y) \mapsto x \cdot y\) and \(x \mapsto x^{-1}\) are both differentiable.

Category-theoretic definition:

A Lie Group is a group object in the category of smooth manifolds.

Examples:

Examples of Matrix Lie Groups:

\(1.\) The 2x2 real invertible matrices form a group under multiplication denoted \(\text{GL}(2,\mathbb{R})\):

\begin{equation} \text{GL}(2,\mathbb{R}) = \Big\{A = \begin{pmatrix}a & b\\ c & d\end{pmatrix}: \text{det}(A) = ad-bc \neq 0\Big\} \end{equation}

\(2.\) The 2x2 rotation matrices \(\text{SO}(2,\mathbb{R}) \subset GL(2,\mathbb{R})\):

\begin{equation} \text{SO}(2,\mathbb{R}) = \Big\{\begin{pmatrix}\text{cos} (\phi) & -\text{sin} (\phi)\\ \text{sin}(\phi) & \text{cos}(\phi) \end{pmatrix}: \phi \in \mathbb{R} \setminus 2\pi \mathbb{Z}\Big\} \end{equation}

Matrix Lie Groups in general:

If we let \(\text{GL}(2,\mathbb{C})\) represent the group of nxn invertible matrices with entries in \(\mathbb{C}\), by the closed-subgroup theorem any closed subgroup of \(\text{GL}(2,\mathbb{C})\) is a Lie Group.

Discussion:

More can be said on Lie Groups and I think that while the objective of this article was to provide a bird’s eye view, future articles will explore Lie Groups through concrete examples motivated by problems in physics. The objective in the medium term is to demonstrate that Lie Groups allow scientists to classify special functions which arise as the solutions to PDEs in theoretical physics [3].

References:

  1. nLab authors. Lie Group. http://ncatlab.org/nlab/show/Lie%20group. 2021.
  2. John Lee. Introduction to Smooth Manifolds. Springer. 2012.
  3. Willard Miller. The Lie theory approach to special functions. 2010.