🔁Lie Algebras and Lie Groups Unit 11 – Physics and Geometry in Lie Theory

Key Concepts and Definitions

• Lie group a smooth manifold equipped with a group structure where the group operations are smooth maps
• Lie algebra the tangent space at the identity element of a Lie group, equipped with a bracket operation
• Exponential map connects the Lie algebra to the Lie group, allowing elements of the algebra to generate group elements
  • Defined as $\exp: \mathfrak{g} \to G$ where $\mathfrak{g}$ is the Lie algebra and $G$ is the Lie group
• Adjoint representation captures how a Lie group acts on its own Lie algebra via the adjoint map $\mathrm{Ad}: G \to \mathrm{Aut}(\mathfrak{g})$
• Roots and weights fundamental objects in the structure theory of semisimple Lie algebras, related to the eigenvalues of the adjoint representation
• Cartan subalgebra a maximal abelian subalgebra of a Lie algebra, plays a key role in the classification of semisimple Lie algebras
• Killing form a symmetric bilinear form on a Lie algebra, used to classify Lie algebras and construct invariant metrics on Lie groups

Historical Context and Development

• Sophus Lie (1842-1899) Norwegian mathematician who laid the foundations of the theory in the late 19th century
  • Motivated by the study of symmetries of differential equations and the desire to extend Galois theory to continuous groups
• Wilhelm Killing (1847-1923) and Élie Cartan (1869-1951) further developed the structure theory of semisimple Lie algebras in the early 20th century
• Hermann Weyl (1885-1955) made significant contributions to the representation theory of Lie groups and its applications to quantum mechanics
• Lie theory has since found applications in various areas of mathematics and physics (differential geometry, algebraic topology, quantum field theory)
• Continues to be an active area of research with connections to other fields (representation theory, harmonic analysis, mathematical physics)

Fundamental Principles of Lie Theory

• Lie groups and Lie algebras provide a unified framework for studying continuous symmetries
• Exponential map allows the local structure of a Lie group to be studied via its Lie algebra
• Adjoint representation encodes how a Lie group acts on its own Lie algebra, revealing important structural information
• Semisimple Lie algebras can be classified using Dynkin diagrams, which encode the root system and Cartan matrix
  • Classification includes the classical Lie algebras ($\mathfrak{sl}_n, \mathfrak{so}_n, \mathfrak{sp}_n$) and the exceptional Lie algebras ($G_2, F_4, E_6, E_7, E_8$)
• Representation theory of Lie groups and Lie algebras plays a crucial role in understanding their structure and applications
• Lie theory provides a natural language for describing symmetries in physics (gauge theories, particle physics, general relativity)

Geometry in Lie Groups

• Lie groups are differentiable manifolds, allowing geometric concepts to be applied to the study of symmetries
• Invariant metrics on Lie groups can be constructed using the Killing form, enabling the study of geometric properties
• Homogeneous spaces arise as quotients of Lie groups by closed subgroups, providing a rich source of examples in differential geometry
  • Examples include spheres ($S^n = \mathrm{SO}(n+1)/\mathrm{SO}(n)$) and projective spaces ($\mathbb{RP}^n = \mathrm{GL}(n+1)/\mathrm{GL}(1) \times \mathrm{GL}(n)$)
• Symmetric spaces are homogeneous spaces with additional symmetry properties, playing a key role in the theory of Riemannian manifolds
• Flag varieties are projective algebraic varieties that arise as quotients of Lie groups, connecting Lie theory to algebraic geometry
• Representation theory of Lie groups can be used to construct vector bundles and study their geometric properties

Physics Applications of Lie Theory

• Lie groups provide a natural framework for describing symmetries in physical systems
• Noether's theorem connects symmetries (Lie groups) to conservation laws in classical and quantum mechanics
• Gauge theories in particle physics are based on Lie groups (U(1), SU(2), SU(3)) and their representations
  • Electroweak theory based on $\mathrm{U}(1) \times \mathrm{SU}(2)$, quantum chromodynamics based on $\mathrm{SU}(3)$
• General relativity can be formulated using the Lorentz group $\mathrm{SO}(3,1)$ and its associated Lie algebra
• Representation theory of Lie groups plays a crucial role in the classification of elementary particles and the construction of quantum field theories
• Lie algebras and their representations are used in the study of integrable systems and exactly solvable models in statistical mechanics

Mathematical Techniques and Tools

• Structure theory of semisimple Lie algebras uses root systems, Weyl groups, and Dynkin diagrams to classify and study their properties
• Representation theory of Lie groups and Lie algebras uses techniques from linear algebra, analysis, and algebraic geometry
  • Highest weight theory, character formulas, Weyl character formula, Borel-Weil theorem
• Cohomology of Lie algebras and Lie groups plays a key role in the study of their structure and representation theory
  • Chevalley-Eilenberg complex, Lie algebra cohomology, group cohomology, Hochschild-Serre spectral sequence
• Harmonic analysis on Lie groups uses techniques from functional analysis and representation theory to study invariant differential operators and eigenfunctions
• Geometric quantization provides a framework for constructing quantum mechanical systems from classical systems with symmetries described by Lie groups
• Categorification of Lie algebras and their representations leads to the study of higher categorical structures (2-groups, 2-representations)

Connections to Other Areas of Mathematics

• Lie theory has deep connections to algebraic geometry through the study of algebraic groups and their representations
  • Flag varieties, Schubert calculus, intersection theory, geometric Langlands program
• Representation theory of Lie groups and Lie algebras is closely related to the theory of automorphic forms and the Langlands program
• Lie theory plays a key role in the study of differential equations, particularly in the theory of integrable systems and the Painlevé equations
• Connections to number theory arise through the study of arithmetic groups and automorphic representations
• Lie groups and their actions on manifolds provide a rich source of examples in algebraic topology and K-theory
• Quantum groups are deformations of universal enveloping algebras of Lie algebras, connecting Lie theory to noncommutative geometry and knot theory

Practical Examples and Problem-Solving

• Classification of simple Lie algebras using Dynkin diagrams and Cartan matrices
  • Calculating roots, weights, and Weyl groups for classical and exceptional Lie algebras
• Constructing representations of Lie groups and Lie algebras using highest weight theory and the Borel-Weil theorem
  • Determining the dimension and character of irreducible representations
• Solving differential equations with symmetries using Lie group methods
  • Reduction of order, invariant solutions, symmetry breaking
• Analyzing physical systems with continuous symmetries using Noether's theorem and conserved quantities
  • Angular momentum conservation from rotational symmetry, energy conservation from time translation symmetry
• Constructing invariant metrics and connections on homogeneous spaces and symmetric spaces
  • Calculating curvature, geodesics, and harmonic forms on classical symmetric spaces
• Applying representation theory to the classification of elementary particles and the construction of gauge theories
  • Determining the particle content and interaction terms from the representation theory of the Standard Model gauge group