Lie Groups and Lie Algebra

Published 2020-12-02

Almost every scientific or engineering discipline deals with ideas of symmetry. Mathematically, symmetry is crafted using Group theory. Simple discrete examples include the possible position of hands of a clock or the manipulations of a Rubik's Cube. Continuous groups can include rotations in 3-D space.

Special cases of continuous groups are the lie groups. Rotations, for example, form a lie group. Unfortunately, lie groups are not a vector space -- namely, we cannot construct a discrete basis and define coordinates. However, the lie algebra of a lie group is a vector space. Here I provide a quick intuition of lie algebras and lie groups.

Unitary group

Consider the Unitary group which consists of $n \times n$ unitary matrices. Let's look at the simple example U(1) which also called the circle group, and consists of complex numbers with absolute value 1. This of course is an infinite group and the identity of this group is just the real number 1.

Every point on the circle, as we know from Euler's equation, can be expressed as $z = e^{i \theta}$. This satisfies the definition of U(1) since $|z| = 1$.

The lie algebra of the group is actually $i\theta$. $i\theta$ are imaginary numbers, where $\theta$ is a real number, $\theta \in \mathcal{R}$. The real numbers over the real numbers form a vector space of dimension 1!

The map between the group and the algebra can be done via $log$ or $exp$:

$$i\theta = log(r) = log(e^{i\theta})$$

And to get back the element of the unitary group $U(1)$:

$$e^{i\theta} = r$$

Now given $i\theta_1$ and $i\theta_2$ we can consider $i\theta = i(\theta_1 + \theta_2)$. Then there a nice mapping to the circle:

$$e^{i\theta} = e^{i(\theta_1 + \theta_2)} = e^{i\theta_1}e^{i\theta_2}$$

In general, this fact is due to the Bake-Campbell-Hausdorff Formula which relates the exponential with commutation relations. And since real numbers commute, the commutator between $\theta_1$ and $\theta_2$ is 0.

Lie Algebra is tangent space at the identity of the Lie Group

Often the lie algebra is described as the tangent space to the identity of the lie group. To see this, let's plot the tangent line to the identity element of U(1), $r = 1$.

Well our lie algebra, which is formed from $i\theta$ where $\theta \in \mathcal{R}$, consists of all complex numbers that have no real component. That is just the imaginary axis in our plane, which is indeed parallel to the tangent!

This property holds true for more complex lie groups. For example, the SU(2) group -- the special unitary group of $2\times 2$ matrices -- has a lie algebra, $su(2)$ which is generated by Pauli matrices, and these indeed live in tangent space at the group's identity.