🔁Elementary Differential Topology Unit 11 – Lie Groups and Algebras in Topology

Key Concepts and Definitions

• Lie group: A smooth manifold that also has a group structure compatible with its smooth structure
• Smooth manifold: A topological space that locally resembles Euclidean space and has a globally defined differential structure
• Group structure: Consists of an identity element, inverse elements, and an associative binary operation satisfying the group axioms
• Lie algebra: A vector space equipped with a non-associative, alternating bilinear operation called the Lie bracket
• Exponential map: A map from the Lie algebra to the Lie group that relates the two structures
  • Defined as $\exp: \mathfrak{g} \to G$, where $\mathfrak{g}$ is the Lie algebra and $G$ is the Lie group
• Adjoint representation: A representation of a Lie group $G$ on its own Lie algebra $\mathfrak{g}$ given by the differential of conjugation
• Baker-Campbell-Hausdorff formula: Relates the Lie bracket in the Lie algebra to the group operation in the Lie group

Historical Context and Development

• Sophus Lie, a Norwegian mathematician, introduced the concept of continuous transformation groups in the late 19th century
• Lie's work was motivated by the study of differential equations and their symmetries
• Élie Cartan further developed the theory of Lie groups and Lie algebras in the early 20th century
  • Introduced the notion of a Cartan subalgebra and the root space decomposition
• Hermann Weyl and Claude Chevalley made significant contributions to the structure and classification of semisimple Lie algebras
• Lie groups and algebras have since found applications in various areas of mathematics and physics
  • Gauge theories, quantum mechanics, and general relativity all utilize Lie groups and algebras
• The development of Lie theory has led to a deeper understanding of symmetries and their role in mathematical structures

Fundamental Structures of Lie Groups

• Smooth manifold structure: Lie groups are first and foremost smooth manifolds
  • The smooth structure allows for the use of differential calculus on Lie groups
• Group operation: Lie groups have a group operation that is compatible with the smooth structure
  • The group operation is a smooth map $G \times G \to G$ satisfying the group axioms
• Identity element: Every Lie group has a unique identity element $e$ such that $ge = eg = g$ for all $g \in G$
• Inverse map: The inverse map $i: G \to G$ sending each element to its inverse is a smooth map
• Left and right translations: For each $g \in G$, the left translation $L_g$ and right translation $R_g$ are diffeomorphisms of $G$
• Tangent space at the identity: The tangent space at the identity element $T_eG$ plays a crucial role in the study of Lie groups
  • $T_eG$ is naturally equipped with a Lie algebra structure, which is the associated Lie algebra of the Lie group

Lie Algebras: Basics and Properties

• Vector space structure: A Lie algebra is a vector space over a field (usually $\mathbb{R}$ or $\mathbb{C}$)
• Lie bracket: The Lie bracket is a binary operation $[\cdot, \cdot]: \mathfrak{g} \times \mathfrak{g} \to \mathfrak{g}$ satisfying certain properties
  • Bilinearity: $[ax+by,z] = a[x,z] + b[y,z]$ and $[z,ax+by] = a[z,x] + b[z,y]$ for all $a,b \in \mathbb{F}$ and $x,y,z \in \mathfrak{g}$
  • Antisymmetry: $[x,y] = -[y,x]$ for all $x,y \in \mathfrak{g}$
  • Jacobi identity: $[x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0$ for all $x,y,z \in \mathfrak{g}$
• Derived series and lower central series: Two descending series of ideals that measure the non-commutativity of a Lie algebra
• Nilpotent and solvable Lie algebras: Classes of Lie algebras with specific properties related to their derived and lower central series
• Semisimple Lie algebras: Lie algebras with no non-zero solvable ideals
  • Characterized by the Cartan criterion, which states that a Lie algebra is semisimple if and only if its Killing form is non-degenerate

Connections Between Lie Groups and Algebras

• Lie algebra of a Lie group: Every Lie group $G$ has an associated Lie algebra $\mathfrak{g}$, which is the tangent space at the identity $T_eG$
  • The Lie bracket on $\mathfrak{g}$ is induced by the commutator of left-invariant vector fields on $G$
• Exponential map: The exponential map $\exp: \mathfrak{g} \to G$ is a local diffeomorphism near the identity
  • Relates the Lie algebra to the Lie group and allows for the transfer of information between the two structures
• Baker-Campbell-Hausdorff formula: Expresses the product of two elements in a Lie group in terms of their Lie algebra counterparts
  • For $X,Y \in \mathfrak{g}$, the BCH formula gives $\exp(X)\exp(Y) = \exp(X+Y+\frac{1}{2}[X,Y]+\cdots)$
• Adjoint representation: A representation of a Lie group $G$ on its Lie algebra $\mathfrak{g}$ given by $\mathrm{Ad}_g = d(C_g)_e$, where $C_g$ is conjugation by $g$
  • The differential of the adjoint representation at the identity gives the adjoint representation of the Lie algebra
• Correspondence between Lie subgroups and Lie subalgebras: Lie subgroups of $G$ correspond to Lie subalgebras of $\mathfrak{g}$
  • This correspondence is not one-to-one, but it provides a useful tool for studying the structure of Lie groups

Applications in Differential Topology

• Lie groups as symmetries of manifolds: Lie groups often appear as symmetry groups of smooth manifolds
  • For example, the isometry group of a Riemannian manifold is a Lie group
• Lie group actions: A Lie group action on a smooth manifold $M$ is a smooth map $G \times M \to M$ satisfying certain properties
  • Lie group actions can be used to study the geometry and topology of the manifold
• Principal bundles: A principal $G$-bundle is a fiber bundle $P \to M$ with a free and transitive right action of a Lie group $G$ on $P$
  • Principal bundles are used to construct vector bundles and other geometric structures on manifolds
• Connections on principal bundles: A connection on a principal $G$-bundle is a way to split the tangent space of the total space into vertical and horizontal subspaces
  • Connections are used to define parallel transport and curvature, which are important in the study of differential geometry
• Characteristic classes: Cohomology classes associated with principal bundles that measure the non-triviality of the bundle
  • Chern classes and Pontryagin classes are examples of characteristic classes that are defined using Lie groups and Lie algebras

Problem-Solving Techniques

• Utilizing the exponential map: The exponential map can be used to solve problems involving the relationship between Lie groups and Lie algebras
  • For example, to find the Lie subgroup corresponding to a given Lie subalgebra, one can exponentiate the subalgebra
• Applying representation theory: Representations of Lie groups and Lie algebras can be used to simplify problems and gain insights
  • The adjoint representation is particularly useful in studying the structure of Lie groups and Lie algebras
• Leveraging the structure theory of Lie algebras: The classification of semisimple Lie algebras and the root space decomposition can be used to solve problems involving specific Lie groups and Lie algebras
• Using the Baker-Campbell-Hausdorff formula: The BCH formula can be used to compute products in Lie groups and to study the relationship between Lie groups and Lie algebras
• Employing the correspondence between Lie subgroups and Lie subalgebras: This correspondence can be used to study the subgroup structure of Lie groups by examining the subalgebras of their associated Lie algebras

Advanced Topics and Current Research

• Infinite-dimensional Lie groups and Lie algebras: The theory of Lie groups and Lie algebras can be extended to infinite-dimensional settings
  • Examples include the group of diffeomorphisms of a manifold and the Virasoro algebra in conformal field theory
• Lie groupoids and Lie algebroids: Generalizations of Lie groups and Lie algebras that allow for a more flexible notion of symmetry
  • Lie groupoids and Lie algebroids have applications in Poisson geometry and mathematical physics
• Quantum groups: Deformations of the algebra of functions on a Lie group that arise in the study of quantum systems
  • Quantum groups have applications in knot theory, representation theory, and mathematical physics
• Kac-Moody algebras and affine Lie algebras: Infinite-dimensional Lie algebras that generalize the structure of semisimple Lie algebras
  • These algebras have applications in conformal field theory and the theory of moduli spaces
• Representation theory of Lie groups and Lie algebras: The study of representations of Lie groups and Lie algebras is an active area of research
  • Representations have applications in harmonic analysis, number theory, and mathematical physics
• Geometric Langlands program: A far-reaching program that connects representation theory, algebraic geometry, and number theory
  • The geometric Langlands program involves the study of Lie groups, Lie algebras, and their representations