Lie groups

Lie groups
Name	Lie groups
Field	Differential geometry, Abstract algebra
Namedafter	Sophus Lie
Related	Topological group, Manifold, Group representation

Lie groups. In mathematics, a Lie group is a smooth manifold that is also a group where the group operations of multiplication and inversion are smooth maps. This elegant fusion of continuous symmetry and algebraic structure makes them a central object in modern geometry and theoretical physics. The theory was founded by the Norwegian mathematician Sophus Lie in the late 19th century to study symmetries of differential equations.

Definition and basic examples

A Lie group is defined as a differentiable manifold \(G\) equipped with a group structure such that the map \(G \times G \to G\) given by \((g, h) \mapsto gh^{-1}\) is smooth. Basic examples include the general linear group \(\text{GL}(n, \mathbb{R})\) of invertible \(n \times n\) real matrices, which is fundamental to Linear algebra. Important subgroups are the special linear group \(\text{SL}(n, \mathbb{R})\) of matrices with determinant one, the orthogonal group \(\text{O}(n)\) preserving the standard inner product, and the unitary group \(\text{U}(n)\) in complex spaces. The circle group \(\mathbb{S}^1\) and the real numbers \(\mathbb{R}\) under addition are simple one-dimensional examples, while the torus \(\mathbb{T}^n\) provides a compact abelian case. The Heisenberg group illustrates a non-commutative, nilpotent structure important in Quantum mechanics.

Lie algebra of a Lie group

To every Lie group \(G\) one can associate a Lie algebra \(\mathfrak{g}\), a vector space equipped with a bilinear, antisymmetric bracket operation satisfying the Jacobi identity. This algebra is constructed as the tangent space at the identity element of \(G\), capturing the infinitesimal structure of the group. For matrix groups like \(\text{GL}(n, \mathbb{C})\), the Lie algebra consists of all \(n \times n\) matrices with the commutator as the bracket, as seen with the special unitary group \(\text{SU}(2)\) and its algebra \(\mathfrak{su}(2)\). The exponential map provides a local diffeomorphism from \(\mathfrak{g}\) into \(G\), connecting infinitesimal generators to group elements, a concept pivotal in the work of Élie Cartan. This correspondence is foundational for classifying and studying Lie groups through their linearized counterparts.

Classification and structure theory

The classification of Lie groups is a monumental achievement in 20th-century mathematics, largely due to the efforts of Wilhelm Killing, Élie Cartan, and Hermann Weyl. Simple Lie groups, which have no non-trivial normal subgroups, are classified into four infinite families—the classical groups like the special orthogonal groups \(\text{SO}(n)\) and the symplectic groups \(\text{Sp}(2n)\)—and five exceptional groups, including \(\text{G}_2\) and \(\text{E}_8\). The structure theory distinguishes between compact groups, such as \(\text{SU}(n)\), and non-compact forms like the Lorentz group \(\text{SO}(1,3)\). Semisimple groups decompose into simple factors, a result formalized by the Levi decomposition theorem. The classification is intimately tied to root systems and Dynkin diagrams, tools developed by Eugene Dynkin.

Representation theory

Representation theory studies how Lie groups act linearly on vector spaces, providing a bridge between abstract symmetry and concrete linear algebra. A fundamental result is the Peter–Weyl theorem, which generalizes Fourier analysis to compact groups. For complex semisimple groups, the theory of highest-weight representations, pioneered by Hermann Weyl and Harish-Chandra, gives a complete classification of irreducible representations. Key examples include the spin representations of the spin groups and the adjoint representation where a group acts on its own Lie algebra. The Langlands program, initiated by Robert Langlands, seeks deep connections between representations of Lie groups and number theory.

Applications in physics

Lie groups are the mathematical language of symmetry in modern theoretical physics. In Classical mechanics, the Galilean group describes the symmetries of Newtonian spacetime, while in special relativity, the Poincaré group—incorporating the Lorentz group and translations—is fundamental. The Standard Model of particle physics is built upon gauge groups like \(\text{SU}(3) \times \text{SU}(2) \times \text{U}(1)\), with \(\text{SU}(3)\) corresponding to quantum chromodynamics and the strong force. The Higgs mechanism involves spontaneous symmetry breaking within such groups. In quantum mechanics, the angular momentum algebra is \(\mathfrak{su}(2)\), and the Heisenberg group underlies the canonical commutation relations.

Homogeneous spaces and symmetric spaces

A homogeneous space is a manifold on which a Lie group acts transitively; such spaces are of the form \(G/H\) where \(H\) is a closed subgroup. These spaces are ubiquitous in geometry, with examples including projective spaces, spheres like \(\mathbb{S}^2 \cong \text{SO}(3)/\text{SO}(2)\), and Grassmannians. A particularly important subclass are symmetric spaces, introduced by Élie Cartan, which possess an involutive geodesic symmetry at each point. They arise as quotients \(G/K\) where \(K\) is a maximal compact subgroup, and include hyperbolic spaces like the Poincaré half-plane and compact examples such as complex projective space. Their study connects differential geometry, Klein's Erlangen program, and the theory of automorphic forms.

Category:Lie groups Category:Differential geometry Category:Group theory