Lie GroupEdit
Lie groups sit at the crossroads of algebra and geometry, providing a language for continuous symmetries that occur across mathematics and physics. They are objects that are simultaneously a group and a differentiable manifold, with the group operations—multiplication and inversion—being smooth maps. This dual nature makes Lie groups indispensable for studying how smooth transformations compose and invert, and how those transformations act on spaces that carry geometric or physical meaning. In short, Lie groups formalize the idea that symmetry behaves in a way that can be analyzed using calculus and geometry.
In practice, the study of Lie groups revolves around the corresponding Lie algebra, the tangent space at the identity equipped with a bracket that encodes infinitesimal commutators. The exponential map links the Lie algebra to the Lie group, turning linear data into genuine group elements via one-parameter subgroups. Representation theory then asks how Lie groups can act linearly on vector spaces, revealing the ways symmetry can manifest in linear spaces and operators. This blend of algebra, geometry, and analysis underpins much of modern mathematics and theoretical physics.
Definitions and basic notions
A Lie group is a set G that is both a group and a differentiable manifold, with the group operations - multiplication m: G × G → G, (g, h) ↦ gh, and - inversion inv: G → G, g ↦ g^{-1} being smooth maps. The manifold structure is finite-dimensional and compatible with the group structure, so G is a smooth manifold whose algebraic and geometric features inform each other.
The Lie algebra g of a Lie group G is the tangent space at the identity element e, equipped with the Lie bracket [X, Y] that arises from the commutator of left-invariant vector fields. The Lie algebra captures the infinitesimal aspect of the group, and many global properties of G are reflected in its Lie algebra. The exponential map - exp: g → G provides a bridge from linear data to the nonlinear group, sending a tangent vector to the endpoint of the corresponding one-parameter subgroup.
Key constructions - Left- and right-invariant vector fields on G, which reflect how the group structure translates infinitesimal information across the manifold. - The distinction between connected components of G; the identity component G_0 is itself a Lie group and plays a central role in classification and representation theory. - The notion of a finite-dimensional Lie group; many fundamental results rely on this finiteness to ensure a well-behaved differential structure.
In the study of Lie groups, one encounters a rich interplay between topology, algebra, and analysis. The global topology of a Lie group interacts with its algebraic structure, leading to phenomena such as covering groups, fundamental groups, and classification results for certain classes of groups.
Examples
Lie groups include many familiar matrix groups, each equipped with a natural smooth structure coming from matrix multiplication and inversion. - the circle group circle group: the unit complex numbers under multiplication; a compact, one-dimensional Lie group. - the general linear group General linear group: all invertible n×n real matrices; dimension n^2. - the special linear group Special linear group: invertible matrices with determinant 1; a characteristic example of a noncompact simple group for n ≥ 2. - the orthogonal group Orthogonal group: real matrices preserving the standard inner product with determinant 1; compact and important in rotation symmetry. - the unitary group Unitary group: complex matrices preserving the standard Hermitian form; compact and central in quantum mechanics. - the special unitary group Special unitary group: unitary matrices with determinant 1; fundamental in particle physics and representation theory. - the symplectic group Symplectic group: matrices preserving a standard symplectic form; arises in classical and quantum mechanics. - the Poincaré group Poincaré group: semidirect product of the Lorentz group with translations; encodes spacetime symmetries in special relativity. - the Heisenberg group Heisenberg group: a nilpotent Lie group central to quantum mechanics and harmonic analysis.
Each example carries a natural smooth manifold structure in addition to its group operation, and the dimension of the group matches the dimension of its Lie algebra. The circle group, for instance, is 1-dimensional, while GL(n, R) is n^2-dimensional.
Lie algebras and the exponential map
The Lie algebra g of a Lie group G is a linear space that encodes infinitesimal symmetries of G. The bracket [X, Y] in g reflects how the corresponding flows fail to commute, translating the nonlinear geometry of G into a manageable algebraic object. The exponential map, exp: g → G, sends a tangent vector to the endpoint of the integral curve defined by the associated left-invariant vector field. In many familiar groups, exp is surjective onto a neighborhood of the identity, and for connected, compact Lie groups it is surjective onto all of G.
A central principle is Lie's correspondence: to each finite-dimensional real Lie algebra, one can (locally) integrate to a Lie group, and, up to covering problems, the global structure of the group is controlled by its algebra. This correspondence provides a two-way dictionary between linear, infinitesimal data and nonlinear, global symmetry groups. For many standard groups, the Lie algebras are explicitly described by matrices with the same bracket given by commutators, linking algebraic and geometric viewpoints.
Representations
A representation of a Lie group G is a homomorphism from G into the group of automorphisms of a vector space, typically realized as a homomorphism into GL(V). Representations illuminate how a group can act by linear transformations on spaces of interest. Unitary representations, in particular, preserve a Hermitian form and are central to quantum mechanics and harmonic analysis.
Key results in representation theory include - complete reducibility for compact Lie groups: every finite-dimensional representation decomposes into irreducible pieces (Peter–Weyl theory). - the role of highest-weight theory for semisimple Lie groups, which leads to a combinatorial classification via root systems and Dynkin diagrams. - the connection between representations and symmetric spaces, geometric quantization, and gauge theories in physics.
Geometry and topology
Lie groups are smooth manifolds on which the group structure imposes a rich geometric framework. Left- and right-invariant vector fields provide natural ways to study differential-geometric properties; the Lie algebra can be realized as the space of left-invariant vector fields with the bracket given by the commutator. The global topology of a Lie group interacts with its algebraic structure, producing phenomena such as: - coverings and universal covers: many Lie groups are not simply connected, but their universal cover is a simply connected Lie group with the same Lie algebra. - classification results in specific families, particularly for compact or semisimple groups, where the Lie algebra structure strongly constrains the global geometry. - homogeneous spaces, formed by quotienting a Lie group by a closed subgroup, which carry natural geometric and topological structures.
Applications in physics and beyond
Lie groups provide the mathematical language for continuous symmetries in physics. Gauge theories use local symmetry groups described by Lie groups; the Standard Model relies on the gauge group SU(3) × SU(2) × U(1) to organize particle interactions. In classical and quantum mechanics, Lie groups model rotational, translational, and more abstract symmetries; in general relativity, spacetime symmetries are studied via Lie groups like the Poincaré group or the Lorentz group. Beyond physics, Lie groups appear in differential equations, geometry, number theory, and data science, whenever a problem exhibits a smooth symmetry that can be analyzed through group actions and their linear representations.
History and development
The subject bears the name of Sophus Lie, whose work on continuous transformation groups laid the groundwork for a systematic study of symmetries. Early developments by Lie and his successors evolved into a full differential-geometric theory under the influence of Elie Cartan and later by Weyl, Chevalley, and others. The modern theory integrates tools from algebra, topology, and analysis, yielding a robust framework for studying symmetry in both abstract mathematics and physical theories. Key figures in the development include Sophus Lie, Elie Cartan, and Hermann Weyl.