2.6 Lie groups and Lie algebras

Lie groups and Lie algebras are fundamental concepts in non-associative algebra, bridging algebra and differential geometry. These mathematical structures describe continuous symmetries, combining smooth manifolds with group operations, and play crucial roles in physics and mathematics.

Lie algebras provide linearized versions of Lie groups, capturing their local structure as infinitesimal generators. The relationship between Lie groups and algebras, explored through exponential maps and adjoint representations, forms a cornerstone of non-associative algebra and differential geometry.

Definition of Lie groups

• Lie groups form a fundamental concept in non-associative algebra combining continuous symmetry with group structure
• These mathematical objects serve as a bridge between algebra and differential geometry
• Lie groups play a crucial role in various branches of mathematics and physics

Properties of Lie groups

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• Smooth manifolds endowed with group operations (multiplication and inversion) that are differentiable
• Closed under group operations ensuring stability and consistency
• Locally Euclidean spaces allowing for the application of calculus techniques
• Possess a Lie algebra as their tangent space at the identity element
• Satisfy the axioms of associativity, identity element, and inverse element

Examples of Lie groups

• General linear group GL(n,R) consists of all invertible n×n real matrices
• Special orthogonal group SO(3) represents rotations in three-dimensional space
• Unitary group U(n) includes all n×n complex unitary matrices
• Heisenberg group describes symmetries of certain quantum mechanical systems
• Lorentz group SO(3,1) characterizes symmetries of special relativity

Continuous symmetry groups

• Describe transformations that can be continuously deformed into the identity transformation
• Allow for infinitesimal changes in parameters leading to smooth transitions between group elements
• Include translations, rotations, and scaling in Euclidean space
• Poincaré group combines Lorentz transformations with spacetime translations
• Gauge groups in particle physics represent internal symmetries of fundamental interactions

Lie algebras

• Lie algebras provide a linearized version of Lie groups, capturing their local structure
• These algebraic structures serve as infinitesimal generators of Lie group elements
• Lie algebras play a crucial role in the study of non-associative algebra and group theory

Tangent space at identity

• Vector space of tangent vectors to the Lie group manifold at the identity element
• Represents infinitesimal transformations or generators of the Lie group
• Dimension of the tangent space equals the dimension of the Lie group
• Elements of the tangent space can be exponentiated to produce group elements
• Provides a local approximation of the Lie group structure

Structure constants

• Coefficients that define the Lie bracket operation in a given basis of the Lie algebra
• Satisfy antisymmetry and Jacobi identity conditions
• Determine the commutation relations between Lie algebra elements
• Encode the non-commutativity of the Lie group operations
• Can be used to classify and distinguish different Lie algebras

Lie bracket operation

• Bilinear, antisymmetric operation on the Lie algebra satisfying the Jacobi identity
• Measures the non-commutativity of infinitesimal group transformations
• For matrix Lie algebras, defined as $[X,Y] = XY - YX$
• Generalizes the notion of commutators in associative algebras
• Determines the structure of the Lie algebra and its corresponding Lie group

Relationship between groups and algebras

• Lie groups and Lie algebras share a deep connection, with each providing insights into the other
• This relationship forms a cornerstone of non-associative algebra and differential geometry
• Understanding this connection allows for powerful techniques in analyzing continuous symmetries

Exponential map

• Maps elements of the Lie algebra to elements of the Lie group
• Defined as $\exp(X) = \sum_{n=0}^{\infty} \frac{X^n}{n!}$ for matrix Lie groups
• Generalizes the exponential function to curved manifolds
• Provides a local diffeomorphism between the Lie algebra and a neighborhood of the identity in the Lie group
• Allows for the study of Lie group elements through their Lie algebra counterparts

Baker-Campbell-Hausdorff formula

• Expresses the product of exponentials in terms of Lie algebra elements
• Given by $\exp(X)\exp(Y) = \exp(X + Y + \frac{1}{2}[X,Y] + \frac{1}{12}[X,[X,Y]] - \frac{1}{12}[Y,[X,Y]] + ...)$
• Accounts for non-commutativity in the multiplication of Lie group elements
• Crucial for understanding the structure of non-abelian Lie groups
• Allows for computations in the Lie algebra that correspond to operations in the Lie group

Adjoint representation

• Action of a Lie group on its Lie algebra through conjugation
• Defined as $Ad_g(X) = gXg^{-1}$ for matrix Lie groups
• Infinitesimal version given by the adjoint action of the Lie algebra on itself
• Preserves the Lie bracket operation
• Provides a faithful representation of the Lie group and its algebra

Matrix Lie groups

• Subgroups of the general linear group GL(n,F) where F is typically R or C
• Represent an important class of Lie groups with concrete realizations
• Play a significant role in various areas of mathematics and physics

Special linear group

• Denoted as SL(n,F) consists of n×n matrices with determinant 1
• Preserves volume in n-dimensional space
• Lie algebra sl(n,F) consists of traceless n×n matrices
• Important in projective geometry and representation theory
• Includes the special unitary group SU(n) as a compact subgroup

Orthogonal group

• Denoted as O(n) consists of n×n orthogonal matrices
• Preserves the Euclidean inner product and distances
• Special orthogonal group SO(n) restricted to matrices with determinant 1
• Lie algebra so(n) consists of skew-symmetric matrices
• Describes rotations in n-dimensional Euclidean space

Unitary group

• Denoted as U(n) consists of n×n unitary matrices
• Preserves the Hermitian inner product in complex vector spaces
• Special unitary group SU(n) restricted to matrices with determinant 1
• Lie algebra u(n) consists of skew-Hermitian matrices
• Crucial in quantum mechanics and gauge theories

Representations of Lie groups

• Describe how Lie group elements act as linear transformations on vector spaces
• Provide a powerful tool for studying the structure and properties of Lie groups
• Form a fundamental aspect of non-associative algebra and group theory

Irreducible representations

• Cannot be decomposed into smaller subrepresentations
• Fundamental building blocks for understanding more complex representations
• Classified by highest weight vectors for semisimple Lie algebras
• Dimension given by the Weyl dimension formula
• Play a crucial role in the representation theory of Lie groups and algebras

Weight diagrams

• Graphical representations of weights in a given representation
• Illustrate the action of Cartan subalgebra elements on weight vectors
• Reveal symmetries and structure of the representation
• Useful for determining branching rules and tensor product decompositions
• Examples include the weight diagram of the adjoint representation of SU(3)

Character theory

• Studies traces of representation matrices
• Characters uniquely determine finite-dimensional representations
• Satisfy orthogonality relations for compact Lie groups
• Weyl character formula expresses characters in terms of weights and roots
• Facilitates the decomposition of tensor products and restrictions of representations

Classification of Lie algebras

• Aims to categorize and understand the structure of all possible Lie algebras
• Focuses primarily on finite-dimensional semisimple Lie algebras
• Provides a foundation for studying more general Lie algebras and their applications

Root systems

• Collection of vectors in a Euclidean space associated with a semisimple Lie algebra
• Encode the structure of the Lie algebra and its representations
• Classified into ADE and BCFG types based on their geometry
• Determine the commutation relations between Lie algebra elements
• Root lattice generated by the root system relates to the weight lattice of representations

Cartan matrix

• Square matrix encoding the geometry of the root system
• Entries given by $A_{ij} = 2\frac{(\alpha_i, \alpha_j)}{(\alpha_i, \alpha_i)}$ where $\alpha_i$ are simple roots
• Determines the Lie algebra up to isomorphism
• Properties include positive definiteness for finite-dimensional semisimple Lie algebras
• Generalizes to Kac-Moody algebras for infinite-dimensional cases

Dynkin diagrams

• Graphical representation of the Cartan matrix
• Nodes represent simple roots, edges indicate their relationships
• Classify simple Lie algebras into the classical series (A_n, B_n, C_n, D_n) and exceptional types (G_2, F_4, E_6, E_7, E_8)
• Reveal symmetries and structure of the Lie algebra
• Extended Dynkin diagrams include information about the highest root

Applications in physics

• Lie groups and algebras provide a mathematical framework for describing symmetries in physical systems
• Their applications span various areas of theoretical physics and beyond
• Understanding these applications highlights the importance of non-associative algebra in physics

Symmetries in quantum mechanics

• Rotational symmetry described by SO(3) or SU(2) for spin systems
• Time reversal symmetry represented by antiunitary operators
• Parity transformations form a discrete symmetry group
• Symmetries lead to conservation laws via Noether's theorem
• Representation theory crucial for understanding angular momentum and selection rules

Gauge theories

• Local symmetries described by Lie groups acting on fiber bundles
• U(1) gauge symmetry in electromagnetism
• SU(3) color symmetry in quantum chromodynamics
• Electroweak theory based on SU(2) × U(1) symmetry
• Gauge transformations generate covariant derivatives and field strengths

Particle physics models

• Standard Model based on SU(3) × SU(2) × U(1) gauge group
• Grand Unified Theories explore larger symmetry groups (SU(5), SO(10))
• Supersymmetry extends spacetime symmetries to include fermionic generators
• String theory utilizes infinite-dimensional Lie algebras and exceptional groups
• Conformal field theories employ the conformal group SO(d+1,1) or its covering group

Infinite-dimensional Lie algebras

• Generalize finite-dimensional Lie algebras to infinite-dimensional vector spaces
• Arise naturally in the study of integrable systems and conformal field theories
• Provide a rich mathematical structure with applications in physics and mathematics

Kac-Moody algebras

• Generalize finite-dimensional semisimple Lie algebras
• Classified by generalized Cartan matrices
• Include affine Lie algebras as an important subclass
• Appear in the study of two-dimensional conformal field theories
• Hyperbolic Kac-Moody algebras conjectured to play a role in M-theory

Virasoro algebra

• Central extension of the Witt algebra of vector fields on the circle
• Generators satisfy $[L_m, L_n] = (m-n)L_{m+n} + \frac{c}{12}(m^3-m)\delta_{m+n,0}$
• Central charge c determines the representation theory
• Fundamental in two-dimensional conformal field theory
• Appears in string theory as symmetries of the worldsheet

Loop algebras

• Lie algebras of maps from the circle to a finite-dimensional Lie algebra
• Form the non-centrally extended part of affine Kac-Moody algebras
• Arise in the study of integrable systems and soliton equations
• Related to current algebras in quantum field theory
• Generalize to multi-loop algebras and twisted versions

Lie group manifolds

• Study the geometric properties of Lie groups as smooth manifolds
• Combine techniques from differential geometry and group theory
• Provide insights into the global structure of Lie groups

Homogeneous spaces

• Quotients of Lie groups by closed subgroups
• Examples include spheres S^n = SO(n+1)/SO(n)
• Possess a transitive group action
• Natural setting for studying symmetric spaces
• Important in the theory of harmonic analysis on Lie groups

Coset spaces

• Special case of homogeneous spaces G/H where H is a closed subgroup of G
• Represent the set of left (or right) cosets of H in G
• Equipped with a natural G-invariant measure
• Play a role in the decomposition of representations
• Examples include Grassmannians and flag manifolds

Fiber bundles

• Geometric structures with Lie groups as structure groups
• Principal bundles have Lie groups as fibers
• Associated bundles constructed from principal bundles and group representations
• Gauge theories formulated in terms of connections on principal bundles
• Characteristic classes measure the non-triviality of fiber bundles

Lie group actions

• Describe how Lie groups act as symmetries on other mathematical objects
• Provide a geometric perspective on group representations
• Connect Lie theory with symplectic geometry and Hamiltonian mechanics

Orbits and stabilizers

• Orbits form homogeneous spaces under the group action
• Stabilizer subgroups measure the symmetry of individual points
• Orbit-stabilizer theorem relates the size of orbits to the order of stabilizers
• Closed orbits correspond to semisimple group elements
• Nilpotent orbits play a special role in representation theory

Moment maps

• Assign Lie algebra elements to points in a symplectic manifold
• Generalize the notion of conserved quantities in Hamiltonian mechanics
• Satisfy equivariance properties with respect to the group action
• Marsden-Weinstein reduction uses moment maps to construct quotient spaces
• Geometric quantization relates moment maps to operators in quantum mechanics

Symplectic reduction

• Procedure for constructing quotient spaces preserving symplectic structure
• Reduces the dimensionality of a system by eliminating symmetries
• Involves taking the quotient of the level set of a moment map
• Produces new symplectic manifolds with reduced symmetry
• Applications in classical mechanics and gauge theory