Lie Algebras: Brackets and Structure Constants

The Lie algebra is the infinitesimal shadow of a Lie group: a vector space equipped with a bracket that encodes, to first order, the group's non-commutativity. Almost every computation in a continuous-symmetry theory — commutation relations, conserved charges, Casimir labels — happens in the algebra rather than the group. This page develops the algebraic structure that lie-groups.md attached to $T_{e}G$, and sets up the classification in lie-classification.md and the representation theory in reps-lie.md.

The Lie bracket

A Lie algebra $\mathfrak{g}$ over a field $\mathbb{F}$ is a vector space with a bilinear bracket $[\cdot ,\cdot ]:\mathfrak{g}\times \mathfrak{g}\to \mathfrak{g}$ that is

• antisymmetric: $[X,Y]=-[Y,X]$, and
• satisfies the Jacobi identity $$[X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]=0.$$

For matrix Lie algebras (the only kind we use) the bracket is the matrix commutator $[X,Y]=XY-YX$; antisymmetry is obvious and the Jacobi identity is a one-line computation. The bracket is the linearisation of the group commutator $gh{g}^{-1}{h}^{-1}$: expanding $\exp (sX)\exp (tY)\exp (-sX)\exp (-tY)=\exp (st[X,Y]+\cdots )$ shows $[X,Y]$ measures the leading failure of $X$ and $Y$ to commute in the group. (This is the infinitesimal content of the Baker–Campbell–Hausdorff series — see lie-groups.md.)

An abelian Lie algebra has $[X,Y]=0$ for all elements — the algebra of a commutative group like ${\mathbb{R}}^n$ or $U(1)$.

Generators and structure constants

Choose a basis $\{T^a{\}}_{a=1}^{\dim \mathfrak{g}}$ of $\mathfrak{g}$: a linearly independent spanning set, so every $X\in \mathfrak{g}$ has a unique expansion $X=X^a{T}^a$. The basis elements are the generators. Near the identity every group element is $g=\exp (i{\theta }^a{T}^a)$, so the $T^a$ "generate" the group via the exponential map. The bracket is fixed by its values on the basis, encoded in the structure constants $f^{abc}$: $$[T^a,T^b]=i{f}^{abc}{T}^c.$$

The factor of $i$ is the physics convention (mathematicians omit it): with it, Hermitian generators $T^a=(T^a{)}^{†}$ produce unitary group elements $\exp (i{\theta }^a{T}^a)$. The structure constants are

• antisymmetric in $a,b$ (from bracket antisymmetry), and
• constrained by a Jacobi identity $f^{abe}{f}^{ecd}+f^{bce}{f}^{ead}+f^{cae}{f}^{ebd}=0$ inherited from the algebra's Jacobi identity.

They determine the entire group structure in a neighbourhood of the identity. Example: $\mathfrak{su}(2)$ has $f^{abc}={ϵ}^{abc}$ (the Levi-Civita symbol); see examples-classical.md.

Subalgebras, ideals, and the derived series

• A subalgebra $\mathfrak{h}\subseteq \mathfrak{g}$ is a subspace closed under the bracket.
• An ideal $\mathfrak{i}$ satisfies $[\mathfrak{g},\mathfrak{i}]\subseteq \mathfrak{i}$ — the algebra analogue of a normal subgroup, and exactly the kernel of a Lie-algebra homomorphism. Quotients $\mathfrak{g}/\mathfrak{i}$ are defined just as for groups.
• $\mathfrak{g}$ is simple if it is non-abelian with no proper non-zero ideals, and semisimple if it is a direct sum of simple algebras — equivalently (Cartan) if its Killing form is non-degenerate.

These notions mirror the finite-group hierarchy of series-solvable.md: the derived series $\mathfrak{g}\supseteq [\mathfrak{g},\mathfrak{g}]\supseteq [[\mathfrak{g},\mathfrak{g}],[\mathfrak{g},\mathfrak{g}]]\supseteq \cdots$ terminating at $0$ defines a solvable algebra, and the semisimple algebras are the "opposite" extreme, classified in lie-classification.md.

The adjoint representation and the Killing form

Every Lie algebra acts on itself by the bracket: the adjoint representation $${\mathrm{ad}}_X:\mathfrak{g}\to \mathfrak{g},{\mathrm{ad}}_X(Y)=[X,Y].$$ The Jacobi identity is exactly the statement that $\mathrm{ad}$ is a homomorphism: ${\mathrm{ad}}_{[X,Y]}=[{\mathrm{ad}}_X,{\mathrm{ad}}_Y]$. In a basis, $({\mathrm{ad}}_{T^a}{)}_{cb}=i{f}^{acb}$ — the structure constants are the adjoint matrices. The corresponding group representation is ${\mathrm{Ad}}_g(Y)=gY{g}^{-1}$; for gauge theory this is the representation the gauge fields themselves live in.

The Killing form is the symmetric bilinear form $$K(X,Y)=\mathrm{tr}\left({\mathrm{ad}}_X{\mathrm{ad}}_Y\right).$$ It is invariant ($K([X,Y],Z)=K(X,[Y,Z])$) and its (non-)degeneracy diagnoses structure: Cartan's criterion says $\mathfrak{g}$ is semisimple iff $K$ is non-degenerate, and solvable iff $K(\mathfrak{g},[\mathfrak{g},\mathfrak{g}])=0$. The Killing form provides the invariant "metric" used to raise/lower algebra indices and to build the quadratic Casimir below.

Casimir operators

A Casimir is an element of the universal enveloping algebra (formal polynomials in the $T^a$) that commutes with every generator. The quadratic Casimir is $$C_2=g_{ab}{T}^a{T}^b,$$ with $g_{ab}$ the inverse Killing metric. By Schur's lemma (see reps-basics.md), a Casimir acts as a scalar on each irreducible representation, so its eigenvalue labels the irrep. The number of independent Casimirs equals the rank of $\mathfrak{g}$ — the dimension of a maximal abelian (Cartan) subalgebra. Examples:

• $\mathfrak{su}(2)$: rank $1$, single Casimir $J^2=J_x^2+J_y^2+J_z^2$ with eigenvalue $j(j+1)$.
• $\mathfrak{su}(3)$: rank $2$, two Casimirs labelling irreps by Dynkin labels $(p,q)$.
• The Poincaré algebra: two Casimirs $P^2$ (mass-squared) and $W^2$ (Pauli–Lubanski, spin) — the labels of Wigner's classification in QFT/preliminaries.md.

References

• Hall, Lie Groups, Lie Algebras, and Representations, Ch. 3, 7.
• Humphreys, Introduction to Lie Algebras and Representation Theory — the standard mathematical text.
• Georgi, Lie Algebras in Particle Physics — structure constants and Casimirs in the physics idiom.