Lie group and algebra homomorphisms are smooth maps that preserve structure between groups or algebras. They're crucial for studying symmetries in physics and math, connecting abstract concepts to concrete transformations. Homomorphisms help us understand group structure through kernels and images.
Representation theory brings abstract structures to life through linear transformations. It's key in quantum mechanics and group analysis. Adjoint representations act on Lie algebras through conjugation, while irreducible representations are the building blocks of more complex ones. Characters provide a compact way to identify representations.
Homomorphisms
Lie Group and Algebra Homomorphisms
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Lie group homomorphism defines a smooth map between two Lie groups preserving group structure
Preserves group operations (multiplication and inversion) and smoothness
Lie algebra homomorphism represents a linear map between Lie algebras respecting bracket operations
Maintains the structure of Lie brackets and vector space operations
Exponential map connects Lie group homomorphisms to Lie algebra homomorphisms
Homomorphisms play crucial roles in studying symmetries and transformations in physics and mathematics
Applications include describing particle interactions in quantum field theory and analyzing geometric symmetries
Properties and Examples of Homomorphisms
Kernel of a Lie group homomorphism forms a normal subgroup of the domain
Image of a Lie group homomorphism constitutes a Lie subgroup of the codomain
Isomorphisms represent bijective homomorphisms with inverses that are also homomorphisms
Automorphisms describe isomorphisms from a Lie group or algebra to itself
Inner automorphisms arise from conjugation by group elements
Outer automorphisms encompass all automorphisms not arising from inner ones
Examples include the exponential map from the Lie algebra of 2x2 matrices to the special linear group SL(2,R)
Representations
Fundamentals of Representation Theory
Representation theory studies abstract algebraic structures through linear transformations
Provides concrete realizations of abstract mathematical objects
Linear representations map group elements to linear transformations on vector spaces
Unitary representations preserve inner products and play key roles in quantum mechanics
Faithful representations have trivial kernels and provide isomorphic copies of the original group
Reducible representations can be decomposed into direct sums of simpler representations
Schur's lemma states that intertwining operators between irreducible representations are either zero or isomorphisms
Adjoint and Irreducible Representations
Adjoint representation acts on the Lie algebra of a Lie group through conjugation
Maps each group element to an automorphism of its Lie algebra
Crucial for understanding the structure of Lie groups and their associated Lie algebras
Irreducible representations cannot be further decomposed into simpler subrepresentations
Form the building blocks for understanding more complex representations
Classified by highest weight theory for semisimple Lie algebras
Weyl character formula provides a powerful tool for computing characters of irreducible representations
Characters and Their Applications
Characters encode essential information about representations in a compact form
Define character as the trace of the representation matrix
Provide a fingerprint for identifying and distinguishing representations
Orthogonality relations of characters simplify calculations in representation theory
Peter-Weyl theorem decomposes functions on compact groups using characters of irreducible representations
Applications include spectral analysis in quantum mechanics and harmonic analysis on groups
Character tables summarize representation theory for finite groups
Measures on Groups
Haar Measure and Its Properties
Haar measure provides a notion of volume on locally compact topological groups
Generalizes Lebesgue measure to non-Euclidean spaces
Left-invariant under group operations, ensuring consistency with group structure
Unique up to scalar multiplication for locally compact groups
Bi-invariant (both left and right-invariant) for compact groups
Normalized Haar measure assigns total volume 1 to compact groups
Enables integration theory on groups, crucial for harmonic analysis and representation theory
Applications and Extensions of Haar Measure
Facilitates averaging over group elements in various mathematical contexts
Peter-Weyl theorem uses Haar measure to decompose L^2 functions on compact groups
Fourier analysis on locally compact abelian groups relies on Haar measure
Pontryagin duality connects a locally compact abelian group with its dual group using Haar measure
Modular function measures the failure of right-invariance for non-unimodular groups
Haar measure on Lie groups relates to left-invariant differential forms
Applications include probability theory on groups and ergodic theory