Lie Algebras Explained: Commutators, SU(2), SO(3), Euc(n)

A Lie algebra is the vector space of infinitesimal generators attached to a Lie group at its identity element, equipped with a bilinear antisymmetric commutator operation. Lie algebras are typically easier to analyze than the Lie groups that generate them, and much of Lie group representation theory is built directly from the algebra rather than the group itself.

Key Takeaways

• A Lie algebra is closed under the commutator operation: the commutator of any two basis elements is a linear combination of basis elements.
• SU(2) and SO(3) have isomorphic Lie algebras, both satisfying [L_i, L_j] = i ε_{ijk} L_k, yet the two groups themselves are not isomorphic.
• SU(2) double-covers SO(3): SU(2) contains a central subgroup {I, −I} isomorphic to Z2, and SO(3) ≅ SU(2)/Z2.
• The Euclidean algebra Euc(n) splits as a semidirect sum T^n ⋊ so(n), where T^n is the translation algebra and so(n) is the rotation algebra.
• For n ≥ 3, so(n) is semisimple, making T^n the radical of Euc(n) and so(n) a Levi subalgebra.
• Every nilpotent Lie algebra is solvable, but the converse does not hold — Euc(2) is a solvable algebra that is not nilpotent.

What Is a Lie Algebra?

A Lie algebra (pronounced “Lee”) is the tangent-space algebra of a Lie group at its identity element. Concretely, it is a vector space of generators that describes infinitesimal transformations near the identity. Lie algebras carry a binary bilinear antisymmetric operation called the commutator, and they are closed under that operation: the commutator of two basis elements is a linear combination of the basis elements.

Lie algebras are often easier to study than the full Lie groups they generate. Much of the theory of Lie groups, especially representation theory, is developed via their Lie algebras. However, some global group properties do not follow from the algebra alone. Non-isomorphic Lie groups can have isomorphic Lie algebras, as demonstrated below by SO(3) and SU(2).

Basic Equations Governing Lie Algebras

Commutator

For matrices, the commutator is defined as:

[A,B] = AB - BA

For linear operators acting on a vector X, the commutator is defined as:

[A,B](X) = A(B(X)) - B(A(X))

Jacobi Identity

The commutator satisfies the Jacobi identity:

[A,[B,C]] + [B,[C,A]] + [C,[A,B]] = 0.

Structure Constants

With a basis L_i for the algebra, the commutator is written as:

[L_i, L_j] = f_{ij}{}^{k} L_k,

using Einstein summation on repeated indices. The numbers f_{ij}{}^{k} are the structure constants of the algebra. From antisymmetry of the commutator and the Jacobi identity, the structure constants must satisfy these two constraints:

f_{ji}{}^{k} = - f_{ij}{}^{k}

f_{ij}{}^{a} f_{ak}{}^{b} + f_{jk}{}^{a} f_{ai}{}^{b} + f_{ki}{}^{a} f_{aj}{}^{b} = 0.

Bilinearity

For linear combinations a^i L_i and b^j L_j, bilinearity of the commutator gives:

[a^i L_i, b^j L_j] = a^i b^j [L_i, L_j] = a^i b^j f_{ij}{}^{k} L_k.

Invariant Form / Killing Form

One can form an invariant bilinear form directly from the structure constants. A common choice is the Killing form (conventions vary between texts):

g_{ij} = f_{i a}{}^{b} f_{j b}{}^{a}.

For a semisimple Lie algebra, the Killing form is nondegenerate, meaning it is invertible.

How Do SU(2) and SO(3) Compare as an Example?

Quaternion Parametrization

Both SU(2) and SO(3) can be parametrized using unit quaternions:

q = (q_0, q_1, q_2, q_3), with q_0^2 + q_1^2 + q_2^2 + q_3^2 = 1.

Group Representations (Matrices)

SU(2) can be expressed in terms of the Pauli matrices:

D(q) = q_0 I_2 + q_1 σ_1 + q_2 σ_2 + q_3 σ_3
      = \begin{pmatrix} q_0 + i q_3 & i q_1 + q_2 \\
                        i q_1 - q_2 & q_0 - i q_3 \end{pmatrix}

SO(3) can be expressed as a 3×3 rotation matrix built from the same quaternion components:

D(q) = \begin{pmatrix}
 q_0^2 + q_1^2 - q_2^2 - q_3^2 & 2 q_1 q_2 + 2 q_0 q_3 & 2 q_1 q_3 - 2 q_0 q_2 \\
 2 q_1 q_2 - 2 q_0 q_3 & q_0^2 - q_1^2 + q_2^2 - q_3^2 & 2 q_2 q_3 + 2 q_0 q_1 \\
 2 q_1 q_3 + 2 q_0 q_2 & 2 q_2 q_3 - 2 q_0 q_1 & q_0^2 - q_1^2 - q_2^2 + q_3^2
\end{pmatrix}

Algebras at the Identity

Taking the derivative at the identity quaternion q = (1,0,0,0) yields the Lie algebra generators for each group.

SU(2) generators, using one convenient basis:

L_1 = 1/2 \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix},
L_2 = 1/2 \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix},
L_3 = 1/2 \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.

SO(3) generators, expressed as antisymmetric 3×3 matrices:

L_1 = -i \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \end{pmatrix},
L_2 = -i \begin{pmatrix} 0 & 0 & -1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix},
L_3 = -i \begin{pmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}.

Both algebras satisfy the same commutation relation:

[L_i, L_j] = i ε_{ijk} L_k,

so the two Lie algebras are isomorphic. The Lie groups SU(2) and SO(3) themselves are not isomorphic, because SU(2) double-covers SO(3): D(-q) = -D(q) holds in SU(2), whereas D(-q) = D(q) holds in SO(3). Equivalently, SU(2) has a central subgroup {I, −I} isomorphic to Z2, and SO(3) ≅ SU(2)/Z2.

What Are Series, Ideals, Solvability, and Nilpotency?

Commutator series from group theory have direct analogues for Lie algebras, used to classify the internal structure of a given algebra G.

Derived Series and Solvability

The derived series of a Lie algebra G is defined by:

G^{(0)} = G,

G^{(n)} = [G^{(n-1)}, G^{(n-1)}].

If the derived series reaches the zero algebra after finitely many steps, the Lie algebra G is called solvable.

Lower Central Series and Nilpotency

The lower central series of a Lie algebra G is defined by:

G_0 = G,

G_n = [G_{n-1}, G].

If the lower central series reaches the zero algebra after finitely many steps, the Lie algebra G is called nilpotent. Every nilpotent Lie algebra is solvable, but not every solvable Lie algebra is nilpotent.

Ideals and Radicals

A subalgebra J of a Lie algebra G is called an ideal if [G,J] ⊂ J. The Lie group generated by an ideal J corresponds to a normal subgroup of the full Lie group. Every finite-dimensional Lie algebra G has a unique maximal solvable ideal, called the radical of G.

Semisimple and Simple Algebras; the Levi Decomposition

If the radical of a Lie algebra G is zero, G is called semisimple. If G has no nontrivial ideals other than {0} and G itself, G is called simple. A semisimple Lie algebra decomposes as a direct sum of simple algebras. The Levi decomposition states that any finite-dimensional Lie algebra equals the semidirect sum of its radical and a semisimple subalgebra, known as a Levi subalgebra.

How Does the Euclidean Algebra Euc(n) Illustrate These Ideas?

Euc(n) denotes the Lie algebra of the Euclidean group in n dimensions, generated by rotations and translations. Euc(n) splits as a semidirect sum of the translation algebra T^n and the rotation algebra so(n):

Euc(n) = T^n ⋊ so(n).

The translation algebra T^n is abelian, and therefore nilpotent, in every dimension. The rotation algebra so(n) behaves differently depending on n:

Behavior of the rotation algebra so(n) by dimension n

Dimension n	Structure of so(n)
n = 1	so(1) is trivial (the zero algebra)
n = 2	so(2) is one-dimensional and abelian, hence nilpotent
n ≥ 3	so(n) is semisimple

These results determine the structure of Euc(n) itself. Euc(1) is nilpotent, since its radical is the whole algebra. Euc(2) is solvable but not nilpotent, and its lower central series stabilizes on the translation subalgebra T^2. For n ≥ 3, the radical of Euc(n) is the translation algebra T^n, and a Levi subalgebra of Euc(n) is so(n).

Glossary

• Lie algebra — the vector space of infinitesimal generators of a Lie group at its identity element, closed under the commutator operation.
• Commutator — the bilinear antisymmetric operation [A,B] = AB − BA that defines the algebra’s structure.
• Structure constants — the coefficients f_{ij}^k expressing the commutator of basis elements as a linear combination of the basis.
• Killing form — an invariant bilinear form built from structure constants, nondegenerate for semisimple algebras.
• Solvable algebra — a Lie algebra whose derived series reaches the zero algebra in finitely many steps.
• Nilpotent algebra — a Lie algebra whose lower central series reaches the zero algebra in finitely many steps.
• Radical — the unique maximal solvable ideal of a finite-dimensional Lie algebra.
• Semisimple algebra — a Lie algebra whose radical is zero, decomposable as a direct sum of simple algebras.
• Levi decomposition — the theorem stating any finite-dimensional Lie algebra is the semidirect sum of its radical and a semisimple Levi subalgebra.

Frequently Asked Questions

Why do SU(2) and SO(3) have the same Lie algebra but different groups?

SU(2) and SO(3) share the commutation relation [L_i, L_j] = i ε_{ijk} L_k, making their Lie algebras isomorphic. The groups differ because SU(2) double-covers SO(3): SU(2) has a central subgroup {I, −I} isomorphic to Z2, and SO(3) ≅ SU(2)/Z2, so quaternion q and −q map to the same SO(3) rotation but distinct SU(2) elements.

Is every nilpotent Lie algebra also solvable?

Yes. Every nilpotent Lie algebra is solvable, since a vanishing lower central series implies a vanishing derived series. The converse is false: solvable algebras such as Euc(2) are not necessarily nilpotent.

What is the radical of a Lie algebra?

The radical is the unique maximal solvable ideal contained within a finite-dimensional Lie algebra G. If the radical is the zero algebra, G is called semisimple; if the radical equals all of G, G is itself solvable.

How does the Euclidean algebra Euc(n) split for n ≥ 3?

For n ≥ 3, Euc(n) splits as T^n ⋊ so(n), where the translation algebra T^n forms the radical and the rotation algebra so(n), which is semisimple in this range, serves as a Levi subalgebra.

What is the difference between a simple and a semisimple Lie algebra?

A simple Lie algebra has no nontrivial ideals other than {0} and itself. A semisimple Lie algebra has a zero radical and decomposes as a direct sum of simple algebras, so every simple algebra is semisimple but not every semisimple algebra is simple.