Lie Algebras — Mathematics Knowledge System

Overview

A Lie algebra is a vector space equipped with a bilinear, antisymmetric bracket operation satisfying the Jacobi identity. Lie algebras are the infinitesimal counterparts of Lie groups: they linearise the group structure at the identity and capture the local geometry of the group. The classification of simple Lie algebras over ℂ (by Killing and Cartan via Dynkin diagrams) is one of the great achievements of nineteenth and twentieth century mathematics. Lie algebras appear throughout physics as the algebras of observables and generators of symmetry transformations.

Intuition

A Lie algebra is the tangent space of a Lie group at the identity, with the group's non-commutativity encoded by the bracket. If you rotate a rigid body first around x then around y, versus first y then x, the infinitesimal difference is a rotation around z — this is the Lie bracket [Lₓ, Lᵧ] = Lz of so(3). More abstractly, the bracket measures the failure of two infinitesimal symmetries to commute. The Jacobi identity is a consistency condition ensuring the bracket behaves like a commutator.

Formal Definition

Definition

A Lie algebra over a field k is a k-vector space g equipped with a bilinear map [·,·]: g × g → g, called the Lie bracket, satisfying antisymmetry and the Jacobi identity.

[X, Y] = -[Y, X] \quad \forall X, Y \in \mathfrak{g}

Antisymmetry

[X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0 \quad \forall X, Y, Z \in \mathfrak{g}

Jacobi identity

\mathfrak{g} = T_e G \text{ with } [X, Y] = \mathcal{L}_X Y

Lie algebra of a Lie group G (Lie bracket = Lie derivative of vector fields)

\mathrm{ad}: \mathfrak{g} \to \mathfrak{gl}(\mathfrak{g}),\quad \mathrm{ad}_X(Y) = [X, Y]

Adjoint representation

\kappa(X, Y) = \mathrm{tr}(\mathrm{ad}_X \circ \mathrm{ad}_Y)

Killing form (invariant bilinear form)

Properties

Classification of simple complex Lie algebras

A_n \cong \mathfrak{sl}(n+1),\; B_n \cong \mathfrak{so}(2n+1),\; C_n \cong \mathfrak{sp}(2n),\; D_n \cong \mathfrak{so}(2n),\; E_6, E_7, E_8, F_4, G_2

Abelian Lie algebras

[X, Y] = 0 \;\forall X, Y \in \mathfrak{g} \iff \mathfrak{g} \text{ is abelian}

Solvable and nilpotent Lie algebras

\mathfrak{g}^{(0)} = \mathfrak{g},\; \mathfrak{g}^{(k+1)} = [\mathfrak{g}^{(k)}, \mathfrak{g}^{(k)}];\; \mathfrak{g} \text{ solvable} \iff \mathfrak{g}^{(N)} = 0 \text{ for some } N

Theorems

Theorem 1: Cartan's Criterion for Semisimplicity

\mathfrak{g} \text{ is semisimple} \iff \kappa \text{ (the Killing form) is non-degenerate.}

Theorem 2: Ado's Theorem

\text{Every finite-dimensional Lie algebra over } \mathbb{R} \text{ or } \mathbb{C} \text{ has a faithful finite-dimensional representation (embeds in } \mathfrak{gl}(n,k)\text{).}

Theorem 3: Weyl's Complete Reducibility Theorem

\text{Every finite-dimensional representation of a semisimple Lie algebra is completely reducible (direct sum of irreducibles).}

Worked Examples

1
Define e = [[0,1],[0,0]], f = [[0,0],[1,0]], h = [[1,0],[0,−1]].
$e = \begin{pmatrix}0&1\\0&0\end{pmatrix},\; f = \begin{pmatrix}0&0\\1&0\end{pmatrix},\; h = \begin{pmatrix}1&0\\0&-1\end{pmatrix}$
2
Compute [h,e] = he − eh:
$[h,e] = \begin{pmatrix}0&2\\0&0\end{pmatrix} = 2e$
3
Compute [h,f] = hf − fh:
$[h,f] = \begin{pmatrix}0&0\\-2&0\end{pmatrix} = -2f$
4
Compute [e,f] = ef − fe:
$[e,f] = \begin{pmatrix}1&0\\0&-1\end{pmatrix} = h$
5
Antisymmetry holds (bracket of matrices is antisymmetric). Jacobi identity holds since we are taking commutators of matrices.

✓ Answer

sl(2, ℝ) = {A ∈ M_2(ℝ) : tr A = 0} with [A,B] = AB − BA is a 3-dimensional Lie algebra with basis e, f, h satisfying [h,e] = 2e, [h,f] = −2f, [e,f] = h.

Practice Problems

Mediumproof writing

Prove that [X,Y] = −[Y,X] implies [X,X] = 0 for any Lie algebra over a field of characteristic ≠ 2.

Hardfree response

What is the Dynkin diagram of A_2 = sl(3, ℂ), and what does it encode?

Common Mistakes

Common Mistake

The Lie bracket is associative like matrix multiplication

The Lie bracket is not associative in general. The Jacobi identity [[X,Y],Z] + [[Y,Z],X] + [[Z,X],Y] = 0 is a weaker, cyclic condition that replaces associativity.

Common Mistake

A Lie algebra determines its Lie group uniquely

A Lie algebra determines a simply connected Lie group uniquely (Lie's third theorem), but multiple Lie groups can share the same Lie algebra — they differ by their fundamental group. For example, SU(2) and SO(3) have isomorphic Lie algebras su(2) ≅ so(3).

Quiz

Historical Background

Lie algebras emerged from Sophus Lie's study of transformation groups (Lie groups) in the 1870s. The term 'Lie algebra' was coined by Hermann Weyl. Wilhelm Killing attempted to classify simple Lie algebras in 1888–1890, finding the classical series A_n, B_n, C_n, D_n and conjecturing the exceptional ones. Élie Cartan in his 1894 thesis corrected Killing's errors and rigorously classified all simple complex Lie algebras. Dynkin diagrams, introduced by Eugene Dynkin in the 1940s, give an elegant graphical encoding of the classification. The representation theory, developed by Cartan and Weyl, is a cornerstone of quantum mechanics.

1874
Lie introduces infinitesimal transformation groups, precursors of Lie algebras
Sophus Lie
1888
Killing attempts classification of simple Lie algebras, conjectures exceptional types
Wilhelm Killing
1894
Cartan completes the classification of simple complex Lie algebras in his doctoral thesis
Élie Cartan
1947
Dynkin introduces root systems and Dynkin diagrams for the classification
Eugene Dynkin

Summary

A Lie algebra (g, [·,·]) is a vector space with an antisymmetric bracket satisfying the Jacobi identity.
The Lie algebra of a Lie group G is g = T_eG with bracket given by the commutator of left-invariant vector fields.
Simple complex Lie algebras are classified by Dynkin diagrams: A_n, B_n, C_n, D_n, and G_2, F_4, E_6, E_7, E_8.
Cartan's criterion: g is semisimple iff the Killing form κ(X,Y) = tr(ad_X ad_Y) is non-degenerate.
Weyl's theorem: every representation of a semisimple Lie algebra is completely reducible.

References

BookHumphreys, J. E. — Introduction to Lie Algebras and Representation Theory, Springer, 1972
BookHall, B. — Lie Groups, Lie Algebras, and Representations, 2nd ed., Springer, 2015, Part II
WebsiteWikipedia — Lie algebra
WebsitenLab — Lie algebra