🔁Lie Algebras and Lie Groups Unit 8 – Compact Lie Groups: Representation Theory

Key Concepts and Definitions

• Compact Lie group defined as a Lie group that is compact as a topological space
• Lie algebra associated to a Lie group consists of the tangent space at the identity element equipped with a Lie bracket operation
• Representation of a Lie group is a homomorphism from the group to the general linear group $GL(V)$ of a vector space $V$
  • Representation of a Lie algebra is a linear map from the algebra to the endomorphism algebra $End(V)$ preserving the Lie bracket
• Irreducible representation cannot be decomposed into smaller subrepresentations
• Character of a representation is the trace of the representation matrix
• Weights are eigenvalues of the representation with respect to a maximal torus of the Lie group
• Root system associated to a Lie algebra encodes its structure and properties
  • Simple roots generate the root system and determine the Lie algebra uniquely

Historical Context and Development

• Lie groups introduced by Sophus Lie in the late 19th century as continuous symmetry groups
• Representation theory of compact Lie groups developed by Hermann Weyl and Élie Cartan in the early 20th century
• Weyl character formula discovered by Hermann Weyl in 1925 expresses characters of irreducible representations in terms of the root system
• Representation theory of Lie algebras formalized by Nathan Jacobson and Eugene Dynkin in the 1940s and 1950s
• Connections between representation theory and mathematical physics explored by Valentine Bargmann, Irving Segal, and others in the mid-20th century
• Langlands program, formulated by Robert Langlands in the 1960s, relates representation theory to number theory and automorphic forms
• Kac-Moody algebras, introduced by Victor Kac and Robert Moody in the 1960s, generalize finite-dimensional Lie algebras and have important applications in string theory

Structure and Properties of Compact Lie Groups

• Compact Lie groups have a unique bi-invariant Haar measure allowing for integration on the group
• Maximal tori play a central role in the structure and representation theory of compact Lie groups
  • Maximal torus is a maximal connected abelian subgroup of a Lie group
• Weyl group acts on the maximal torus and its Lie algebra by reflections determined by the root system
• Classification of compact simple Lie groups corresponds to the classification of complex simple Lie algebras (Killing-Cartan classification)
  • Classical Lie groups: $SU(n)$, $SO(n)$, $Sp(n)$
  • Exceptional Lie groups: $G_2$, $F_4$, $E_6$, $E_7$, $E_8$
• Compact Lie groups have a finite center and a finite fundamental group
• Representation ring of a compact Lie group is a complete invariant of the group

Representation Theory Fundamentals

• Representations of compact Lie groups are completely reducible into irreducible representations
  • Every representation can be decomposed as a direct sum of irreducible representations
• Schur's lemma states that homomorphisms between irreducible representations are either zero or isomorphisms
• Characters of irreducible representations are orthonormal with respect to the Haar measure on the group
• Highest weight theory classifies irreducible representations by their highest weight vector
  • Highest weight vector is annihilated by all positive root vectors of the Lie algebra
• Weyl character formula expresses the character of an irreducible representation in terms of its highest weight and the root system
• Tensor product decomposition of representations governed by the Littlewood-Richardson rule
• Representation theory of compact Lie groups has applications in harmonic analysis and the study of invariant differential operators

Important Theorems and Proofs

• Peter-Weyl theorem asserts that the matrix coefficients of irreducible representations form a complete orthonormal basis for the Hilbert space $L^2(G)$
  • Provides a generalization of Fourier analysis to compact Lie groups
• Weyl integral formula expresses integration over a compact Lie group in terms of integration over a maximal torus and the Weyl group
• Borel-Weil-Bott theorem realizes irreducible representations as cohomology groups of line bundles over the flag manifold
• Kostant's convexity theorem describes the projection of orbits in the representation space onto the Lie algebra of a maximal torus
• Freudenthal's formula recursively computes the dimensions of irreducible representations
• Weyl's dimension formula expresses the dimension of an irreducible representation in terms of its highest weight and the root system
• Harish-Chandra's character formula generalizes the Weyl character formula to non-compact semisimple Lie groups

Applications in Physics and Mathematics

• Representation theory of $SU(2)$ and $SO(3)$ fundamental in quantum mechanics for describing angular momentum and spin
• Representation theory of $SU(3)$ describes the quark model and the Eightfold Way in particle physics
• Gauge theories in physics formulated using principal bundles with structure group a compact Lie group ($U(1)$, $SU(2)$, $SU(3)$)
• Representation theory of Virasoro algebra and Kac-Moody algebras plays a central role in conformal field theory and string theory
• Representations of compact Lie groups appear in the study of symmetric spaces and homogeneous spaces in differential geometry
• Representation theory has applications in algebraic combinatorics, notably in the study of symmetric functions and Schur polynomials
• Langlands program connects representation theory to number theory, automorphic forms, and arithmetic geometry

Computational Techniques and Tools

• Weight diagrams and Dynkin diagrams used to visualize the structure of representations and Lie algebras
• Weyl character ring implemented in computer algebra systems (Maple, Mathematica, SageMath) for computations with characters and representations
• Littelmann path model provides a combinatorial description of representation theory using piecewise-linear paths in the weight space
• Crystal bases, introduced by Kashiwara, give a combinatorial model for representations of quantum groups and Kac-Moody algebras
• Gelfand-Tsetlin patterns encode weight bases for irreducible representations of classical Lie algebras
• Kostant's partition function describes the weight multiplicities in irreducible representations
• Computational methods for branching rules and tensor product decompositions using combinatorial algorithms and algebraic manipulations

Advanced Topics and Current Research

• Affine Lie algebras and their representations have applications in conformal field theory and integrable systems
• Quantum groups are deformations of universal enveloping algebras of Lie algebras with connections to knot theory and low-dimensional topology
• Categorification of representation theory replaces vector spaces and linear maps with categories and functors
  • Khovanov homology is a categorification of the Jones polynomial in knot theory
• Geometric representation theory studies representations using techniques from algebraic geometry and symplectic geometry
• Langlands program and its geometric counterpart, the geometric Langlands program, are active areas of research connecting representation theory to number theory and algebraic geometry
• Representation stability and FI-modules investigate the behavior of representations under inclusions of groups or categories
• Modular representation theory studies representations over fields of positive characteristic, with applications to finite groups and algebraic groups