2.4 Classification of simple Lie algebras

Lie algebras are essential structures in non-associative algebra, providing a framework for studying continuous symmetries. They consist of vector spaces with a bilinear operation called the Lie bracket, which satisfies specific properties like skew-symmetry and the Jacobi identity.

The classification of simple Lie algebras is a cornerstone of the field. It organizes these algebras into four infinite families (classical Lie algebras) and five exceptional cases, based on their root systems and Dynkin diagrams. This classification reveals the underlying structure and properties of these fundamental algebraic objects.

Foundations of Lie algebras

• Non-associative algebra encompasses Lie algebras as fundamental structures
• Lie algebras provide a framework for studying continuous symmetries in mathematics and physics
• Understanding Lie algebras forms the basis for classification of simple Lie algebras

Definition and properties

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• Lie algebras consist of vector spaces equipped with a bilinear operation called the Lie bracket
• Lie bracket satisfies skew-symmetry and the Jacobi identity
• Closure property ensures the Lie bracket of two elements remains within the algebra
• Homomorphisms between Lie algebras preserve the Lie bracket structure

Structure constants

• Define the Lie bracket operation in terms of basis elements
• Appear in the expansion $[e_i, e_j] = \sum_k c_{ij}^k e_k$
• Satisfy antisymmetry and Jacobi identity conditions
• Determine the algebraic structure and properties of the Lie algebra

Killing form

• Symmetric bilinear form defined on a Lie algebra
• Calculated as $K(x,y) = \text{tr}(\text{ad}(x) \circ \text{ad}(y))$
• Plays a crucial role in the classification of semisimple Lie algebras
• Invariance under the adjoint action of the Lie algebra

Root systems

• Provide a geometric representation of the structure of semisimple Lie algebras
• Enable the classification of simple Lie algebras through their root system properties
• Form the foundation for understanding the weight space decomposition of representations

Root space decomposition

• Decomposes a semisimple Lie algebra into direct sum of root spaces
• Root spaces correspond to eigenspaces of the adjoint action of the Cartan subalgebra
• Roots represent non-zero weights in the adjoint representation
• Allows for a systematic study of the algebra's structure

Positive and negative roots

• Partition the root system into positive and negative subsets
• Choice of positive roots determines a specific Borel subalgebra
• Negative roots obtained by negating positive roots
• Weyl chambers defined by the choice of positive roots

Simple roots

• Minimal set of positive roots that generate all positive roots
• Form a basis for the root system
• Number of simple roots equals the rank of the Lie algebra
• Determine the Cartan matrix and Dynkin diagram of the Lie algebra

Cartan subalgebra

• Maximal abelian subalgebra of a Lie algebra
• Plays a central role in the structure theory of semisimple Lie algebras
• Diagonalizable in the adjoint representation

Definition and significance

• Nilpotent subalgebra equal to its own normalizer
• Dimension of Cartan subalgebra defines the rank of the Lie algebra
• Allows for the root space decomposition of the Lie algebra
• Provides a natural basis for describing the weight space decomposition of representations

Cartan matrix

• Encodes the relationships between simple roots
• Entries given by $A_{ij} = \frac{2(\alpha_i, \alpha_j)}{(\alpha_i, \alpha_i)}$
• Determines the Lie algebra up to isomorphism
• Properties include symmetrizability and positive definiteness

Dynkin diagrams

• Graphical representation of the Cartan matrix
• Nodes represent simple roots
• Edges indicate angles between simple roots
• Completely classify simple Lie algebras up to isomorphism

Classification theorem

• Culmination of the theory of simple Lie algebras
• Provides a complete list of all possible simple Lie algebras over algebraically closed fields
• Demonstrates the power of root system analysis in algebra classification

Statement of theorem

• Simple Lie algebras over algebraically closed fields of characteristic zero classified into four infinite families and five exceptional cases
• Infinite families: A_n, B_n, C_n, D_n
• Exceptional cases: G_2, F_4, E_6, E_7, E_8
• Classification based on properties of root systems and Dynkin diagrams

Proof outline

• Reduction to the study of root systems
• Classification of irreducible root systems
• Construction of simple Lie algebras from root systems
• Uniqueness of simple Lie algebras for each root system
• Verification of completeness of the classification

Historical context

• Developed over several decades in the late 19th and early 20th centuries
• Key contributions from Wilhelm Killing, Élie Cartan, and Hermann Weyl
• Unified various strands of research in group theory, differential geometry, and algebra
• Laid the foundation for further developments in representation theory and mathematical physics

Classical Lie algebras

• Form the four infinite families in the classification of simple Lie algebras
• Correspond to matrix Lie groups of linear transformations
• Play crucial roles in various areas of mathematics and physics

A_n series (SL(n+1))

• Special linear Lie algebra of (n+1) × (n+1) matrices with trace zero
• Root system consists of vectors in n-dimensional space
• Dynkin diagram forms a simple chain with n nodes
• Corresponds to the group of linear transformations with determinant 1

B_n series (SO(2n+1))

• Special orthogonal Lie algebra of (2n+1) × (2n+1) matrices
• Root system includes both long and short roots
• Dynkin diagram has n nodes with a double edge at one end
• Represents rotations in odd-dimensional Euclidean space

C_n series (Sp(2n))

• Symplectic Lie algebra of 2n × 2n matrices
• Root system similar to B_n but with long and short roots interchanged
• Dynkin diagram has n nodes with a double edge at one end
• Corresponds to transformations preserving a symplectic form

D_n series (SO(2n))

• Special orthogonal Lie algebra of 2n × 2n matrices
• Root system consists of vectors in n-dimensional space
• Dynkin diagram forms a "Y" shape for n ≥ 4
• Represents rotations in even-dimensional Euclidean space

Exceptional Lie algebras

• Five simple Lie algebras not part of the classical infinite families
• Discovered during the classification process of simple Lie algebras
• Exhibit unique properties and structures not found in classical Lie algebras
• Play important roles in various areas of mathematics and theoretical physics

G_2 algebra

• Smallest exceptional Lie algebra with rank 2 and dimension 14
• Root system consists of 12 roots in a hexagonal pattern
• Dynkin diagram has two nodes connected by a triple edge
• Appears in the study of octonions and certain geometrical structures

F_4 algebra

• Exceptional Lie algebra of rank 4 and dimension 52
• Root system combines features of B_4 and C_4 systems
• Dynkin diagram has four nodes with one double edge
• Connected to the symmetries of the 24-cell in four dimensions

E_6, E_7, E_8 algebras

• Form a family of exceptional Lie algebras with increasing complexity
• E_6: rank 6, dimension 78, Dynkin diagram forms a "T" shape
• E_7: rank 7, dimension 133, Dynkin diagram extends E_6 with an additional node
• E_8: largest exceptional Lie algebra, rank 8, dimension 248
• E_8 root system exhibits remarkable symmetry and connections to various mathematical structures

Representation theory

• Studies how Lie algebras act on vector spaces
• Provides tools for understanding the structure and properties of Lie algebras
• Connects Lie algebra theory to applications in physics and other areas of mathematics

Weights and weight spaces

• Weights generalize the concept of eigenvalues for Lie algebra representations
• Weight spaces decompose the representation space into subspaces
• Highest weight determines the structure of irreducible representations
• Weight lattice encodes the possible weights in representations

Highest weight theory

• Classifies irreducible representations of semisimple Lie algebras
• Highest weight vector generates the entire representation
• Dominant integral weights correspond to finite-dimensional irreducible representations
• Weyl character formula expresses characters of irreducible representations

Character formulas

• Encode information about the structure of representations
• Weyl character formula provides a general expression for characters of irreducible representations
• Freudenthal formula allows for recursive computation of weight multiplicities
• Kostant multiplicity formula gives an alternating sum expression for weight multiplicities

Applications in physics

• Lie algebras provide a mathematical framework for describing symmetries in physical systems
• Understanding of Lie algebras crucial for advanced topics in theoretical physics
• Applications span multiple areas of physics from fundamental particles to cosmology

Particle physics

• SU(3) Lie algebra describes quark flavors in the eightfold way classification
• Standard Model based on the product of SU(3), SU(2), and U(1) Lie groups
• Gauge theories formulated using Lie algebra-valued connection forms
• Symmetry breaking mechanisms involve representations of Lie algebras

Quantum mechanics

• Angular momentum operators form representations of SO(3) Lie algebra
• Symmetry groups of Hamiltonians described by Lie algebras
• Coherent states in quantum optics related to representations of Heisenberg-Weyl algebra
• Supersymmetry involves Z_2-graded extensions of Lie algebras

String theory

• Exceptional Lie algebras (E_8) appear in heterotic string theory
• Conformal field theory uses representations of Virasoro and Kac-Moody algebras
• M-theory involves E_11 Lie algebra as a proposed symmetry
• AdS/CFT correspondence relates string theory to conformal field theories

Computational methods

• Develop algorithms and tools for working with Lie algebras and their representations
• Enable efficient calculations and analysis of complex Lie algebraic structures
• Facilitate applications of Lie algebra theory in various fields

Root system algorithms

• Implement methods for generating and manipulating root systems
• Algorithms for finding positive roots, simple roots, and Weyl groups
• Efficient computation of root strings and reflection operators
• Implement root system isomorphism tests and classification algorithms

Weyl group calculations

• Develop algorithms for generating Weyl group elements
• Implement efficient methods for Weyl group operations (reflections, products)
• Calculate orbits of weights under Weyl group action
• Compute Weyl character formula using Weyl group elements

Software tools

• LiE: Specialized computer algebra system for Lie algebra calculations
• GAP: System for computational discrete algebra with Lie algebra packages
• SageMath: Open-source mathematics software with Lie algebra functionality
• Custom Python libraries for root system manipulation and representation theory calculations