Fact-checked by Grok 2 weeks ago

Killing form

The Killing form is a symmetric bilinear form on a finite-dimensional Lie algebra \mathfrak{g} over a field of characteristic zero, defined by \kappa(X, Y) = \operatorname{tr}(\operatorname{ad}_X \operatorname{ad}_Y) for X, Y \in \mathfrak{g}, where \operatorname{ad}_X denotes the adjoint endomorphism \operatorname{ad}_X(Z) = [X, Z] and \operatorname{tr} is the trace in the adjoint representation.^[1]^[2] Introduced by the German mathematician Wilhelm Killing in his seminal papers published between 1888 and 1890 in Mathematische Annalen, the form emerged from his investigations into the symmetries of non-Euclidean geometries and the structure of continuous transformation groups.^[3] Killing's work laid the groundwork for the classification of simple Lie algebras, identifying the four infinite families of classical types (A_\ell, B_\ell, C_\ell, D_\ell) and the five exceptional types (G_2, F_4, E_6, E_7, E_8), though his proofs contained gaps that were later rigorized by Élie Cartan in his 1894 doctoral thesis.^[3] The Killing form, often called the Cartan-Killing form in recognition of Cartan's contributions, is invariant under Lie algebra automorphisms, meaning \kappa(\phi(X), \phi(Y)) = \kappa(X, Y) for any automorphism \phi.^[1] A cornerstone property is its non-degeneracy: a Lie algebra \mathfrak{g} is semisimple if and only if the Killing form is non-degenerate, providing a key criterion to distinguish semisimple algebras from solvable or nilpotent ones.^[1]^[2] For the Lie algebra of a compact connected Lie group with finite center, the Killing form is negative definite, enabling the construction of invariant Riemannian metrics on the corresponding group manifold.^[1] Beyond classification, the form plays a vital role in representation theory, root systems, and the study of real forms of complex semisimple Lie algebras, influencing applications in physics such as quantum mechanics and gauge theories.^[2]^[3]

Historical Development

Origin and Early Contributions

The origins of the Killing form trace back to the late 19th-century efforts to classify simple Lie algebras, beginning with the work of Wilhelm Killing. In his series of papers published between 1888 and 1890 in Mathematische Annalen, Killing undertook the ambitious task of classifying all finite-dimensional simple Lie algebras over the complex numbers, motivated by his studies in non-Euclidean geometry and continuous transformation groups. Notably, in his 1888 paper, Killing first identified the exceptional simple Lie algebras, including those now denoted as G₂, F₄, E₆, E₇, and E₈, alongside the classical series. Although he did not explicitly define a bilinear form, Killing observed key trace-related invariants, such as the trace of the square of the adjoint representation operator Tr((ad_X)^2), which played a crucial role in distinguishing the structure of these algebras and verifying their simplicity. These invariants allowed him to outline the root systems and Dynkin diagrams implicitly, though his proofs contained gaps and relied on geometric intuition rather than fully algebraic rigor.^[4]^[5] Building directly on Killing's classification, Élie Cartan provided a rigorous foundation in his 1894 doctoral thesis, Sur la structure des groupes de transformations finis et continus, published in Nouvelles Annales de Mathématiques. Cartan reworked Killing's results with greater algebraic precision, introducing the explicit trace form on the adjoint representation—now known as the Killing form, defined as B(X, Y) = Tr(ad_X ad_Y)—to systematically classify semisimple Lie algebras over the complex numbers. This form enabled Cartan to complete the classification by confirming the existence and uniqueness of the simple components, dividing them into the four infinite classical families (A_n, B_n, C_n, D_n) and the five exceptional ones identified by Killing. In the same thesis, Cartan proved that the non-degeneracy of the Killing form implies semisimplicity, demonstrating that if the form is non-degenerate, the algebra has no non-trivial solvable ideals and thus providing a fundamental characterization linking the form's properties to the structure of the algebra. This proof marked a pivotal step in Lie theory, offering a practical test for semisimplicity and influencing the development of representation theory. Cartan further extended these ideas to real Lie algebras and their compact forms in his subsequent publications of 1913–1914 on continuous groups of transformations.^[5]^[3]

Naming and Later Recognition

The term "Killing form" was first introduced by Armand Borel in 1951 during one of his reports at the Séminaire Bourbaki, where he used it to acknowledge Wilhelm Killing's pioneering classification of simple Lie algebras in the late 19th century. Borel later reflected that the naming was a misnomer, as Killing himself did not study this bilinear form; instead, it had been defined earlier by Élie Cartan in his 1894 doctoral thesis on the structure of Lie algebras. Prior to Borel's usage, Cartan described the form without assigning it a specific name, referring to it generally as the bilinear form associated to the Lie algebra, derived from the trace in the adjoint representation. This terminology persisted in early 20th-century works, including Claude Chevalley's influential 1946 monograph Theory of Lie Groups, where the form appears as a key invariant but is not termed the "Killing form"; Chevalley credits Cartan for its introduction while emphasizing its role in characterizing semisimple algebras. Mid-20th-century texts, such as those in the Bourbaki seminars, began shifting attribution toward Killing, reflecting a broader recognition of his foundational insights into Lie algebra structure. By the 1950s, the name "Killing form" had become standard in Lie theory nomenclature, influencing subsequent textbooks and expositions. For instance, it is routinely employed in Nathan Jacobson's Lie Algebras (1962) and Sigurdur Helgason's Differential Geometry, Lie Groups, and Symmetric Spaces (1962), solidifying its place as the conventional term for this invariant bilinear form. This adoption underscored the form's centrality in modern treatments of Lie algebras, bridging historical developments with contemporary applications.

Mathematical Context

Lie Algebras and Representations

A Lie algebra \mathfrak{g} over a field K of characteristic zero is defined as a vector space over K equipped with a bilinear map [\cdot, \cdot]: \mathfrak{g} \times \mathfrak{g} \to \mathfrak{g}, known as the Lie bracket, satisfying two axioms: skew-symmetry, [x, y] = -[y, x] for all x, y \in \mathfrak{g}, and the Jacobi identity, [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 for all x, y, z \in \mathfrak{g}.^[6] This structure captures infinitesimal symmetries and arises naturally from the tangent spaces at the identity of Lie groups, though the abstract definition suffices for algebraic study.^[7] The characteristic zero assumption ensures that the axioms behave well under scalar multiplication and avoids complications in representation theory, such as those in positive characteristic where the bracket may not generate the algebra properly.^[6] Representations provide a way to study Lie algebras through their actions on vector spaces. A representation of \mathfrak{g} is a Lie algebra homomorphism \rho: \mathfrak{g} \to \mathfrak{gl}(V), where \mathfrak{gl}(V) denotes the Lie algebra of all linear endomorphisms of a finite-dimensional vector space V over K, equipped with the commutator bracket [A, B] = AB - BA.^[6] This means \rho([x, y]) = [\rho(x), \rho(y)] = \rho(x)\rho(y) - \rho(y)\rho(x) for all x, y \in \mathfrak{g}, allowing \mathfrak{g} to act linearly on V. Homomorphisms between Lie algebras preserve the bracket operation, enabling the classification of structures via faithful representations, such as embeddings into matrix algebras.^[7] Among representations, irreducible ones are fundamental, consisting of those where the only \mathfrak{g}-invariant subspaces of V are \{0\} and V itself.^[6] A representation is semisimple if it decomposes as a direct sum of irreducible representations. The adjoint representation, which acts on \mathfrak{g} itself, exemplifies a canonical representation but is explored further elsewhere. For semisimple Lie algebras, every finite-dimensional representation is completely reducible, meaning it splits into irreducibles, a key result in the theory.^[6] A Lie algebra \mathfrak{g} is semisimple if it decomposes as a direct sum of simple Lie algebras, where a simple Lie algebra has no nontrivial ideals.^[6] Equivalently, the radical of \mathfrak{g}—its maximal solvable ideal—is zero, ensuring no abelian or nilpotent substructures dominate the algebra.^[7] Semisimple Lie algebras over algebraically closed fields of characteristic zero are precisely the direct sums of simple ones, facilitating their classification via root systems and Dynkin diagrams.^[6]

Adjoint Representation

The adjoint representation of a Lie algebra \mathfrak{g} over a field of characteristic zero is the linear map \mathrm{ad}: \mathfrak{g} \to \mathfrak{gl}(\mathfrak{g}) defined by \mathrm{ad}_x(y) = [x, y] for all x, y \in \mathfrak{g}, where [\cdot, \cdot] is the Lie bracket and \mathfrak{gl}(\mathfrak{g}) is the Lie algebra of all linear endomorphisms of the vector space \mathfrak{g}.^[1] This construction endows \mathfrak{g} with a canonical representation on itself, where each element x acts as the endomorphism \mathrm{ad}_x that captures the linearization of the bracket operation around x.^[8] The map \mathrm{ad} is a Lie algebra homomorphism, meaning it preserves the bracket structure: \mathrm{ad}_{[x,y]} = [\mathrm{ad}_x, \mathrm{ad}_y] for all x, y \in \mathfrak{g}, with the bracket on the right denoting the commutator [\mathrm{ad}_x, \mathrm{ad}_y] = \mathrm{ad}_x \circ \mathrm{ad}_y - \mathrm{ad}_y \circ \mathrm{ad}_x in \mathfrak{gl}(\mathfrak{g}).^[1] This property ensures that the image \mathrm{ad}(\mathfrak{g}) forms a Lie subalgebra of \mathfrak{gl}(\mathfrak{g}), consisting of the inner derivations of \mathfrak{g}.^[8] Additionally, each \mathrm{ad}_x satisfies the derivation property \mathrm{ad}_x([y,z]) = [\mathrm{ad}_x(y), z] + [y, \mathrm{ad}_x(z)] for all y, z \in \mathfrak{g}, which is a direct consequence of the Jacobi identity in \mathfrak{g}.^[1] The kernel of \mathrm{ad} coincides with the center Z(\mathfrak{g}) = \{ x \in \mathfrak{g} \mid [x, y] = 0 \ \forall \, y \in \mathfrak{g} \}, the ideal of elements that commute with everything in \mathfrak{g}.^[8] Thus, \mathrm{ad} induces an isomorphism \mathfrak{g} / Z(\mathfrak{g}) \cong \mathrm{ad}(\mathfrak{g}).^[1] Regarding the action, the subspace generated by all images under the adjoint maps, \{ [x, y] \mid x, y \in \mathfrak{g} \}, is precisely the derived subalgebra [\mathfrak{g}, \mathfrak{g}], the Lie ideal spanned by all commutators.^[9]

Definition and Construction

Formal Definition

The Killing form on a finite-dimensional Lie algebra \mathfrak{g} over a field K of characteristic zero is defined as the bilinear map B: \mathfrak{g} \times \mathfrak{g} \to K given by

B(x, y) = \operatorname{tr}(\operatorname{ad}_x \circ \operatorname{ad}_y)

for all x, y \in \mathfrak{g}, where \operatorname{ad}_x: \mathfrak{g} \to \mathfrak{g} denotes the adjoint map \operatorname{ad}_x(y) = [x, y] and \operatorname{tr} is the trace in the adjoint representation.^[10]^[11] This form is bilinear because the trace of the composition of linear endomorphisms is bilinear in the endomorphisms, and \operatorname{ad}_x is linear in x.^[10] The requirement that \operatorname{char} K = 0 ensures the trace is well-defined and the adjoint representation acts appropriately on \mathfrak{g}, while K is typically taken to be algebraically closed (such as \mathbb{C}) to facilitate the study of semisimple Lie algebras.^[11]^[10] An equivalent expression arises in a basis \{t_a\}_{a=1}^{\dim \mathfrak{g}} of \mathfrak{g}, where the Lie bracket is expressed via structure constants [t_a, t_b] = \sum_c C^c_{ab} t_c; then

B(t_a, t_b) = \sum_{c,d} C^c_{a d} C^d_{b c}.

This summation form computes the Killing form directly from the algebra's structure constants.^[11]

Matrix Representation

In a finite-dimensional Lie algebra \mathfrak{g} over a field K of characteristic zero, equipped with a basis \{e_i\}_{i=1}^n, the Lie bracket is expressed via structure constants as [e_i, e_j] = \sum_{k=1}^n c_{ij}^k e_k, where c_{ij}^k \in K.^[12] The Killing form B, defined as the trace in the adjoint representation, yields matrix elements B(e_i, e_j) given by B(e_i, e_j) = \sum_{k,l=1}^n c_{i k}^l c_{j l}^k.^[12] This expression arises from computing the trace \operatorname{Tr}(\operatorname{ad}_{e_i} \operatorname{ad}_{e_j}), where the matrices of \operatorname{ad}_{e_i} and \operatorname{ad}_{e_j} are determined by the structure constants. In matrix form, the Killing form corresponds to the symmetric bilinear form with components B_{ij} = B(e_i, e_j), often denoted as the Cartan-Killing metric tensor.^[12] For semisimple Lie algebras, where the Killing form is non-degenerate, the matrix (B_{ij}) is invertible, allowing it to serve as a metric to raise and lower indices in tensor expressions involving the structure constants. For instance, the contravariant structure constants can be defined using the inverse metric B^{ij} such that c^{ijk} = B^{il} c_{lm}^k B^{mj}.^[13] A concrete example is the special linear Lie algebra \mathfrak{sl}(2, K), with basis \{h, x, y\} where h = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, x = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, y = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, satisfying [h, x] = 2x, [h, y] = -2y, [x, y] = h. The structure constants are c_{xy}^h = 1, c_{yx}^h = -1, c_{hx}^x = 2, c_{xh}^x = -2, c_{hy}^y = -2, c_{yh}^y = 2, and zero otherwise. The Killing form matrix in this basis is \begin{pmatrix} B(h,h) & B(h,x) & B(h,y) \\ B(x,h) & B(x,x) & B(x,y) \\ B(y,h) & B(y,x) & B(y,y) \end{pmatrix} = \begin{pmatrix} 8 & 0 & 0 \\ 0 & 0 & 4 \\ 0 & 4 & 0 \end{pmatrix}, with B(h,h) = 8, B(x,y) = B(y,x) = 4, and all other entries zero.^[8]

Intrinsic Properties

Symmetry and Invariance

The Killing form on a finite-dimensional Lie algebra \mathfrak{g} over a field of characteristic zero, defined by B(x, y) = \operatorname{tr}(\operatorname{ad}_x \circ \operatorname{ad}_y) for x, y \in \mathfrak{g}, possesses fundamental algebraic properties that underscore its role as a canonical invariant bilinear form. Chief among these is its symmetry. Specifically, B(x, y) = B(y, x) for all x, y \in \mathfrak{g}. This follows directly from the cyclicity of the trace: \operatorname{tr}(\operatorname{ad}_x \circ \operatorname{ad}_y) = \operatorname{tr}(\operatorname{ad}_y \circ \operatorname{ad}_x), as the trace of a composition of linear operators is unchanged under cyclic permutation.^[14] A key consequence of the derivation property of the adjoint representation is the associativity of the Killing form, expressed as B([z, x], y) = B(x, [z, y]) for all x, y, z \in \mathfrak{g}. To verify this, note that \operatorname{ad}_{[z, x]} = [\operatorname{ad}_z, \operatorname{ad}_x] = \operatorname{ad}_z \circ \operatorname{ad}_x - \operatorname{ad}_x \circ \operatorname{ad}_z. Thus,

B([z, x], y) = \operatorname{tr}([\operatorname{ad}_z, \operatorname{ad}_x] \circ \operatorname{ad}_y) = \operatorname{tr}(\operatorname{ad}_z \circ \operatorname{ad}_x \circ \operatorname{ad}_y - \operatorname{ad}_x \circ \operatorname{ad}_z \circ \operatorname{ad}_y).

Applying cyclicity of the trace yields

\operatorname{tr}(\operatorname{ad}_z \circ \operatorname{ad}_x \circ \operatorname{ad}_y) = \operatorname{tr}(\operatorname{ad}_x \circ \operatorname{ad}_y \circ \operatorname{ad}_z), \quad \operatorname{tr}(\operatorname{ad}_x \circ \operatorname{ad}_z \circ \operatorname{ad}_y) = \operatorname{tr}(\operatorname{ad}_z \circ \operatorname{ad}_y \circ \operatorname{ad}_x).

However, the full equality B([x, y], z) = B(x, [y, z]) (with indices adjusted) holds by the general lemma that \operatorname{tr}([\operatorname{ad}_x, \operatorname{ad}_y] \circ \operatorname{ad}_z) = \operatorname{tr}(\operatorname{ad}_x \circ [\operatorname{ad}_y, \operatorname{ad}_z]), which follows from the Jacobi identity in the universal enveloping algebra and properties of the trace on derivations. This associativity reflects the compatibility of the Killing form with the Lie bracket structure.^[15] The Killing form is further invariant under the action of Lie algebra automorphisms. For any automorphism \phi \in \operatorname{Aut}(\mathfrak{g}), B(\phi(x), \phi(y)) = B(x, y) for all x, y \in \mathfrak{g}. The proof relies on the intertwining property \operatorname{ad}_{\phi(x)} = \phi \circ \operatorname{ad}_x \circ \phi^{-1}, so

B(\phi(x), \phi(y)) = \operatorname{tr}(\operatorname{ad}_{\phi(x)} \circ \operatorname{ad}_{\phi(y)}) = \operatorname{tr}((\phi \circ \operatorname{ad}_x \circ \phi^{-1}) \circ (\phi \circ \operatorname{ad}_y \circ \phi^{-1})) = \operatorname{tr}(\phi \circ \operatorname{ad}_x \circ \operatorname{ad}_y \circ \phi^{-1}).

Since the trace is invariant under conjugation by invertible linear maps, \operatorname{tr}(\phi \circ \operatorname{ad}_x \circ \operatorname{ad}_y \circ \phi^{-1}) = \operatorname{tr}(\operatorname{ad}_x \circ \operatorname{ad}_y) = B(x, y).^[16] As a direct consequence, the Killing form is preserved under the adjoint action of the corresponding Lie group G with Lie algebra \mathfrak{g}. For g \in G, the adjoint map \operatorname{Ad}_g: \mathfrak{g} \to \mathfrak{g} is an automorphism, so B(\operatorname{Ad}_g(x), \operatorname{Ad}_g(y)) = B(x, y). This group-level invariance extends the algebraic properties to the geometric setting of the Lie group, facilitating applications in representation theory and differential geometry.^[16]

Non-degeneracy and Cartan's Criterion

The Killing form B on a finite-dimensional Lie algebra \mathfrak{g} over a field of characteristic zero is non-degenerate if and only if \mathfrak{g} is semisimple.^[17]^[18] This equivalence forms the basis of Cartan's semisimplicity criterion, which characterizes semisimple Lie algebras through the properties of this invariant bilinear form. Non-degeneracy means that the only element x \in \mathfrak{g} satisfying B(x, y) = 0 for all y \in \mathfrak{g} is x = 0.^[19] A related result is Cartan's criterion of solvability, which states that \mathfrak{g} is solvable if and only if B degenerates on the derived algebra, meaning B([ \mathfrak{g}, \mathfrak{g} ], \mathfrak{g} ) = 0.^[16]^[17] In particular, for solvable \mathfrak{g}, [ \mathfrak{g}, \mathfrak{g} ] lies in the kernel of the form, since B([ \mathfrak{g}, \mathfrak{g} ], \mathfrak{g} ) = 0.^[16] This criterion highlights the connection between the algebraic structure of \mathfrak{g} and the degeneracy of B, with degeneracy implying solvability and non-degeneracy ensuring semisimplicity. For ideals in \mathfrak{g}, the Killing form exhibits orthogonality properties: if \mathfrak{i} is an ideal of \mathfrak{g}, then both \mathfrak{i} and the quotient \mathfrak{g}/\mathfrak{i} are orthogonal to the orthogonal complement of the derived algebra [ \mathfrak{g}, \mathfrak{g} ]^\perp.^[19] More precisely, the orthogonal complement of an ideal is itself an ideal, and the restriction of B to an ideal \mathfrak{i} coincides with the Killing form on \mathfrak{i}.^[16] These properties ensure that ideals behave compatibly under the form, aiding in the decomposition of \mathfrak{g}. The radical of the Killing form, defined as \operatorname{Rad}(B) = \{ x \in \mathfrak{g} \mid B(x, y) = 0 \ \forall y \in \mathfrak{g} \}, is precisely the solvable radical of \mathfrak{g}.^[18]^[19] This radical is itself an ideal, and it is solvable, so quotienting \mathfrak{g} by \operatorname{Rad}(B) yields a semisimple Lie algebra on which B induces a non-degenerate form.^[16] Thus, the non-degeneracy of B provides a practical algebraic tool for classifying semisimple Lie algebras and identifying their solvable components.

Extensions and Connections

Trace Forms and Generalizations

The trace form associated to a finite-dimensional representation \rho: \mathfrak{g} \to \mathfrak{gl}(V) of a Lie algebra \mathfrak{g} over an algebraically closed field of characteristic zero is the symmetric bilinear form defined by

B_\rho(x, y) = \operatorname{tr}(\rho(x) \rho(y))

for all x, y \in \mathfrak{g}.^[20] This form is invariant under the adjoint action, meaning B_\rho([z, x], y) + B_\rho(x, [z, y]) = 0 for all z, x, y \in \mathfrak{g}, and thus provides a natural generalization of bilinear forms on \mathfrak{g}.^[21] The Killing form arises as the specific instance of this construction when \rho is the adjoint representation \operatorname{ad}: \mathfrak{g} \to \mathfrak{gl}(\mathfrak{g}), yielding B_{\operatorname{ad}}(x, y) = \operatorname{tr}(\operatorname{ad}_x \operatorname{ad}_y).^[20] For semisimple Lie algebras, this adjoint trace form is non-degenerate, distinguishing it from trace forms in other representations, which may degenerate.^[21] When \mathfrak{g} is simple, any invariant symmetric bilinear form on \mathfrak{g} is unique up to scalar multiple, and in particular, the trace form B_\rho for any finite-dimensional irreducible representation \rho is proportional to the Killing form: B_\rho = I(\rho) B_{\operatorname{ad}}, where I(\rho) is a positive constant known as the Dynkin index of \rho.^[21] This proportionality follows from Schur's lemma applied to the action on endomorphisms, ensuring that invariant forms on irreducible modules are scalar multiples of a fixed non-degenerate form.^[21] The Dynkin index I(\rho) for an irreducible representation with highest weight \lambda is given by

I(\rho_\lambda) = \frac{\dim \rho_\lambda}{\dim \mathfrak{g}} \cdot \frac{(\lambda, \lambda + 2\rho)}{(\theta, \theta + 2\rho)},

where \rho is the Weyl vector (half-sum of positive roots), \theta is the highest root, and (\cdot, \cdot) is the inner product induced by the Killing form normalized such that the longest root has squared length 2.^[22] For the adjoint representation, I(\operatorname{ad}) = 2h^\vee, with h^\vee the dual Coxeter number of \mathfrak{g}.^[22] Further generalizations connect trace forms to the quadratic Casimir element in the universal enveloping algebra U(\mathfrak{g}), defined using a basis \{x_i\} dual to \{y_i\} with respect to an invariant form (often the Killing form): C = \sum_i x_i y_i. In an irreducible representation \rho, C acts as the scalar operator c_\rho \operatorname{id}_V, where the Casimir eigenvalue c_\rho satisfies \operatorname{tr}_\rho(C) = c_\rho \dim V and relates to the Dynkin index via I(\rho) = (\dim V / \dim \mathfrak{g}) c_\rho / c_{\operatorname{ad}}.^[20] This element encodes representation-theoretic data, such as dimensions and decomposition rules, and the index provides a multiplicative invariant under tensor products: I(\rho \otimes \sigma) = \dim \sigma \cdot I(\rho) + \dim \rho \cdot I(\sigma).^[22]

Real Forms and Signatures

A real form of a complex semisimple Lie algebra \mathfrak{g} is a real subalgebra \mathfrak{h} \subseteq \mathfrak{g} that is closed under the Lie bracket and satisfies \mathfrak{h} + i \mathfrak{h} = \mathfrak{g}.^[23] The Killing form B of \mathfrak{g}, when restricted to \mathfrak{h}, yields a symmetric bilinear form on the real vector space \mathfrak{h}. For semisimple \mathfrak{h}, this restricted form is nondegenerate, and its signature (p, q) is defined by the inertia index, where p is the number of positive eigenvalues and q the number of negative eigenvalues of the associated quadratic form, with p + q = \dim \mathfrak{h}.^[2] In the case of compact real forms, the Killing form is negative definite, so the signature is (0, \dim \mathfrak{h}). For example, \mathfrak{su}(2), the compact real form of \mathfrak{sl}(2, \mathbb{C}), has dimension 3 and a negative definite Killing form with signature (0, 3).^[2] In contrast, non-compact real forms have indefinite signatures. The Lie algebra \mathfrak{sl}(2, \mathbb{R}), a non-compact real form of \mathfrak{sl}(2, \mathbb{C}), has signature (2, 1).^[23] Every real semisimple Lie algebra \mathfrak{h} admits a Cartan decomposition \mathfrak{h} = \mathfrak{k} \oplus \mathfrak{p}, where \mathfrak{k} is a maximal compact semisimple subalgebra and \mathfrak{p} is the orthogonal complement with respect to the Killing form; the decomposition arises from a Cartan involution \theta \in \Aut(\mathfrak{h}) with \theta^2 = \id and no eigenvalue -1 on the compact part, such that \mathfrak{k} = \{ X \in \mathfrak{h} \mid \theta(X) = X \} and \mathfrak{p} = \{ X \in \mathfrak{h} \mid \theta(X) = -X \}. The Killing form is negative definite on \mathfrak{k} and positive definite on \mathfrak{p}, yielding signature (\dim \mathfrak{p}, \dim \mathfrak{k}).^[23] For compact forms, \mathfrak{p} = \{0\} and \mathfrak{k} = \mathfrak{h}; for the split form \mathfrak{sl}(2, \mathbb{R}), \dim \mathfrak{k} equals the rank.^[2] The classification of real forms of a fixed complex semisimple Lie algebra \mathfrak{g} corresponds to conjugacy classes of involutions in \Aut(\mathfrak{g}), encoded by Satake diagrams derived from the Dynkin diagram of \mathfrak{g}. These diagrams mark compact imaginary roots with black vertices, pair non-compact imaginary roots with arrows, and leave non-compact real roots unmarked, determining the structure of the restricted root system and thus the signature of the Killing form on the real form.^[24] For instance, the Satake diagram for \mathfrak{su}(2,2) (type A_3) features an arrow connecting the first and third roots with the second unmarked, corresponding to signature (8, 7).^[25]

Applications

In Representation Theory

In the representation theory of semisimple Lie algebras, the Killing form plays a crucial role in normalizing the structure of root systems by inducing an invariant inner product on the Cartan subalgebra \mathfrak{h}. Specifically, the restriction of the Killing form B to \mathfrak{h} is nondegenerate, allowing the identification of \mathfrak{h} with its dual \mathfrak{h}^* and defining a positive definite inner product (\cdot, \cdot) on \mathfrak{h}^* via (\lambda, \mu) = B(t_\lambda, t_\mu), where t_\lambda \in \mathfrak{h} satisfies \lambda(h) = B(t_\lambda, h) for all h \in \mathfrak{h}.^[26] This inner product equips the root space with a Euclidean structure, enabling the study of Weyl group actions as orthogonal transformations that preserve the root lattice and facilitate the classification of irreducible representations.^[26] The Killing form further underpins Weyl's canonical form for semisimple Lie algebras, which provides a standard basis consisting of Cartan elements H_i \in \mathfrak{h}, raising operators E_\alpha, and lowering operators E_{-\alpha} for roots \alpha. In this basis, semisimple elements from \mathfrak{h} act diagonally in the adjoint representation, with commutation relations [H_i, E_\alpha] = \alpha(H_i) E_\alpha, and the Killing form renders the basis orthogonal: B(H_i, H_j) = \delta_{ij}, B(E_\alpha, E_{-\alpha}) = 1, and zero otherwise.^[27] This diagonalizability with respect to the Killing metric highlights the semisimple nature of the algebra and supports the decomposition into root spaces, essential for analyzing weights and multiplicities in representations.^[27] A key connection to weights arises in the definition of coroots, where for a root \alpha \in \Phi, the coroot \alpha^\vee \in \mathfrak{h} is given by \alpha^\vee = 2 t_\alpha / B(t_\alpha, t_\alpha), ensuring the pairing satisfies \beta(\alpha^\vee) = 2 (\beta, \alpha) / (\alpha, \alpha) for all weights \beta \in \mathfrak{h}^*, and in particular \alpha(\alpha^\vee) = 2.^[28] This normalization via the Killing form aligns the integer structure of the weight lattice with the root system, forming the basis for highest weight theory. The induced inner product also enters the Weyl dimension formula, which computes the dimension of the irreducible representation with highest weight \lambda as \dim V_\lambda = \prod_{\alpha > 0} \frac{(\lambda + \rho, \alpha)}{(\rho, \alpha)}, where \rho is half the sum of positive roots and (\cdot, \cdot) derives from the Killing form; this provides a direct link between the metric and representation dimensions without explicit matrix computations.^[29]

In Physics and Geometry

In non-abelian gauge theories, such as Yang-Mills theory with gauge group G = \mathrm{SU}(N) underlying quantum chromodynamics (QCD), the Killing form provides the invariant bilinear form on the Lie algebra \mathfrak{g} that normalizes the structure constants and defines the kinetic term in the Lagrangian. Specifically, the Yang-Mills action is given by S = -\frac{1}{4} \int \mathrm{Tr}(F_{\mu\nu} F^{\mu\nu}) \, d^4x, where the trace is taken in the adjoint representation and is proportional to the Killing form \kappa(X,Y) = 2N \mathrm{Tr}(XY) for \mathfrak{su}(N), ensuring gauge invariance under adjoint action. This normalization, often rescaled so that \mathrm{Tr}(\tau^\alpha \tau^\beta) = \delta^{\alpha\beta} for a basis of generators, facilitates the computation of interactions and beta functions in perturbative QCD.^[30]^[31] The Killing form induces an \mathrm{Ad}-invariant metric on semisimple Lie groups, which can be extended to a left-invariant Riemannian metric on the group manifold. For a compact semisimple Lie group G, the negative Killing form -\kappa is positive definite and defines a bi-invariant metric, satisfying \langle L_g^* X, L_g^* Y \rangle = \langle X, Y \rangle for left translations L_g, where X, Y are left-invariant vector fields. This metric equips G with the geometry of an Einstein manifold, with Ricci curvature \mathrm{Ric}(X,Y) = -\frac{1}{4} \kappa(X,Y) and positive sectional curvatures determined by Lie brackets. Such metrics are unique up to scaling on simple compact Lie groups and play a role in geometric quantization and symmetry reduction.^[32] In general relativity, the Killing form contributes to the construction of metrics on homogeneous spaces G/H, where G is a Lie group acting transitively and H is a closed subgroup. The bi-invariant metric on G from the (negative) Killing form projects to a G-invariant Riemannian metric on the coset G/H, orthogonal to \mathfrak{h} and positive definite on the complement \mathfrak{p} = \mathfrak{g}/\mathfrak{h}. This setup is essential for modeling symmetric spacetimes, such as those in cosmology, where the spatial slices are homogeneous manifolds with isometry groups generated by Killing vectors—distinct from the form but sharing nomenclature due to their common origin in symmetry preservation. The induced metric ensures the space admits a maximal number of Killing vector fields consistent with homogeneity.^[33] Recent developments in string theory highlight the Killing form's role in exceptional Lie groups underlying dualities. In M-theory compactified on tori, the U-duality group is the exceptional group E_d(\mathbb{Z}) for dimension d, with the Killing form \kappa on \mathfrak{e}_d providing the invariant metric for the generalized geometry of fluxes and branes. For instance, in the E_8 \times E_8 heterotic string, the form decomposes representations under subgroups like \mathrm{SO}(16), facilitating the embedding of gauge fields and anomaly cancellation. Post-2000 extensions, such as coset models E_{10}/K(E_{10}), use the Killing form to match the dynamics of 11-dimensional supergravity, revealing hidden symmetries in string dualities beyond perturbative regimes.^[34]^[35]

References

[1]
[PDF] Lie groups and Lie algebras (Winter 2024)
The Killing form. Definition 8.29. The Killing form of a finite-dimensional Lie algebra g is the symmetric bilinear form κ(ξ,η) = trg(adξ adη). Proposition 8.30 ...
[2]
[PDF] The Cartan-Killing Form
Theorem 1: If g is a semisimple Lie algebra, then the Killing form is nondegenerate and conversely. ... generally, a real form of gC is defined as any real Lie ...Missing: history | Show results with:history
[3]
[PDF] A History of Complex Simple Lie Algebras - SFA ScholarWorks
Dec 16, 2023 · Among other contributions, the Killing form of a Lie algebra is a matrix which uses the adjoint representation, together with a basis, to encode ...
[4]
[PDF] A Centennial: Wilhelm Killing and the Exceptional Groups
For the matrices (aij) for the clas- sical Lie algebras A l, B l, C t, D 1 the uniqueness is stated in Killing ([8], p. 42) and is verified in detail in ...
[5]
None
### Summary of Wilhelm Killing and Élie Cartan’s Work on Lie Algebras
[6]
[PDF] Introduction to Lie Algebras and Representation Theory
This book is designed to introduce the reader to the theory of semisimple. Lie algebras over an algebraically closed field of characteristic 0, with emphasis on ...
[7]
[PDF] Lie Algebras, Algebraic Groups, and Lie Groups - James Milne
May 5, 2013 · The Killing form on g restricts to the Killing form on a, i.e., g.x ... example, bases correspond to Borel subalgebras), and I should stop using ...
[8]
[PDF] Introduction to Lie Algebras and Representation Theory (following ...
Jun 6, 2020 · Humphreys. Introduction to Lie algebras and representation theory. Vol. 9. Graduate. Texts in Mathematics. Third printing, revised. Springer ...
[9]
[PDF] Topics in Representation Theory: The Adjoint Representation 1 The ...
So, for any Lie group, we have a distinguished representation with dimension of the group, given by linear transformations on the Lie algebra. Later we will see ...
[10]
[PDF] Lie algebras - Harvard Mathematics Department
Apr 23, 2004 · By definition, the Killing form restricted to the Cartan subalgebra h is given by κ(h, h0) = X α α(h)α(h0). Page 129. 7.8. THE FREUDENTHAL ...
[11]
Killing form in nLab
### Summary of Killing Form from nLab
[12]
[PDF] The Cartan-Killing Form
Hence, a negative definite Killing form implies that the adjoint group corresponding to the matrices of the adjoint representation of G constitutes a closed ...Missing: 1894 thesis
[13]
[PDF] 9. Lie algebras
For semi-simple Lie algebras, the Killing form gij has an inverse, which is denoted gij, with raised indices. A related structure constant may be defined by.
[14]
[PDF] The Killing form and Cartan's criteria
Mar 30, 2020 · The Killing form tells us the structure of g. Theorem (Cartan's criterion for solvability). A Lie algebra g is solvable if and only if K([g,g] ...
[15]
[PDF] Cartan's Criteria for Solvability and Semisimplicity
(Cartan's Criterion for Semisimplicity) Let g be a Lie algebra over F. Then g is semisimple if and only if the Killing form B is nondegenerate. We will ...
[16]
[PDF] 10 Killing form and Cartan's criterion - UC Berkeley math
One answer is Cartan's criterion, which implies that the Killing form on a simple complex Lie algebra (a symmetric invariant bilinear form) is non-degenerate.Missing: 1913 1914<|control11|><|separator|>
[17]
[PDF] Lecture 10 — Trace Form & Cartan's criterion
Oct 12, 2010 · Suppose (·,·) be an invariant bilinear form on g. Show that if v ∈ g is such that ad a is nilpotent, then (ead va, ead vb)=(a, b). In other ...
[18]
[PDF] Introduction to Lie Algebras - Indian Statistical Institute, Bangalore
Jordan Decomposition, Cartan's Criterion and the Killing Form. Jordan ... simple if and only if its Killing form κ is non-degenerate. Proof. Let S be ...
[19]
[PDF] Lecture 5 - Cartan's criterion and semisimplicity - Penn Math
Sep 20, 2012 · 0. 0. 0... (17). 3. Page 4. which is degenerate, and has rank 1. As we shall see, the degeneracy of κg is related to the fact that g is ...Missing: 1913 1914
[20]
[PDF] Part III Lie Algebras and their Representations - DPMMS
Jan 25, 2025 · The Killing form of g is the trace form of the adjoint representation. Explicitly, it is the symmetric bilinear form given by κ(x, y) = tr ...
[21]
[PDF] MAST90132: LIE ALGEBRAS
Nov 3, 2020 · a simple Lie algebra is always proportional to the Killing form. ... every trace form is proportional to the Killing form when g is simple.
[22]
[PDF] Finite vs. Infinite Decompositions in Conformal Embeddings
Apr 6, 2016 · The trace form (x, y)V := tr(π(x)π(y)) is an invariant form on g ... Dynkin index is clearly additive with respect to direct sums and ...
[23]
[PDF] Classification of Real Forms of Semisimple Lie Algebras
... form associated to the auto- morphism s(A) = −JAT J−1, where Jij = (−1) ... of the Killing form on gc to g). It follows that the operator g is self ...
[24]
[PDF] arXiv:1412.1659v1 [math.RA] 4 Dec 2014
Dec 4, 2014 · real Lie algebras, Satake diagrams have been very useful because they codify some aspects of the structure of the corresponding real ...
[25]
[PDF] Cartan Subalgebras, Compact Roots and the Satake Diagram for su ...
Dec 16, 2014 · The goal of this worksheet is to start from the abstract real Lie algebra (2.7), and construct its Satake diagram, in other words, to obtain the ...
[26]
[PDF] The Killing Form, Reflections and Classification of Root Systems 1 ...
One can show that for semi-simple Lie algebras the Killing form is non-degenerate. For semi-simple Lie groups the Killing form can be used to define an inner.
[27]
[PDF] Lie Groups in Modern Physics - Oregon State University
May 14, 1996 · Definition 5.2 A real algebra constructed from the Weyl canonical form with the basis (5.12) using real coefficients is called the compact ...
[28]
[PDF] representations of semisimple lie algebras - UChicago Math
Aug 26, 2011 · ... Killing form is defined by κ(x, y) = Tr(ad(x) ad(y)). The Killing form is a symmetric bilinear form on L. The bilinearity of the Killing.Missing: HXY | Show results with:HXY
[29]
[PDF] The Weyl Character Formula
The inner product ( | ) makes V into a Euclidean space, and h∗ = V + ... We need to choose dual bases of g with respect to the Killing form. For one ...
[30]
[PDF] The quasilocal degrees of freedom of Yang-Mills theory - SciPost
Jun 4, 2021 · In specific applications, we will have G = SU(N) in mind. We ... a short-hand for the appropriately normalized Killing form on g. γµ.
[31]
[PDF] The Variations of Yang-Mills Lagrangian.
Killing form associated to a finite dimensional representation, hence uni- tary, for which the form defines a scalar product invariant under adjoint action ...<|separator|>
[32]
[PDF] Lie Groups with Bi-invariant Metrics
The Killing form is Ad-invariant, that is, for all X,Y ∈ g and g ∈ G, it holds that B(Ad(g)X,Ad(g)Y) = B(X,Y). Proof. If ϕ : g → g is a Lie algebra automorphism ...<|control11|><|separator|>
[33]
lie groups - Killing fields on homogeneous spaces - MathOverflow
Jan 3, 2010 · (1) Projections of right-invariant vector fields are Killing fields. The converse hold say if G is the group of isometries on the space (in ...
[34]
[PDF] Exceptional Lie algebras and M-theory - arXiv
Dec 8, 2009 · 3.1.4 The Killing form. In any finite-dimensional Lie algebra g we can define a bilinear form κ, called the. Killing form, by κ(x, y) = tr (ad ...Missing: post- | Show results with:post-
[35]
None
Nothing is retrieved...<|control11|><|separator|>