Lie Algebras

A Lie algebra is the infinitesimal algebra of a Lie group. It records the first-order generators of continuous transformations and the commutator structure that measures how infinitesimal transformations fail to commute.

In quantum mechanics, Lie algebras appear as:

• angular-momentum commutators;
• generators of translations, rotations, and time evolution;
• Pauli-matrix and spin algebras;
• selection-rule machinery;
• the local algebra behind continuous symmetry groups.

The Lie group is the global smooth symmetry object. The Lie algebra is the linearized object near the identity.

Abstract Definition

A Lie algebra over a field such as $\mathbb R$ or $\mathbb C$ is a vector space $\mathfrak g$ with a bilinear bracket

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

satisfying antisymmetry

$$[X,Y]=-[Y,X],$$

and the Jacobi identity

$$[X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]] = 0$$

for all $X,Y,Z\in\mathfrak g$.

The bracket is not ordinary multiplication. It is a new operation that captures infinitesimal noncommutativity.

Matrix Lie Algebras

For matrices, the standard bracket is the commutator:

$$[X,Y] = XY-YX.$$

This bracket is bilinear, antisymmetric, and satisfies the Jacobi identity.

Matrix Lie algebras are the main examples in quantum mechanics. If $G$ is a matrix Lie group, its Lie algebra consists of matrices $X$ for which

$$g(t) = I+tX+O(t^2)$$

lies in $G$ to first order. Equivalently, $X$ is a tangent vector at the identity.

From Lie Group to Lie Algebra

For a Lie group $G$, the Lie algebra is the tangent space at the identity:

$$\mathfrak g = T_eG.$$

The vector-space structure comes from the tangent space. The bracket comes from the way infinitesimal transformations fail to commute. For matrix groups, that bracket reduces to the matrix commutator.

The local picture is:

$$X\in\mathfrak g \quad\Longrightarrow\quad \exp(tX)\in G$$

for small real $t$.

The exponential map converts an infinitesimal direction into a one-parameter subgroup:

$$g_X(t)=\exp(tX).$$

Physics Generator Convention

Mathematics and physics often use different generator conventions.

For the unitary group $U(n)$, the mathematical Lie algebra is

$$\mathfrak u(n) = \{X:X^\dagger=-X\},$$

the anti-Hermitian matrices. A one-parameter subgroup is

$$U(t)=\exp(tX).$$

Physics usually writes the same subgroup using a Hermitian generator $G$:

$$U(t) = \exp \left( -\frac{i}{\hbar}tG \right).$$

The relation is

$$X=-\frac{i}{\hbar}G.$$

Both conventions describe the same transformation. The difference is where the factor of $i$ and the physical unit $\hbar$ are placed.

Structure Constants

Choose a basis $\{T_a\}$ of a finite-dimensional Lie algebra. The bracket of basis elements can be expanded as

$$[T_a,T_b] = \sum_c f_{ab}^{\ \ c}T_c.$$

The numbers $f_{ab}^{\ \ c}$ are the structure constants in that basis.

With Hermitian physics generators, it is common to write

$$[T_a,T_b] = i\sum_c f_{ab}^{\ \ c}T_c,$$

with real $f_{ab}^{\ \ c}$ for compact unitary examples. The extra factor of $i$ reflects the Hermitian-generator convention.

Structure constants depend on the chosen basis, but the Lie algebra structure does not.

Abelian Lie Algebras

A Lie algebra is abelian if every bracket vanishes:

$$[X,Y]=0$$

for all $X,Y\in\mathfrak g$.

Examples:

• the Lie algebra of $(\mathbb R^d,+)$ is abelian;
• the Lie algebra of $U(1)$ is one-dimensional and abelian;
• commuting translation generators satisfy zero brackets.

Abelian Lie algebras describe continuous symmetries whose infinitesimal transformations commute.

The Lie Algebra of U(1)

The group $U(1)$ consists of phases $e^{i\theta}$. Mathematically, its Lie algebra is

$$\mathfrak u(1) = i\mathbb R.$$

A typical element is $i\alpha$ with $\alpha\in\mathbb R$, and

$$\exp(i\alpha) \in U(1).$$

Since the algebra is one-dimensional, all brackets vanish. In physics, one often uses a Hermitian charge generator $Q$ and writes

$$U(\theta) = e^{-i\theta Q}.$$

The sign and normalization depend on the convention for the charge.

The Lie Algebra of SO(3)

The Lie algebra of $SO(3)$ is denoted

$$\mathfrak{so}(3).$$

It consists of real antisymmetric $3\times3$ matrices:

$$X^T=-X.$$

As a real vector space, it is three-dimensional. Its bracket is the matrix commutator.

In physics, rotations are usually represented by angular momentum generators $J_i$ satisfying

$$[J_i,J_j] = i\hbar\sum_k\epsilon_{ijk}J_k.$$

The factors of $i$ and $\hbar$ come from the Hermitian operator convention.

The Lie Algebra of SU(2)

The mathematical Lie algebra $\mathfrak{su}(2)$ consists of traceless anti-Hermitian $2\times2$ matrices:

$$X^\dagger=-X, \qquad \operatorname{Tr}X=0.$$

In the Hermitian physics convention, a standard basis is

$$J_i=\frac12\sigma_i,$$

where $\sigma_i$ are the Pauli matrices. These obey

$$[J_i,J_j] = i\sum_k\epsilon_{ijk}J_k.$$

With physical spin operators

$$S_i=\frac{\hbar}{2}\sigma_i,$$

the same algebra becomes

$$[S_i,S_j] = i\hbar\sum_k\epsilon_{ijk}S_k.$$

This is the algebraic heart of spin-$1/2$ and angular momentum.

Lie Algebra Representations

A representation of a Lie algebra assigns a linear operator to each algebra element while preserving brackets. If $\rho$ is a representation, then

$$\rho([X,Y]) = [\rho(X),\rho(Y)].$$

For a Lie group representation $R(g)$, differentiating near the identity gives a Lie algebra representation. Conversely, a Lie algebra representation may or may not integrate to a global group representation, depending on topology and domain issues.

This is why the distinction between $SO(3)$ and $SU(2)$ matters. They have closely related Lie algebras, but their global group structures differ, and quantum spinors detect that difference. The global comparison is developed in SU(2) versus SO(3).

Why Lie Algebras Are Useful

Lie algebras are useful because they turn continuous symmetry into linear algebra.

They let one:

• compute commutators of generators;
• classify finite-dimensional angular-momentum multiplets;
• build ladder operators;
• derive selection-rule constraints;
• identify conserved quantities from continuous symmetry;
• work locally near the identity without parameterizing the entire group.

The tradeoff is that Lie algebras do not always remember global topology. They see infinitesimal structure, not every global feature of the Lie group.

Common Mistakes

• Confusing a Lie group with its Lie algebra.
• Forgetting that $\mathfrak u(n)$ is anti-Hermitian in the mathematics convention.
• Dropping the factor of $i$ when using Hermitian physics generators.
• Treating the exponential map as globally one-to-one.
• Assuming identical Lie algebras imply identical global groups.
• Ignoring $\hbar$ in angular-momentum commutators when units are not set to one.
• Thinking the bracket is ordinary multiplication.
• Forgetting that a Lie algebra representation may have global integrability conditions.

Cross-Links

• Lie Groups
• Groups
• Group Actions
• Representations
• SO(3)
• SU(2) versus SO(3)
• Heisenberg Group
• Commutators and Anticommutators
• Matrix Functions and Exponentials
• Pauli Matrices
• SU(2)
• Angular Momentum Algebra
• Ladder Operators as Lie Algebra Tools
• Generators
• Commutators and Conservation Laws
• Spin Rotations

References

• B. C. Hall, Lie Groups, Lie Algebras, and Representations, 2nd ed., Springer, 2015.
• H. Georgi, Lie Algebras in Particle Physics, 2nd ed., Westview Press, 1999.
• J. F. Cornwell, Group Theory in Physics, Vol. 1, Academic Press, 1984.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.

Exercises

1. Show that the matrix commutator is antisymmetric.

Solution

By definition,

$$[X,Y]=XY-YX.$$

Then

$$[Y,X]=YX-XY=-(XY-YX)=-[X,Y].$$

2. Let $J_i=\sigma_i/2$. Use $[\sigma_i,\sigma_j]=2i\sum_k\epsilon_{ijk}\sigma_k$ to compute $[J_i,J_j]$.

Solution

Compute

$$[J_i,J_j] = \frac14[\sigma_i,\sigma_j] = \frac{i}{2} \sum_k\epsilon_{ijk}\sigma_k.$$

Since $J_k=\sigma_k/2$,

$$[J_i,J_j] = i\sum_k\epsilon_{ijk}J_k.$$

3. Why is the Lie algebra of $U(1)$ abelian?

Solution

It is one-dimensional. Any two elements are scalar multiples of the same basis element, so their bracket is bilinear and antisymmetric:

$$[aX,bX] = ab[X,X] = 0.$$

Thus every bracket vanishes.

4. Explain the relation between an anti-Hermitian mathematical generator $X$ and a Hermitian physics generator $G$ in $U(t)=\exp(-itG/\hbar)$.

Solution

The mathematical one-parameter subgroup is $U(t)=\exp(tX)$ with $X^\dagger=-X$. Comparing with

$$U(t) = \exp \left( -\frac{i}{\hbar}tG \right)$$

gives

$$X=-\frac{i}{\hbar}G.$$

If $G$ is Hermitian, then $X$ is anti-Hermitian.

5. Why can two groups with the same Lie algebra still have different quantum behavior?

Solution

The Lie algebra captures local infinitesimal structure near the identity, but it does not fully determine global topology. The groups $SU(2)$ and $SO(3)$ have closely related Lie algebras, yet $SU(2)$ is a double cover of $SO(3)$. Spinor representations detect this global difference through the sign change under a $2\pi$ rotation.