Lie Algebra Basics: Definitions, Equations & Examples

Definition / Summary

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 (for example, SO(3) and SU(2)).

Basic equations

Commutator

For matrices, the commutator is

[A,B] = AB - BA

For linear operators acting on a vector X,

[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

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

using Einstein summation on repeated indices. The numbers f_{ij}{}^{k} are the structure constants. From antisymmetry of the commutator and the Jacobi identity we obtain the 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 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 from structure constants. A common choice is the Killing form (up to conventions):

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

For a semisimple Lie algebra this form is nondegenerate (invertible).

Example: SU(2) and SO(3)

Quaternion parametrization

Both SU(2) and SO(3) can be parametrized by 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) in terms of 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) as a 3×3 rotation matrix built from 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:

SU(2) generators (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 (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

[L_i, L_j] = i ε_{ijk} L_k,

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

Series, ideals, solvability and nilpotency

Commutator series from group theory have direct analogues for Lie algebras.

Derived series (solvability)

Define

G^{(0)} = G,

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

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

Lower central series (nilpotency)

Define

G_0 = G,

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

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

Ideals and radicals

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

Semisimple and simple algebras; Levi decomposition

If the radical of G is zero, G is semisimple. If G has no nontrivial ideals except {0} and G itself, it is simple. A semisimple 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 (a Levi subalgebra).

Example: Euclidean algebra Euc(n)

Let Euc(n) denote the Lie algebra of the Euclidean group in n dimensions (generators of 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).

Here T^n is abelian (hence nilpotent). The algebra so(n) has the following behaviour:

• n = 1: so(1) is trivial (zero algebra).
• n = 2: so(2) is one-dimensional and abelian (hence nilpotent).
• n ≥ 3: so(n) is semisimple.

Therefore:

• Euc(1) is nilpotent (its radical is itself).
• Euc(2) is solvable but not nilpotent; its lower central series stabilizes on the translations T^2.
• For n ≥ 3, the radical of Euc(n) is T^n and a Levi subalgebra is so(n).

See also / References

• Lie algebras — a walkthrough (Physics Forums Insights) (3 parts)
• Journey: SU(2) and manifolds (Physics Forums Insights) (2 parts)
• Comment thread: “What is Lie algebra?”