Blog post: ajb: A first look at Lie theory

A friend of mine made a request...

As your friendly neighbourhood mathematician I will try to oblige.

 

Disclaimer What I do is give an informal overview and not worry too much about details and proper proofs. Proofs you can find in textbooks. Rather I want to present the ideas and sketch some constructions.

 

I will build this account up over the period of a few weeks.

 

Rough Plan

The things I would like to cover are the following.

Abstract Lie algebras

Lie groups

The Lie algebra of a Lie group

Lie's theorems

Some odds and ends (Maybe a few words about Lie groupoids etc)

There maybe some changes here as the work develops.

 

I will also include some simple exercises for those that are interested. I will post solutions at the end.

 

Part 0: Introduction

Anybody who reads anything about modern physics will encounter the terms 'Lie group' and 'Lie algebra'. Lie theory is all about the relation between these two structures.

 

A Lie group is a group that also has a smooth manifold structure, importantly the group operations are compatible with this smooth structure. Groups represent transformations and symmetries of mathematical objects. Lie groups are the mathematical framework for studying continuous symmetries of mathematical objects. Thus, Lie groups are fundamental in geometry and theoretical physics.

 

Now, every Lie group has associated with it a Lie algebra, whose vector space structure is the tangent space of the Lie group at the identity element. The Lie algebra describes the local structure of the group. Informally one can think of the Lie algebra as describing the elements of the Lie group that are 'very close to the identity element'.

 

The theory of Lie groups and Lie algebras was initiated by Sophus Lie, and hence the nomenclature. Lie's motivation was to extend Galois theory, which proved useful in the study of algebraic equations, to cope with continuous symmetries of differential equations. Lie laid down much of the basic theory of continuous symmetry groups.

 

The plan is with these notes is to sketch the relation between Lie groups and Lie algebras. I will stick to the finite dimensional case for this first look.

 

Part I: Abstract Lie algebras

Let us start with a completely algebraic set-up. Informally, a Lie algebra is a vector space with a non-associative product, known as a 'bracket' that satisfies some nice properties. We will only consider algebras over the reals or complex here, though everything will generalise to more arbitrary fields (with some minor modifications if necessary).

 

Definition

A Lie algebra is a vector space $latex mathfrak{g}$ together with a bilinear operation $latex [bullet,bullet]: mathfrak{g} times mathfrak{g} rightarrow mathfrak{g}$, that satisfies the following conditions

Skewsymmetry

$latex [x,y] = -[y,x]$

Jacobi identity

$latex [x,[y,z]] + [z,[x,y]] +[y,[z,x]]=0$

for all $latex x,y, z in mathfrak{g}$.

 

Note that Lie algebras are non-associative. Thinking of the bracket as a form of multiplication we the Jacobi identity is related to the 'associator' which is non-zero in general

 

$latex [x,[y,z]] -[[x,y],z]= [x,[y,z]] + [z,[x,y]] = [[z,x],y] neq 0$.

 

The dimension of a Lie algebra is defined to be the dimension of the underlying vector space. Elements of a Lie algebra are said to generate that Lie algebra if they form the smallest subalgebra that contains these elements is the Lie algebra itself.

 

Example Any vector space equipped with a vanishing bracket $latex [x,y]=0$, is a Lie algebra. We call any Lie algebra with a vanishing bracket an abelian Lie algebra.

 

Example The (real) vector space of all n×n skew-hermitian matrices together with the standard commutator is Lie algebra. This Lie algebra is denoted $latex mathfrak{u}(n)$.

 

Example The Heisenberg algebra is the Lie algebra generated by three elements x,y,z and the Lie brackets are defined as

$latex [x,y] =z$, $latex [x,z] =0$ and $latex [y,z] =0$.

 

Given a set of generators $latex {T_{a}}$ we can define the Lie algebra in terms of its structure constants. As the Lie bracket of any pair of generators must be a linear combination of the generators we have

 

$latex [T_{a}, T_{b}] = C^{c}_{ab}: T_{c}$,

 

and so the Lie algebra is determined by the structure constants $latexC^{c}_{ab}$.

 

Exercise How many one dimensional Lie algebras are there up to isomorphisms?

 

Exercise There are exactly two Lie algebras of dimension two over the real numbers, up to isomorphism. Can you write these down in terms of generators?

 

Exercise What conditions do the structure constants need to satisfy in order to have a Lie algebra? (Hint: think about the two defining conditions of a Lie algebra)

 

 

People study Lie algebras in their own right, but historically they arose from the study of Lie groups. From my own perspective, it is the fact that Lie algebras are 'infinitesimal Lie groups' that makes them interesting and useful. In the next section I will move on to groups and in particular Lie groups.

 

Part II: Lie groups

To be continued...
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