For a connected simply connected source Lie group, every Lie algebra homomorphism integrates uniquely to a Lie group homomorphism. Along a path from the identity, transport its left logarithmic velocity by and solve the corresponding left-invariant differential equation in the target. The Maurer-Cartan equation and preservation of brackets make the endpoint invariant under fixed-endpoint homotopies. Simple connectedness removes path dependence, and path concatenation proves multiplicativity.
For a connected source, first integrate on its universal covering Lie group. The resulting map descends to the original group exactly when it kills the discrete kernel of the covering map. For the circle group, a real-linear map integrates to a circle endomorphism exactly when , since must map into itself. This is a global period condition invisible to the zero bracket of the circle's Lie algebra.
For a smooth homomorphism of Lie groups, naturality gives . Therefore locally. Apply this to the group commutator and differentiate its two curve parameters to obtain the displayed identity for the Lie bracket from local group commutators. In particular the differential of a group representation is a Lie algebra representation.
The representation ring of finite-dimensional modules for a complex semisimple Lie algebra is the abelian group generated by module classes, with , and multiplication . Weyl complete reducibility theorem makes the classes of irreducible representations a free integral basis. The formal character map identifies this ring with the Weyl group invariant part of the group ring of a weight lattice. Injectivity follows from distinct highest weights; surjectivity follows by subtracting the character with each maximal dominant weight, whose highest-weight coefficient is one, from a Weyl-invariant finite sum.
// Target: semisimple-lie-algebra.bigb
For a Lie subalgebra , its defining representation is its given action on the vector space . The Lie bracket identity verifies the representation property. For the special linear Lie algebra, diagonal trace-zero matrices have coordinate weights of a representation on the standard basis; for a Special orthogonal Lie algebra, the same action preserves the defining symmetric bilinear form.
For a unitary derived representation , its generators are anti-Hermitian. Define Hermitian observables ; then their ordinary operator commutator is . Thus real Lie algebra structure constants and Hermitian-operator commutation relations differ by an . Products in a universal enveloping algebra require one conversion factor for every generator. In particular, the abstract bilinear Pauli-Lubanski element has representation image minus the pseudovector formed from the Hermitian physical generators.
A linear map between Lie algebra representations is a representation homomorphism when it commutes with the action of every element. Its kernel and image of a linear map are invariant subspaces. This is a map of modules, distinct from a Lie algebra homomorphism between the acting algebras themselves.
A Lie algebra representation on induces one on each exterior power by acting on every factor and summing. The tensor product of Lie algebra representations preserves the defining alternating relations, so this descends to the quotient in every characteristic of a field. In characteristic two define the exterior algebra by the relations , rather than by dividing a tensor antisymmetrizer by .
The abelian Lie algebra , for , is a nilpotent Lie algebra but contains the nonnilpotent identity map. Thus nilpotence of the abstract Lie algebra does not imply that a particular Lie algebra representation consists of nilpotent endomorphisms. Engel theorem requires nilpotence of every represented endomorphism to obtain strict upper triangularity.
The tensor product of Lie algebra representations has action . The Lie algebra representation identity follows because operators on the two different tensor factors commute. This action descends to exterior powers and symmetric powers.
The summands have Dynkin labels , and . Convolving the triplet and octet weight diagrams and applying highest-weight character subtraction removes these three characters in that order. Their dimensions match the product.
// Target: quantum-field-theory.bigb
For the defining two-dimensional sl2 Lie algebra module with coordinates , put . Its three highest-weight vectors are , and , of weights five, three and one. Lowering the last once gives , an explicit second vector of its two-dimensional irreducible summand.
Give its Adjoint representation. The representation identity gives , so the action map is a Lie algebra representation homomorphism. For a nontrivial simple module it is surjective. If the acting algebra is semisimple and the module is finite dimensional, Weyl complete reducibility theorem splits the surjection and makes a direct summand.
Take the Verma module for the sl2 Lie algebra of highest weight , with basis and , , . It is simple. In , the highest vectors of weight are multiples of . Every homomorphism to kills , whose weight is zero, and hence kills . Every embedded copy of would have its highest vector proportional to , so no projection onto such a copy can exist. Thus semisimplicity of the Lie algebra alone does not provide an action summand in infinite dimension.
For , , the tensor product of Lie algebra representations for the special linear Lie algebra splits as displayed. The exterior product map has an equivariant right inverse obtained by dividing its three-term alternating insertion by three. Its kernel is the Schur functor of shape , with highest weight and dimension . The other summand has highest weight for and is trivial for .
The weight space of has dimension one when . The weight space of has dimension three for distinct indices. Restricting these functionals to the traceless diagonal Cartan subalgebra creates no additional coincidences; for the latter weight is zero.
For the complex special linear Lie algebra , tensoring an irreducible highest-weight representation by the defining representation gives the displayed sum, omitting terms with negative Dynkin labels. It follows by multiplying its Weyl character formula by . In particular , with dimensions .
For a finite-dimensional real Lie algebra, a finite-dimensional Lie algebra representation integrates uniquely to a smooth representation of the connected simply connected Lie group with that Lie algebra. For another connected group with the same Lie algebra, integration is possible exactly when the representation of its universal cover is trivial on the covering kernel. This separates local Lie-algebra classification from global representation restrictions.
A bilinear form on a Lie algebra representation is invariant when it satisfies the displayed identity. Equivalently, is an intertwining operator from the representation to its dual Lie algebra representation. For a finite-dimensional irreducible representation over an algebraically closed field, any nonzero such form is nondegenerate, and the Schur lemma makes all invariant forms proportional. In characteristic different from two, transposing twice then proves that the form is a symmetric bilinear form or an alternating bilinear form.
The dual of a Lie algebra representation on acts on its dual space by . The minus sign makes this a Lie algebra homomorphism: taking transposes reverses the order in a commutator. It is the infinitesimal version of the dual representation of a group.
For a finite-dimensional irreducible representation of a complex semisimple Lie algebra, its dual Lie algebra representation has highest weight , where is the longest Weyl-group element. Indeed duality negates all weights, and is the lowest weight of the original module. In particular makes every such irreducible module self-dual.
The trivial representation sends every element of the Lie algebra to the zero linear map. For a complex semisimple Lie algebra, the irreducible representation of highest weight zero is the one-dimensional trivial representation.
For Lie algebra representations and , the Hom representation on isIts invariant vectors are exactly the intertwining maps.
A branching rule decomposes a representation into irreducible representations after restriction from a Lie group or Lie algebra to a subgroup or subalgebra.
A root string of length gives an irreducible sl2 Lie algebra module when restricting the Adjoint representation to the sl2 subalgebra associated with a root. Include also the root's own and one commuting Cartan direction . For either simple root gives . For , the long root gives and the short root gives . For , the long root gives and the short root gives .
An oriented three-dimensional Lie algebra has its antisymmetric structure constants encoded by the displayed matrix, whose symmetric part is and whose antisymmetric part uniquely defines the covector . In a volume-normalized basis, the inverse is . The only independent triple in the Jacobi identity has coefficient , so Jacobi holds exactly when . This decomposition is useful in classifying three-dimensional Lie algebras.
Lower the output index of the structure constants of a Lie algebra with its Killing form. The Lie bracket makes the result antisymmetric in the first two indices, while invariance gives , making it antisymmetric in the last two indices. These two adjacent transpositions imply total antisymmetry. Nondegeneracy of the Killing form is unnecessary.
// Target: lie-theory.bigb
Every finite-dimensional Lie algebra over a field of characteristic zero has a finite-dimensional faithful Lie algebra representation. Thus it embeds as a Lie subalgebra of a matrix algebra, using the matrix commutator as bracket. Faithfulness is crucial for detecting central elements: the Adjoint representation alone kills the center of a Lie algebra.
A nonzero Lie algebra representation is irreducible when it has no proper nonzero invariant subspace.
The adjoint representation is . For a semisimple Lie algebra, its nonzero weights are the roots and its zero-weight space is the Cartan subalgebra.
The complex Adjoint representation of SU(3) is the action by conjugation on traceless matrices. With and , its highest-weight vector is of Dynkin labels . The six off-diagonal matrix units have weights , and the two-dimensional Cartan subalgebra has weight zero. In coordinates , the weight diagram is a regular hexagon together with a central point of weight multiplicity two. This representation describes both the gluon colour multiplet and the approximate flavour baryon octet, under different physical SU(3) symmetries.
For traceless matrices transforming by conjugation under SU(3), the trace supplies a singlet, the commutator an antisymmetric octet, and the traceless anticommutator a symmetric octet. Symmetrizing the two upper and two lower indices and removing traces supplies the 27. Contracting an antisymmetric lower pair with the invariant alternating tensor and symmetrizing the remaining three upper indices supplies the 10, and the conjugate construction supplies the conjugate 10. These nonzero equivariant maps exhaust the dimensions: the symmetric part has , the antisymmetric part .
For the general linear Lie algebra of all matrices over a field of characteristic zero, write on the matrix space. The traces of and are , whereas both mixed traces are . Expanding gives the displayed Killing form. Its radical is the scalar matrices because for every forces . Thus the full general linear algebra is not semisimple, even though its traceless summand is semisimple for .
// Target: lie-theory.bigb
Every Lie algebra automorphism obeys . The cyclic property of the trace therefore makes the Killing form invariant under . For a matrix Lie group, is conjugation by , proving the displayed identity even when the group is disconnected. This directly proves gauge invariance of Killing-form contractions of adjoint fields.
For , , and , the Adjoint representation matrices give the displayed Killing form. Cross traces vanish. Nonzero make it nondegenerate and hence the algebra semisimple. If the three constants have the same sign it is negative-definite; mixed signs give signature . In particular is the compact case, whereas is the split real traceless two-by-two algebra. This gives a direct compactness criterion from the Killing form in a small explicit example.
For an ideal of a Lie algebra and nondegenerate Killing form, invariance makes an ideal. An element of commutes with , so the intersection is an abelian ideal. Abelian ideals lie in the radical of the Killing form, forcing the intersection to vanish. Thus as commuting ideals. Iterating minimal nonzero ideals gives the direct-sum formulation of a semisimple Lie algebra.
The radical of a bilinear form of the Killing form is the ideal of a Lie algebra . It is zero for a complex semisimple Lie algebra. Every element of the center of a Lie algebra lies in this radical.
A finite-dimensional real Lie algebra has negative-definite Killing form exactly when it is compact semisimple. In the compact direction, average an inner product over a compact group; all adjoint maps are skew, so , and semisimplicity removes the center. Conversely, negative-definiteness gives the positive invariant metric , embeds the faithful adjoint algebra in an orthogonal algebra, and nondegeneracy gives semisimplicity. This is an algebraic compactness criterion, not a claim that every global group with an Abelian compact-type algebra is compact.
If is an abelian ideal of a Lie algebra, then maps into and vanishes on whenever . Its trace is zero, so . Every nonzero solvable ideal has a last nonzero term in its derived series of a Lie algebra, which is such an abelian ideal. Consequently a nondegenerate Killing form forces the solvable radical to vanish.
The radical of the Killing form is a solvable ideal. Invariance makes it an ideal, and the Cartan solvability criterion applied to its adjoint image proves solvability.
A finite-dimensional complex Lie algebra is semisimple if and only if its Killing form is nondegenerate.
For a finite-dimensional complex Lie algebra , the radical of its Killing form is a Lie algebra ideal by invariance. For , their adjoint actions vanish on , so . The Cartan solvability criterion makes the image solvable; its kernel is the abelian center, so is solvable. A semisimple Lie algebra has no nonzero solvable ideal, forcing its Killing radical to vanish.
Every invariant bilinear form on a finite-dimensional complex simple Lie algebra is a scalar multiple of its Killing form. A nondegenerate invariant form identifies the algebra with its dual; comparing this identification with the Killing form gives an endomorphism of the irreducible adjoint representation, so Schur lemma makes it scalar.
The usual uniqueness theorem assumes a complex simple Lie algebra, or a real form with simple complexification. For a complex simple algebra regarded as real, both and define a real symmetric invariant bilinear form on a Lie algebra, and they are linearly independent. The real Killing form is . For and , and , proving the independence. This real algebra is simple: its real complexification is two complex simple ideals exchanged by conjugation, and a real ideal would complexify to a conjugation-stable sum of these ideals, hence either zero or the whole algebra.
For ,
Every invariant subspace of the Adjoint representation of a Lie algebra is an ideal. Hence the adjoint representation of a simple Lie algebra is irreducible.
For a finite-dimensional Lie algebra representation , its trace form is the symmetric bilinear formIt is invariant: .
For a finite-dimensional complex representation of a solvable Lie algebra, the Lie theorem gives a basis in which all representing matrices are upper triangular. Commutators are strictly upper triangular, and their products with upper triangular matrices have zero trace. Thus the derived algebra lies in the radical of a bilinear form. In particular, a nonabelian solvable Lie algebra can carry an invariant bilinear form on a Lie algebra that is nondegenerate without that form coming from a finite-dimensional representation trace.
// Target: semisimple-lie-algebra.bigb
The cubic trace tensor of a Lie algebra representation is cyclic by the cyclic property of the trace. It need not be fully antisymmetric. Its antisymmetric part in the first two arguments is fixed by the Trace form of a Lie algebra representation:
// Target: lie-theory.bigb
Lowering and raising structure constants with the Killing form gives . Since is antisymmetric in , only the commutator part of the cubic trace tensor of a Lie algebra representation contributes, proving . In an adapted compact basis with and , the resulting sign is positive: . Defining the lowered structure constants with a positive Euclidean metric instead would reverse this sign.
// Target: lie-theory.bigb
// Target: lie-theory.bigb
For a nontrivial finite-dimensional anti-Hermitian Lie algebra representation of a compact Lie algebra that is simple, its Trace form of a Lie algebra representation is a positive multiple of the Killing form. Define by . Invariance makes commute with the Adjoint representation of a Lie algebra. Symmetry makes self-adjoint for , so its eigenspaces are ideals. Simplicity implies . The kernel of the representation is an ideal and hence vanishes; therefore for nonzero , proving .
// Target: lie-theory.bigb
For a finite-dimensional representation , the trilinear form is invariant under simultaneous adjoint action. This follows by writing the sum of its three infinitesimal variations as the trace of a commutator.
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