Simple Lie algebra

A simple Lie algebra is a non-abelian Lie algebra over a field of characteristic zero that contains no nonzero proper ideals, serving as the fundamental indecomposable building block for more general semisimple Lie algebras.^[1]^[2] A Lie algebra itself is a vector space equipped with a bilinear, antisymmetric bracket operation satisfying the Jacobi identity, which encodes infinitesimal symmetries akin to those of Lie groups.^[3] Simple Lie algebras possess a trivial center and admit a root space decomposition relative to a Cartan subalgebra, where the roots form an irreducible root system that fully characterizes their structure.^[1] Semisimple Lie algebras, which are finite direct sums of simple ones, are precisely those with a nondegenerate Killing form—a symmetric, invariant bilinear form defined by the trace of adjoint operators—and this property distinguishes them from solvable or nilpotent algebras via Cartan's criterion.^[2] Over the complex numbers, the classification of finite-dimensional simple Lie algebras, achieved through the work of Killing, Cartan, and Dynkin in the early 20th century, identifies four infinite families of classical types—A_n (corresponding to sl(n+1, ℂ) for n ≥ 1), B_n (so(2n+1, ℂ) for n ≥ 2), C_n (sp(2n, ℂ) for n ≥ 3), and D_n (so(2n, ℂ) for n ≥ 4)—along with five exceptional types: G_2, F_4, E_6, E_7, and E_8, each represented uniquely by connected Dynkin diagrams.^[1]^[4] This classification extends to real simple Lie algebras via Satake diagrams, linking them to representations and symmetry groups in geometry, physics, and representation theory.^[1]

Fundamentals

Definition

A Lie algebra \mathfrak{g} over a field K of characteristic zero is defined to be simple if it is non-abelian and admits no nontrivial ideals, meaning that the only ideals of \mathfrak{g} are the zero ideal \{0\} and \mathfrak{g} itself.^[5] This condition ensures that \mathfrak{g} cannot be decomposed into smaller Lie subalgebras in a nontrivial way via ideals, capturing its indivisibility as a structure. The non-abelian requirement distinguishes simple Lie algebras from the trivial abelian case, where the Lie bracket [x, y] vanishes for all elements.^[6] Simple Lie algebras typically arise over algebraically closed fields of characteristic zero, such as the complex numbers \mathbb{C}, where their structure is particularly well-understood.^[5] In this setting, the simplicity condition aligns with deeper properties like the nondegeneracy of the Killing form, though the ideal-based definition remains the primary one. Over more general fields of characteristic zero, the notion extends similarly, but classifications become more involved.^[6] In contrast, a semisimple Lie algebra is one that decomposes as a direct sum of simple ideals, positioning simple Lie algebras as the fundamental, indecomposable components of semisimple structures.^[7] This relationship underscores the role of simple Lie algebras in building more complex semisimple ones, such as those associated with classical groups.

Basic Properties

Simple Lie algebras exhibit rigid structural properties stemming from their lack of nontrivial ideals. Consider the adjoint representation \operatorname{ad}: \mathfrak{g} \to \mathfrak{gl}(\mathfrak{g}) defined by \operatorname{ad}_x(y) = [x, y] for x, y \in \mathfrak{g}. For a simple Lie algebra \mathfrak{g} over \mathbb{C}, this representation is injective, as its kernel is the center Z(\mathfrak{g}), which vanishes for simple Lie algebras.^[8] The Killing form B(x, y) = \operatorname{tr}(\operatorname{ad}_x \operatorname{ad}_y) provides a key invariant: it is a non-degenerate symmetric bilinear form on \mathfrak{g}. Simple Lie algebras are semisimple, possessing a non-degenerate Killing form, which distinguishes them from solvable or nilpotent cases where the form degenerates.^[9] Over \mathbb{C}, a finite-dimensional Lie algebra is simple if and only if it is semisimple with no proper direct summands, meaning semisimple Lie algebras decompose uniquely as direct sums of simple ideals. In characteristic zero, no nontrivial abelian Lie algebra is simple, as any abelian Lie algebra of dimension greater than one admits proper ideals, violating simplicity.^[10]^[11] The minimal dimension of a simple Lie algebra over \mathbb{C} is 3, achieved by \mathfrak{sl}(2, \mathbb{C}), the Lie algebra of $2 \times 2 trace-zero matrices.^[12]

Complex Simple Lie Algebras

Classification

The Killing–Cartan classification provides a complete description of all finite-dimensional simple Lie algebras over the complex numbers. According to this theorem, every such Lie algebra is isomorphic to the Lie algebra of one of the following types: the classical series A_n (corresponding to \mathfrak{sl}_{n+1}(\mathbb{C}) for n \geq 1), B_n (corresponding to \mathfrak{so}_{2n+1}(\mathbb{C}) for n \geq 2), C_n (corresponding to \mathfrak{sp}_{2n}(\mathbb{C}) for n \geq 3), and D_n (corresponding to \mathfrak{so}_{2n}(\mathbb{C}) for n \geq 4); or one of the five exceptional types G_2, F_4, E_6, E_7, or E_8.^[13]^[14] This classification is unique up to isomorphism: for each type and each admissible rank, there exists exactly one simple Lie algebra of that form, and no other finite-dimensional simple Lie algebras over \mathbb{C} exist.^[13] The classical types arise from the special linear, orthogonal, and symplectic groups, while the exceptional types do not correspond to any broader infinite families and were discovered through systematic analysis of possible structures.^[14] The development of this classification began with Wilhelm Killing's work in the late 1880s, where he introduced the quadratic form now known as the Killing form and attempted to classify simple Lie algebras by studying their invariant bilinear forms and root-like structures, though his proofs contained gaps.^[14] Élie Cartan provided the rigorous completion in his 1894 doctoral thesis, verifying the existence and uniqueness of all types, including the exceptional ones, by constructing explicit matrix realizations and filling in the logical deficiencies in Killing's approach.^[14]^[13] Each simple Lie algebra in the classification has a well-defined rank r (the dimension of a Cartan subalgebra) and total dimension \dim \mathfrak{g}, which vary by type as summarized in the following table:

Type	Rank r	Dimension \dim \mathfrak{g}
A_n	n	n(n+2)
B_n	n	n(2n+1)
C_n	n	n(2n+1)
D_n	n	n(2n-1)
G_2	2	14
F_4	4	52
E_6	6	78
E_7	7	133
E_8	8	248

These dimensions count the Lie algebra as a vector space over \mathbb{C}.^[13]

Root Systems and Dynkin Diagrams

A Cartan subalgebra \mathfrak{h} of a complex semisimple Lie algebra \mathfrak{g} is a maximal toral subalgebra, meaning it is abelian and consists of semisimple elements, with the adjoint representation of \mathfrak{h} on \mathfrak{g} being diagonalizable.^[15] The root space decomposition expresses \mathfrak{g} as \mathfrak{g} = \mathfrak{h} \oplus \bigoplus_{\alpha \in \Phi} \mathfrak{g}_\alpha, where \Phi \subset \mathfrak{h}^* is the set of roots, each \mathfrak{g}_\alpha is the one-dimensional root space corresponding to a nonzero linear functional \alpha: \mathfrak{h} \to \mathbb{C}, and the Lie bracket satisfies [h, x_\alpha] = \alpha(h) x_\alpha for h \in \mathfrak{h} and x_\alpha \in \mathfrak{g}_\alpha.^[15] This decomposition highlights the semisimple structure, with \dim \mathfrak{g}_\alpha = 1 for each root \alpha.^[16] The root system \Phi is a finite, reduced crystallographic root system in the real Euclidean space \mathfrak{h}_\mathbb{R}^*, equipped with a positive definite inner product (\cdot, \cdot) invariant under the adjoint action of \mathfrak{g}.^[15] It satisfies properties such as: for each \alpha \in \Phi, the reflection s_\alpha(\beta) = \beta - 2 \frac{(\beta, \alpha)}{(\alpha, \alpha)} \alpha maps \Phi to itself; the only scalar multiples of roots in \Phi are \pm \alpha; and $2 \frac{(\beta, \alpha)}{(\alpha, \alpha)} \in \mathbb{Z} for all \alpha, \beta \in \Phi.^[16] A subset of simple roots \Delta \subset \Phi is a basis for the \mathbb{R}-span of \Phi such that every root is a unique integer linear combination of elements of \Delta with all coefficients nonnegative or nonpositive.^[15] The positive roots \Phi^+ are those with nonnegative coefficients in this basis, and \Phi = \Phi^+ \sqcup -\Phi^+.^[16] The Weyl group W of \Phi is the finite Coxeter group generated by the reflections s_\alpha for \alpha \in \Delta, acting faithfully on \mathfrak{h}^* and preserving the inner product.^[15] It permutes the roots and acts simply transitively on the Weyl chambers, with the fundamental chamber being the connected component of \{ \lambda \in \mathfrak{h}_\mathbb{R}^* \mid (\lambda, \alpha) > 0 \ \forall \alpha \in \Delta \} containing the origin in its closure.^[16] Dynkin diagrams encode the root system via its simple roots, with one node per simple root in \Delta, and edges representing the angles between them: no edge if the angle is 90°; a single bond if 120°; a double bond if 135° (with an arrow indicating shorter root); and a triple bond if 150°.^[15] The irreducible diagrams, up to isomorphism, are the finite ADE and BCFG series: A_n (n ≥ 1) is a chain of n single bonds; B_n (n ≥ 2) is a chain of n-1 single bonds with a double bond arrow at the end; C_n (n ≥ 3) mirrors B_n with the arrow reversed; D_n (n ≥ 4) is a chain of n-3 single bonds with two branches at the end; E_6, E_7, E_8 form a chain with a single branch at the third node from the end (length 1,2,3 respectively); F_4 has a double and triple bond in the chain; G_2 has a triple bond.^[16] These diagrams classify the irreducible root systems, and thus the complex simple Lie algebras.^[15] The Cartan matrix A = (a_{ij}) associated to \Delta = \{\alpha_1, \dots, \alpha_l\} has entries a_{ij} = 2 \frac{(\alpha_i, \alpha_j)}{(\alpha_j, \alpha_j)}, which are integers encoding the off-diagonal structure (2 on diagonal, ≤0 off-diagonal).^[16] For root systems with equal-length roots, the inner product can be normalized so (\alpha_i, \alpha_i) = 2, simplifying to a_{ij} = 2 (\alpha_i, \alpha_j); unequal lengths require the general form.^[15] Each Dynkin diagram realizes a unique simple Lie algebra through the Chevalley basis, generated by elements e_i, f_i, h_i (i=1 to l) satisfying the Serre relations: [h_i, e_j] = a_{ij} e_j, [h_i, f_j] = -a_{ij} f_j, [e_i, f_j] = \delta_{ij} h_i, and ad-nilpotent relations like [\text{ad } e_i]^{1 - a_{ij}} e_j = 0 for i ≠ j, with symmetric relations for f_i.^[16] This presentation via the diagram ensures the algebra is simple and finite-dimensional over \mathbb{C}.^[15]

Real Simple Lie Algebras

Real Forms of Complex Lie Algebras

A real form of a complex Lie algebra \mathfrak{g}_\mathbb{C} is a real Lie subalgebra \mathfrak{g} \subset \mathfrak{g}_\mathbb{C} such that \mathfrak{g}_\mathbb{C} = \mathfrak{g} \otimes_\mathbb{R} \mathbb{C}. Equivalently, it is defined by an involution \sigma: \mathfrak{g}_\mathbb{C} \to \mathfrak{g}_\mathbb{C} satisfying \sigma^2 = \mathrm{id} and fixed-point set \mathfrak{g} = \{ x \in \mathfrak{g}_\mathbb{C} \mid \sigma(x) = x \}.^[17] Every real simple Lie algebra is a real form of a unique complex simple Lie algebra (up to isomorphism).^[17] For non-compact real forms, there exists a Cartan decomposition \mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}, where \mathfrak{k} is a compact Lie subalgebra (with \mathrm{ad}-skew-symmetric bracket) and \mathfrak{p} is the orthogonal complement with respect to the Killing form, satisfying [\mathfrak{k}, \mathfrak{p}] \subset \mathfrak{p} and [\mathfrak{p}, \mathfrak{p}] \subset \mathfrak{k}. The Killing form B on \mathfrak{g} is negative definite on \mathfrak{k} and positive definite on \mathfrak{p}.^[17] The signature of the Killing form distinguishes real forms: compact forms have negative definite Killing form, while split forms have neutral signature (equal number of positive and negative eigenvalues).^[17] Representative examples include \mathfrak{sl}(2, \mathbb{R}) as the split real form of \mathfrak{sl}(2, \mathbb{C}) and \mathfrak{su}(2) as its compact real form. Another example is \mathfrak{so}(2,1), the Lie algebra of the Lorentz group, serving as a non-compact real form of \mathfrak{so}(3, \mathbb{C}).^[17]

Classification and Examples

The classification of real simple Lie algebras corresponds to the real forms of the complex simple Lie algebras classified by their root systems, with each real form uniquely determined by a Satake diagram derived from the Dynkin diagram of the complex type.^[18] In a Satake diagram, nodes are painted black to indicate compact (imaginary) roots, and pairs of non-compact roots related by the Cartan involution are connected by arrows, providing a complete labeling of the real forms up to isomorphism.^[17] This classification, originally due to Élie Cartan, yields a finite number of real forms for each complex type, varying from 2 to n+1 depending on the rank n and the series.^[19] The compact real form is unique up to isomorphism for each complex simple Lie algebra and corresponds to the Lie algebra of the corresponding compact simple Lie group. Examples include \mathfrak{su}(n+1) for type A_n (dimension n^2 + 2n), \mathfrak{so}(2n+1) for B_n (dimension n(2n+1)), \mathfrak{sp}(n) for C_n (dimension n(2n+1)), \mathfrak{so}(2n) for D_n (dimension n(2n-1)), and the compact exceptional forms \mathfrak{g}_2, \mathfrak{f}_4, \mathfrak{e}_6, \mathfrak{e}_7, \mathfrak{e}_8. These forms have negative definite Killing form and admit only finite-dimensional unitary representations.^[17] Split real forms represent the maximally non-compact case, where the real rank equals the complex rank, and they arise from Satake diagrams with no painted nodes except possibly at the ends. Representative examples are \mathfrak{sl}(n+1, \mathbb{R}) for A_n (dimension n^2 + 2n), \mathfrak{so}(n+1, n) for B_n (dimension n(2n+1)), \mathfrak{sp}(2n, \mathbb{R}) for C_n (dimension n(2n+1)), \mathfrak{so}(n, n) for D_n (dimension n(2n-1)), \mathfrak{g}_{2(2)} for G_2 (dimension 14), \mathfrak{f}_{4(4)} for F_4 (dimension 52), \mathfrak{e}_{6(6)} for E_6 (dimension 78), \mathfrak{e}_{7(7)} for E_7 (dimension 133), and \mathfrak{e}_{8(8)} for E_8 (dimension 248). These forms are associated with symmetry groups of hyperbolic geometries and have a Cartan decomposition with maximal abelian nilpotent part.^[18] Intermediate non-compact real forms fill the spectrum between compact and split, labeled by Satake diagrams with partial paintings and arrows; for type A_n, these are \mathfrak{su}(p, q) with p + q = n+1 and p \geq q > 0, while for B_n and D_n they include \mathfrak{so}(p, q) variants with appropriate signatures. The total number of real forms per complex type varies: for example, type A_3 (corresponding to \mathfrak{sl}(4, \mathbb{C})) has 4 forms—\mathfrak{su}(4), \mathfrak{su}(3,1), \mathfrak{su}(2,2), and \mathfrak{sl}(4, \mathbb{R})—while G_2 has 2 (compact \mathfrak{g}_2 and split \mathfrak{g}_{2(2)}). For exceptional types, F_4 has 4 real forms, E_6 has 4, E_7 has 3, and E_8 has 2.^[17] Illustrative examples include the compact form \mathfrak{su}(2) of type A_1, with dimension 3 and basis given by the Pauli matrices; and the non-compact form \mathfrak{so}(3,1) of type D_2, with dimension 6, serving as the Lie algebra of the Lorentz group SO(3,1) in special relativity.^[19]