Chapter 105. Lie Algebras

Lie algebras study infinitesimal symmetry. They arise when one replaces a continuous group of transformations by its tangent structure at the identity.

The subject belongs to algebra, geometry, and analysis at the same time. Algebraically, a Lie algebra is a vector space with a special product called a bracket. Geometrically, it describes infinitesimal motions. Analytically, it appears through differential equations, flows, and continuous transformation groups.

A Lie algebra is a vector space $\mathfrak g$ with a bilinear operation called the Lie bracket, written $[x,y]$ , satisfying alternation and the Jacobi identity. The bracket is generally non-associative. A standard source of examples comes from associative algebras by using the commutator bracket $[X,Y]=XY-YX$ .

105.1 Motivation

Many mathematical objects have symmetries.

A circle has rotations. A vector space has invertible linear transformations. A differential equation may have transformations preserving its solutions. A physical system may have conservation laws associated with continuous symmetry.

Groups describe symmetry globally. Lie algebras describe symmetry infinitesimally.

For example, the rotation group in the plane consists of matrices

R(\theta) = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}.

At $\theta=0$ , this is the identity matrix. The derivative at $\theta=0$ is

A = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}.

This matrix is the infinitesimal generator of rotations. The whole rotation family can be recovered by exponentiation:

R(\theta)=e^{\theta A}.

Lie theory studies this relation between global transformations and infinitesimal generators.

105.2 Definition

Let $F$ be a field.

A Lie algebra over $F$ is a vector space $\mathfrak g$ over $F$ together with a bilinear operation

[\cdot,\cdot] : \mathfrak g \times \mathfrak g \to \mathfrak g

called the Lie bracket, satisfying two identities.

First, the bracket is alternating:

[x,x]=0

for every $x\in\mathfrak g$ .

Second, the bracket satisfies the Jacobi identity:

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

for all $x,y,z\in\mathfrak g$ .

The operation $[x,y]$ measures how two infinitesimal transformations fail to commute.

105.3 Antisymmetry

Over a field of characteristic not equal to $2$ , alternation implies antisymmetry.

Since

[x+y,x+y]=0,

bilinearity gives

[x,x]+[x,y]+[y,x]+[y,y]=0.

The first and last terms vanish, so

[x,y]+[y,x]=0.

Thus

[x,y]=-[y,x].

In most real and complex examples, Lie brackets are therefore antisymmetric.

This means that reversing the order of the inputs reverses the sign.

105.4 The Jacobi Identity

The Jacobi identity is the defining structural law of Lie algebras:

[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0.

It replaces associativity.

A Lie bracket usually does not satisfy

[x,[y,z]]=[[x,y],z].

Instead, the Jacobi identity controls how nested brackets interact.

One useful interpretation is that the operation

\operatorname{ad}_x(y)=[x,y]

acts as a derivation:

\operatorname{ad}_x([y,z]) = [\operatorname{ad}_x(y),z] + [y,\operatorname{ad}_x(z)].

In bracket notation,

[x,[y,z]] = [[x,y],z]+[y,[x,z]].

This form shows that bracketing with $x$ differentiates the bracket.

105.5 First Examples

The zero bracket gives the simplest Lie algebra.

Let $V$ be any vector space. Define

[x,y]=0

for all $x,y\in V$ .

Then $V$ is a Lie algebra. Such a Lie algebra is called abelian.

The bracket contains no interaction between elements.

Another basic example is $\mathbb{R}^3$ with the cross product:

[x,y]=x\times y.

The cross product is bilinear and antisymmetric. It also satisfies the Jacobi identity. Thus $\mathbb{R}^3$ with the cross product is a Lie algebra.

This Lie algebra is closely related to rotations in three-dimensional space.

105.6 Matrix Lie Algebras

Let $M_n(F)$ be the vector space of all $n\times n$ matrices over $F$ .

Define

[A,B]=AB-BA.

This is called the commutator bracket.

It is bilinear because matrix multiplication distributes over addition and scalar multiplication. It is alternating because

[A,A]=AA-AA=0.

It satisfies the Jacobi identity because matrix multiplication is associative. The commutator bracket is the standard way an associative algebra gives rise to a Lie algebra.

The resulting Lie algebra is denoted

\mathfrak{gl}_n(F).

It is the general linear Lie algebra.

105.7 Computing a Commutator

Let

A= \begin{bmatrix} 0 & 1\\ 0 & 0 \end{bmatrix}, \qquad B= \begin{bmatrix} 0 & 0\\ 1 & 0 \end{bmatrix}.

Then

AB= \begin{bmatrix} 1 & 0\\ 0 & 0 \end{bmatrix}, \qquad BA= \begin{bmatrix} 0 & 0\\ 0 & 1 \end{bmatrix}.

Therefore,

[A,B] = AB-BA = \begin{bmatrix} 1 & 0\\ 0 & -1 \end{bmatrix}.

The commutator is zero exactly when $A$ and $B$ commute. Thus the bracket measures noncommutativity.

105.8 The General Linear Lie Algebra

The Lie algebra

\mathfrak{gl}_n(F)

consists of all $n\times n$ matrices over $F$ , with bracket

[A,B]=AB-BA.

Its dimension is

n^2.

It is associated with the general linear group

GL_n(F),

the group of invertible $n\times n$ matrices.

The Lie algebra $\mathfrak{gl}_n(F)$ describes infinitesimal invertible linear transformations.

A path of matrices near the identity has the form

I+tA+O(t^2).

The matrix $A$ is the infinitesimal part.

105.9 The Special Linear Lie Algebra

The special linear Lie algebra is

\mathfrak{sl}_n(F) = \{A\in M_n(F):\operatorname{tr}(A)=0\}.

It is closed under the commutator bracket because

\operatorname{tr}(AB-BA)=\operatorname{tr}(AB)-\operatorname{tr}(BA)=0.

Therefore $\mathfrak{sl}_n(F)$ is a Lie subalgebra of $\mathfrak{gl}_n(F)$ .

Its dimension is

n^2-1.

It corresponds to infinitesimal transformations preserving volume.

105.10 The Orthogonal Lie Algebra

The orthogonal group consists of matrices $Q$ satisfying

Q^TQ=I.

Its Lie algebra consists of matrices $A$ satisfying

A^T+A=0.

This Lie algebra is denoted

\mathfrak{so}_n(F).

Its elements are skew-symmetric matrices.

To see why, consider a curve $Q(t)$ in the orthogonal group with

Q(0)=I, \qquad Q'(0)=A.

Since

Q(t)^TQ(t)=I,

differentiate at $t=0$ :

A^T+A=0.

Thus skew-symmetric matrices are infinitesimal rotations.

105.11 The Three-Dimensional Rotation Algebra

The Lie algebra

\mathfrak{so}_3(\mathbb{R})

consists of all real $3\times 3$ skew-symmetric matrices.

Every such matrix has the form

\begin{bmatrix} 0 & -a_3 & a_2\\ a_3 & 0 & -a_1\\ -a_2 & a_1 & 0 \end{bmatrix}.

This matrix corresponds to the vector

a= \begin{bmatrix} a_1\\ a_2\\ a_3 \end{bmatrix}.

Under this identification, the matrix commutator corresponds to the cross product:

[A(a),A(b)] = A(a\times b).

Thus the cross product Lie algebra on $\mathbb{R}^3$ is another form of $\mathfrak{so}_3(\mathbb{R})$ .

105.12 Lie Subalgebras

A subspace $\mathfrak h\subseteq\mathfrak g$ is a Lie subalgebra if it is closed under the bracket:

[x,y]\in\mathfrak h

for all $x,y\in\mathfrak h$ .

For example,

\mathfrak{sl}_n(F) \subseteq \mathfrak{gl}_n(F)

is a Lie subalgebra.

The skew-symmetric matrices form a Lie subalgebra of all matrices.

Closure under the bracket is essential. A vector subspace alone does not necessarily form a Lie algebra.

105.13 Ideals

An ideal of a Lie algebra $\mathfrak g$ is a subspace $\mathfrak i\subseteq\mathfrak g$ such that

[x,y]\in\mathfrak i

for every $x\in\mathfrak g$ and every $y\in\mathfrak i$ .

Equivalently,

[\mathfrak g,\mathfrak i]\subseteq\mathfrak i.

Ideals are the Lie-algebra analogue of normal subgroups and ring ideals.

If $\mathfrak i$ is an ideal, then the quotient vector space

\mathfrak g/\mathfrak i

inherits a Lie algebra structure.

105.14 Homomorphisms

A Lie algebra homomorphism is a linear map

\phi:\mathfrak g\to\mathfrak h

such that

\phi([x,y])=[\phi(x),\phi(y)]

for all $x,y\in\mathfrak g$ .

It preserves the bracket structure.

The kernel

\ker\phi

is an ideal of $\mathfrak g$ .

The image

\operatorname{im}\phi

is a Lie subalgebra of $\mathfrak h$ .

Thus Lie algebra homomorphisms behave much like homomorphisms in group theory and ring theory.

105.15 The Adjoint Representation

For each $x\in\mathfrak g$ , define a linear map

\operatorname{ad}_x:\mathfrak g\to\mathfrak g

\operatorname{ad}_x(y)=[x,y].

This gives a map

\operatorname{ad}:\mathfrak g\to\mathfrak{gl}(\mathfrak g).

The Jacobi identity implies that $\operatorname{ad}$ is a Lie algebra homomorphism:

\operatorname{ad}_{[x,y]} = [\operatorname{ad}_x,\operatorname{ad}_y].

Here the bracket on the right is the commutator of linear maps.

The adjoint representation is one of the central constructions in Lie theory.

105.16 Center

The center of a Lie algebra is

Z(\mathfrak g) = \{x\in\mathfrak g:[x,y]=0\text{ for all }y\in\mathfrak g\}.

Elements of the center commute with every element under the Lie bracket.

A Lie algebra is abelian exactly when

Z(\mathfrak g)=\mathfrak g.

The center is always an ideal.

It measures the part of the Lie algebra that has no bracket interaction with the rest.

105.17 Derived Algebra

The derived algebra of $\mathfrak g$ is

[\mathfrak g,\mathfrak g] = \operatorname{span}\{[x,y]:x,y\in\mathfrak g\}.

It records all elements obtainable from brackets.

[\mathfrak g,\mathfrak g]=0,

then $\mathfrak g$ is abelian.

The derived algebra is an ideal. It measures the noncommutative part of the Lie algebra.

A Lie algebra whose derived series eventually becomes zero is called solvable.

105.18 Nilpotent Lie Algebras

A Lie algebra is nilpotent if repeated bracketing eventually produces zero.

Define the lower central series by

\mathfrak g_1=\mathfrak g,

and

\mathfrak g_{k+1}=[\mathfrak g,\mathfrak g_k].

\mathfrak g_m=0

for some $m$ , then $\mathfrak g$ is nilpotent.

Nilpotent Lie algebras are important because their structure is close to abelian structure, but still permits nontrivial brackets.

Strictly upper triangular matrices form a standard example.

105.19 Solvable Lie Algebras

A Lie algebra is solvable if its derived series eventually becomes zero.

Define

\mathfrak g^{(0)}=\mathfrak g,

and

\mathfrak g^{(k+1)} = [\mathfrak g^{(k)},\mathfrak g^{(k)}].

\mathfrak g^{(m)}=0

for some $m$ , then $\mathfrak g$ is solvable.

Every nilpotent Lie algebra is solvable, but not every solvable Lie algebra is nilpotent.

Solvable Lie algebras appear in the structure theory of Lie algebras and in the study of differential equations.

105.20 Simple Lie Algebras

A nonzero Lie algebra $\mathfrak g$ is simple if it has no nontrivial ideals and is not abelian.

That means its only ideals are

0 \quad\text{and}\quad \mathfrak g.

Simple Lie algebras are the basic building blocks of semisimple Lie algebras.

Examples include many forms of

\mathfrak{sl}_n, \qquad \mathfrak{so}_n, \qquad \mathfrak{sp}_{2n}.

The classification of finite-dimensional complex simple Lie algebras is one of the major results of modern algebra.

105.21 Structure Constants

Let

e_1,\ldots,e_n

be a basis of $\mathfrak g$ .

Since $[e_i,e_j]\in\mathfrak g$ , there are scalars $c_{ij}^k$ such that

[e_i,e_j] = \sum_{k=1}^n c_{ij}^k e_k.

The scalars

c_{ij}^k

are called the structure constants of the Lie algebra relative to the chosen basis.

Antisymmetry gives

c_{ij}^k=-c_{ji}^k.

The Jacobi identity imposes quadratic equations on the structure constants.

Thus a finite-dimensional Lie algebra can be encoded by a finite table of constants.

105.22 Lie Algebras and Lie Groups

A Lie group is a group that is also a smooth manifold, with smooth multiplication and inversion.

Every Lie group has an associated Lie algebra. It is the tangent space at the identity element:

\mathfrak g = T_eG.

The Lie bracket comes from commutators of left-invariant vector fields.

Thus the Lie algebra records infinitesimal information about the Lie group.

For matrix Lie groups, the Lie algebra often appears as the set of matrices $A$ such that

e^{tA}\in G

for all real $t$ near zero.

105.23 Exponential Map

For matrix Lie algebras, the exponential map is the ordinary matrix exponential:

\exp(A) = I+A+\frac{A^2}{2!}+\frac{A^3}{3!}+\cdots.

It sends Lie algebra elements to Lie group elements.

For example, if $A\in\mathfrak{so}_n(\mathbb{R})$ , then

e^A\in SO_n(\mathbb{R}).

Thus skew-symmetric matrices exponentiate to rotations.

The exponential map connects infinitesimal generators with finite transformations.

105.24 Representations

A representation of a Lie algebra $\mathfrak g$ on a vector space $V$ is a Lie algebra homomorphism

\rho:\mathfrak g\to\mathfrak{gl}(V).

This means

\rho([x,y]) = [\rho(x),\rho(y)].

A representation lets the abstract Lie algebra act as linear transformations on $V$ .

Representation theory studies all possible ways a Lie algebra can act on vector spaces.

This is central in physics, geometry, and algebra.

105.25 Example: $\mathfrak{sl}_2$

The Lie algebra $\mathfrak{sl}_2(F)$ consists of all $2\times 2$ trace-zero matrices.

A standard basis is

e= \begin{bmatrix} 0 & 1\\ 0 & 0 \end{bmatrix}, \qquad f= \begin{bmatrix} 0 & 0\\ 1 & 0 \end{bmatrix}, \qquad h= \begin{bmatrix} 1 & 0\\ 0 & -1 \end{bmatrix}.

The bracket relations are

[h,e]=2e,

[h,f]=-2f,

and

[e,f]=h.

These three relations determine the Lie algebra.

The algebra $\mathfrak{sl}_2$ is the fundamental example in the representation theory of semisimple Lie algebras.

105.26 Applications

Lie algebras appear in many areas.

Area	Role of Lie algebras
Geometry	Infinitesimal symmetries
Differential equations	Flows and vector fields
Physics	Conservation laws and particles
Quantum mechanics	Commutators of observables
Robotics	Rigid-body motion
Control theory	Reachability and controllability
Representation theory	Linear actions of symmetry

The same algebraic structure appears because many systems involve continuous transformations and their infinitesimal generators.

105.27 Summary

A Lie algebra is a vector space with a bracket operation satisfying bilinearity, alternation, and the Jacobi identity.

Concept	Meaning
Lie bracket	Product $[x,y]$ measuring infinitesimal noncommutativity
Abelian Lie algebra	Bracket always zero
Matrix Lie algebra	Lie algebra using $[A,B]=AB-BA$
Lie subalgebra	Subspace closed under bracket
Ideal	Subspace stable under bracketing with all elements
Adjoint map	$\operatorname{ad}_x(y)=[x,y]$
Lie group relation	Lie algebra as tangent space at identity

Lie algebras provide the linearized language of symmetry. They reduce continuous transformation problems to algebraic problems involving brackets, subalgebras, ideals, representations, and infinitesimal generators.