Lie group and Lie algebra

Jundong LiuOhio University

References

• Wiki– Lie group – Homogenous space – Left-invariant fields– One parameter groups

• Everything 2 “lie group”• http://physics.stackexchange.com/questions/133758/

intuitive-meaning-of-the-exponential-form-of-an-unitary-operator-in-quantum-mech

• Book – Introduction to smooth manifolds

Lie theory: an introduction

• Quite possibly the most beautiful marriage of concepts in all of mathematics, a lie group is an object endowed with rich structure and great symmetry.

• It is also of great use to physicists in all fields, due to its intimate connection with physical symmetries.

• Understanding the great power and beauty of lie groups requires some mathematical rigor. However, the basic idea is not too difficult to grasp.

Some background

• Algebra/ring: * and +

• Vector fields: can be regarded as the directives/velocity or distance field (or coordinates).

Lie Group: groups & manifolds

• Group:

• Manifold:– Locally Euclidean

Lie group: smooth manifold + group

• A Lie group is a manifold that is also a group such that the group operations are smooth.

Lie group: properties

• A Lie group is a homogeneous space in the sense that left translation by a group element g is a diffeomorphism of the group onto itself that maps the identity element to g.

• Therefore, locally the group looks the same around any point. – To study the local structure of a Lie group, it is enough

to examine a neighborhood of the identity element. – It is not surprising that the tangent space at the

identity of a Lie group should play a key role.

Lie groups: a few examples

• (R^n, +) are lie groups

Example subgroups of GL(n, R)

Example subgroups of GL(n, C)

The tangent space of a Lie group & Lie algebra

• If you think about U(1) as the unit circle in C, you can see this tangent space very explicitly: it's the line tangent to the unit circle at 1, which can be naturally identified with the "imaginary line" iR.

Left-invariant vector field

• Now, here's a cool fact about Lie groups: if you pick a velocity at the identity element, it naturally extends to a velocity field defined everywhere in the group! Here's how.– If you're walking along a Lie group G, and you find yourself

getting lonely, you can imagine you have a twin walking a path which is just like yours, but "translated" by an element g of G: when you're at h, your twin is at gh.

• A velocity field that comes from a bunch of people walking in lockstep like this is called a left-invariant vector field. Using this vocabulary, we can restate the cool fact from before by saying that...

Left-invariant vector field & homogenous space

… this is called a left-invariant vector field. Using this vocabulary, we can restate the cool fact from before by saying that...• In a Lie group, picking a velocity at the

identity is the same as picking a left-invariant vector field.

• For a concrete example, let's see how a velocity at the identity of U(1) extends to a velocity field defined everywhere on U(1):

Lie group exponential• Now, let's say you have an element v of the Lie algebra u(1), which

we just saw is the same as having a left-invariant vector field on U(1).

• If you're standing at the identity in U(1), and you're bored, you can entertain yourself by trying to walk so that your velocity always matches the velocity field v. It turns out there's always exactly one way to do this, so if you do it for time t, you'll always end up at the same place.

• This group element is called exptv. You can think of expt as a function that turns Lie algebra elements into Lie group elements; it's called the Lie group exponential. Writing an element a of U(1) in exponential form means finding a left-invariant vector field v∈u(1) that will take you to a in one unit of time, so a= exp1v.

For U(1), we can solve exptv

• with v as the velocity at the identity of U(1), the velocity of z C is ∈ vz

• telling people at z to move with velocity vz

An ODE to solve (easy)

This U(1) example and one-parameter (sub)groups

• One-parameter subgroup: – definition is abstract and complicated. – Example: the equator of earth, while the

parameter is the latitude• Every element of the associated Lie algebra

defines (an) exponential map. – In the case of matrix groups, it is given by the

matrix exponential

From the group to the algebra

• Through Taylor expansion

• How about V (or matrix A)?

Examples of Lie group Lie algebra

Examples of Lie group Lie algebra

Lie group Lie algebra

How about the Lie algebra for…

• GL+(n, R)?• GL+(n, C)?• SGL+(n, R)?• SGL+(n, C)?

• Zhewei, figure this out.