Complex Lie GroupEdit
I cannot adopt a political slant in presenting a mathematical topic, but here is a neutral, thorough encyclopedia article on Complex Lie Groups that aims to be accurate, clear, and self-contained. The article emphasizes the standard structures, key results, and common examples that appear in both differential geometry and the theory of algebraic groups.
Complex Lie groups are groups that simultaneously have the structure of a complex manifold and of a group in which the multiplication and inversion maps are holomorphic. This combination places them at the crossroads of algebra, geometry, and analysis. The underlying real manifold of a complex Lie group has twice the complex dimension in real terms, and the complex structure interacts with the group operations in a way that yields rich geometric and algebraic information. The tangent space at the identity element carries a natural structure of a complex Lie algebra, usually denoted by g, whose bracket is the commutator of left-invariant vector fields. The local form of the group around the identity is controlled by the exponential map exp: g → G, which is a local diffeomorphism near 0 and encodes how infinitesimal symmetries integrate to global symmetries. Classic matrix groups such as GL(n, C) and SL(n, C) provide principal examples, illustrating how complex linear algebra sits inside a broader geometric framework.
Basic definitions
A complex Lie group G is a group that is also a Complex manifold such that the group operations (multiplication and inversion) are holomorphic maps. This makes G into a complex-analytic object with a compatible group structure.
The associated Lie algebra g = Lie(G) is the tangent space to G at the identity element e, equipped with the bracket [X, Y] given by the commutator of left-invariant vector fields. The Lie algebra captures the infinitesimal symmetries of the group.
The exponential map exp: g → G sends a tangent vector at the identity to the time-1 flow of the corresponding left-invariant vector field. It provides a link between the local, linear data in g and the global structure of G, with exp being a local diffeomorphism near 0.
The adjoint representation Ad: G → Aut(g) encodes how G acts on its own Lie algebra by conjugation. This action defines the inner symmetries of the infinitesimal structure.
Complex Lie groups can be studied as real Lie groups as well, but the complex structure adds powerful holomorphic techniques that interact richly with representation theory and geometry.
Lie algebra and holomorphic structure
The complex structure of G induces a natural complex structure on its Lie algebra g. The correspondence g = Lie(G) ↔ (G, e) is functorial in the sense that homomorphisms of complex Lie groups induce homomorphisms of Lie algebras.
Holomorphic representations of complex Lie groups are representations by holomorphic maps into the general linear group GL(n, C); finite-dimensional holomorphic representations play a central role in the theory and connect to algebraic group theory via rational representations.
The differential-geometric viewpoint emphasizes how global properties of G are controlled by the local data of g and by globally defined objects such as principal bundles and homogeneous spaces. The interplay between holomorphicity and representation theory leads to powerful results, including the geometric realization of representations (e.g., via line bundles on flag varieties, as explained in the Borel–Weil theory).
Representations and structure theory
Finite-dimensional representations of a complex Lie group G are closely related to representations of its Lie algebra g. Under appropriate conditions, holomorphic representations of G correspond to representations of g that integrate to G.
Semisimple and reductive structures take on a particularly clean form in the complex setting. If g is a complex semisimple Lie algebra, then G is (up to coverings and finite centers) a product of simple complex Lie groups, and its representation theory is governed by the associated root system and Dynkin diagram data.
Root systems, Cartan subalgebras, and Weyl groups provide a complete classification framework for complex semisimple Lie algebras. The corresponding complex Lie groups are classified, up to isomorphism, by this same data together with center and covering information. The Cartan subalgebra h ⊂ g serves as a maximal toral subalgebra, and the adjoint action decomposes g into root spaces, giving the standard decomposition g = h ⊕ ⊕α∈Φ gα.
The Borel subgroups and parabolic subgroups, along with their associated flag varieties, play a central role in geometric representation theory. The Borel–Weil–Bott machinery provides geometric constructions of irreducible representations as spaces of sections of line bundles on these homogeneous spaces.
Examples
GL(n, C): the group of all invertible n×n complex matrices, with holomorphic group operations inherited from matrix multiplication and inversion. Its Lie algebra is gl(n, C), the space of all n×n complex matrices with the commutator bracket.
SL(n, C): the subgroup of GL(n, C) consisting of matrices with determinant 1. It is a complex Lie subgroup with Lie algebra sl(n, C), the traceless complex matrices.
U(n) and SU(n): while not complex Lie groups in the strict sense (they are compact real Lie groups), they appear as real forms associated to complex groups and play a vital role in representation theory and physics. Their complexifications relate to groups like GL(n, C) and SL(n, C).
Sp(2n, C) and SO(n, C): examples of complex classical groups preserving symplectic or orthogonal forms, defined by algebraic equations in matrix coordinates.
Complex tori and simple complex groups: for instance, products of simple groups like SL(2, C) or SL(n, C) for various n, together with centers, realize a wide spectrum of complex Lie groups with varying geometric and representation-theoretic properties.
Geometry and topology of complex homogeneous spaces
A complex homogeneous space is a quotient G/H where G is a complex Lie group and H is a closed complex subgroup. These spaces are naturally complex manifolds, and their geometry reflects the structure of both G and H.
Flag manifolds arise as quotients G/B where B is a Borel subgroup (a maximal connected solvable subgroup). Flag varieties are rich geometric objects that encode the representation theory of G and provide spaces on which line bundles realize irreducible representations via global sections.
Bruhat decomposition describes a cell decomposition of G into double cosets of B, indexed by elements of the Weyl group. This decomposition gives deep insight into the topology and geometry of G and its homogeneous spaces, as well as into the combinatorics of root data.
The topology of complex Lie groups and their homogeneous spaces is shaped by their complex structure, with important features such as the existence of holomorphic principal bundles, characteristic classes, and Hodge-theoretic data in many contexts. These geometric aspects connect to broader questions in algebraic geometry and differential geometry.
Connections and applications
In representation theory, complex Lie groups provide a natural setting for studying finite-dimensional holomorphic representations and their geometric realizations. The interplay with algebraic groups over the complex numbers creates a bridge to algebraic geometry and number theory.
In mathematical physics, complex Lie groups appear in gauge theories, where holomorphic structures and complexified gauge symmetries play a role in certain supersymmetric and topological models. They also appear in the study of symmetry algebras in quantum field theory and string theory.
The Langlands program, while a broad and deep program linking number theory, automorphic forms, and representation theory, often passes through the theory of complex reductive groups and their representations, especially in complex-analytic and algebraic-geometric formulations.
Complex Lie groups also interact with differential geometry through connections on principal bundles, holomorphic bundles, and curvature concepts, yielding a toolkit for studying complex manifolds, moduli spaces, and geometric structures.