11.3 Homomorphisms and representations

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