- "In mathematics, the mathematician Sophus Lie ( LEE) initiated lines of study involving... Lie theory."
Study of mathematical structures called Lie groups and Lie algebras.
Algebraic structures: Lie algebras are a linear algebraic representation of Lie groups, and they provide the foundation for modern Lie theory. Understanding the basic properties of algebraic structures such as groups, rings, vector spaces, and homomorphisms is important.
Differential Equations: Lie theory is closely connected to the theory of differential equations, as it plays an important role in the study of differential geometry, which is concerned with the geometry of smooth curves and surfaces.
Representation Theory: The theory of representations investigates the algebraic and geometric properties of group actions on vector spaces.
Symmetry: Lie groups play a fundamental role in the study of symmetry, particularly in the context of physics. Lie groups and their representations are employed in quantum mechanics, quantum field theory, and general relativity.
Geometry: Lie theory is particularly interested in the geometric structure of Lie groups and Lie algebras, which is studied in differential geometry.
Algebraic Topology: Algebraic topology is the study of topological spaces using algebraic tools. Lie theory often employs these tools to understand the topology of Lie groups.
Quantum mechanics: The study of the behavior of matter and energy at a scale smaller than atomic scales. In quantum mechanics, Lie groups play an important role in understanding the symmetries of quantum systems.
Differential geometry: Differential geometry is the mathematical discipline that deals with the study of curvature and other geometric properties of surfaces and manifolds.
Homological algebra: Homological algebra is a branch of mathematics concerned with the study of algebraic structures via the use of exact sequences and cohomology.
Topology: Topology is the mathematical discipline that studies the properties of objects that remain unchanged under continuous transformations, such as stretching and bending.
Category Theory: Category theory is a branch of mathematics concerned with the formal study of structure, composition, and transformation. Lie theory makes extensive use of category theory to study the properties of Lie groups.
Lie automorphisms: Lie automorphisms are morphisms between Lie groups that preserve the Lie bracket. They have important applications in algebraic topology, representation theory, and mathematical physics.
Cohomology theory: Cohomology theory is a mathematical discipline that studies the properties of differential forms on a manifold, and their relationship to cohomology groups.
Lie superalgebras: Lie superalgebras are algebras that generalize Lie algebras by introducing a superstructure on the underlying vector space. They have important applications in physics and representation theory.
Representation categories: Representation categories are categories with additive and linear structures that encode the representation theory of a given algebraic structure, such as Lie groups.
Quantum groups: Quantum groups are a family of algebraic structures that generalize Lie groups and are defined as deformations of the enveloping algebra of a Lie algebra.
Noncommutative geometry: Noncommutative geometry is a mathematical discipline that studies the properties of spaces whose coordinates do not commute. Lie groups and their representations have important applications in noncommutative geometry.
Geometric quantization: Geometric quantization is a mathematical procedure that associates a Hilbert space to a given classical phase space. Lie groups and their representations play an important role in geometric quantization.
Symplectic geometry: Symplectic geometry is the study of smooth manifolds with a special class of differential forms, called symplectic forms, which encode the geometric aspects of Hamiltonian mechanics.
Kac-Moody algebras: Kac-Moody algebras are a class of infinite-dimensional Lie algebras that generalize finite-dimensional simple Lie algebras. They have important applications in representation theory and mathematical physics.
Classical Lie Theory: This is the study of finite-dimensional real Lie algebras and Lie groups.
Representation Theory: The study of representations of Lie groups and Lie algebras, including unitary representations on Hilbert spaces.
Differential Geometry: This studies the geometric structures on manifolds that are preserved by Lie groups.
Geometry of Lie Algebras: This studies the structure of Lie algebras and their representations, including Cartan subalgebras, root systems, and Weyl groups.
Quantum Groups: This deals with noncommutative analogues of Lie groups and Lie algebras, which arise in mathematical physics and statistical mechanics.
Symplectic Lie Groups: This studies the geometry of symplectic manifolds that are equipped with a Hamiltonian action of a Lie group, including moment maps and the symplectic reduction.
Homological Algebra: This uses algebraic topology and category theory to study the cohomology and K-theory of Lie groups and Lie algebras.
Mathematical Quantum Field Theory: This focuses on the mathematical foundations of quantum field theory, including the renormalization of Feynman integrals and the notion of a field algebra.
Lie Theory in Number Theory: This applies Lie groups and Lie algebras to the study of number theory, including automorphic forms, zeta functions, and the Langlands program.
Lie Theory and Differential Equations: This uses Lie groups and Lie algebras to study the integrability and symmetry of differential equations, including the Painleve equations and the Korteweg-de Vries equation.
Lie Theory and Harmonic Analysis: This studies the representations of Lie groups and the Fourier analysis on homogeneous spaces, including the Plancherel and Paley–Wiener theorems.
- "...involving integration of differential equations, transformation groups, and contact of spheres."
- "For instance, the latter subject is Lie sphere geometry."
- "...wilhelm Killing and Élie Cartan."
- "The foundation of Lie theory is the exponential map relating Lie algebras to Lie groups which is called the Lie group–Lie algebra correspondence."
- "The subject is part of differential geometry since Lie groups are differentiable manifolds."
- "The structure of a Lie group is implicit in its algebra."
- "...the structure of the Lie algebra is expressed by root systems and root data."
- "Lie theory has been particularly useful in mathematical physics since it describes the standard transformation groups..."
- "...the Galilean group, the Lorentz group, the Poincaré group and the conformal group of spacetime."
- "Sophus Lie initiated lines of study involving integration of differential equations, transformation groups, and contact of spheres..."
- "...was worked out by Wilhelm Killing and Élie Cartan."
- "The exponential map relating Lie algebras to Lie groups which is called the Lie group–Lie algebra correspondence."
- "Lie groups are differentiable manifolds."
- "The tangent vectors to one-parameter subgroups generate the Lie algebra."
- "The structure of a Lie group is implicit in its algebra."
- "The structure of the Lie algebra is expressed by root systems and root data."
- "For instance, the latter subject is Lie sphere geometry."
- "...the Galilean group, the Lorentz group, the Poincaré group and the conformal group of spacetime."
- "Lie theory has been particularly useful in mathematical physics..."