5.3 Classification of semisimple Lie algebras

The classification of semisimple Lie algebras is a cornerstone of Lie theory. It groups these algebras into four infinite families and five exceptional cases, providing a complete picture of their structure and properties.

This classification ties directly to root systems and Dynkin diagrams, which are key tools in understanding Lie algebras. By studying these diagrams, we can uncover important information about the algebra's structure and representations.

Classification of Semisimple Lie Algebras

The Classification Theorem

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• States that every semisimple Lie algebra over the complex numbers is a direct sum of simple Lie algebras, each belonging to one of four infinite families (An, Bn, Cn, Dn) or five exceptional cases (G2, F4, E6, E7, E8)
  • Simple Lie algebras are non-abelian Lie algebras whose only ideals are the zero ideal and the algebra itself
• Provides a complete and exhaustive classification, ensuring every semisimple Lie algebra over the complex numbers falls into one of these categories
• Guarantees uniqueness up to isomorphism, meaning two semisimple Lie algebras are isomorphic if and only if they have the same direct sum decomposition into simple Lie algebras
• Involves the study of root systems and Dynkin diagrams associated with semisimple Lie algebras in its proof

Properties and Implications of the Classification

• Allows for the determination of dimension, rank, and other invariants of a semisimple Lie algebra based on its decomposition into simple Lie algebras
• Provides information about the representation theory of a semisimple Lie algebra, including the dimensions and characters of its irreducible representations, through its root system and Dynkin diagram
• Enables the study of subalgebras, ideals, and automorphisms of semisimple Lie algebras, as well as their relations to other algebraic structures (Lie groups, symmetric spaces)
• Has applications in various areas of mathematics and physics, such as the study of exceptional groups, the classification of simple singularities, and the construction of certain gauge theories and supergravity models

Infinite Families vs Exceptional Lie Algebras

The Four Infinite Families

• An (n ≥ 1): The special linear Lie algebra sl(n+1, C), consisting of (n+1) × (n+1) matrices with trace zero
• Bn (n ≥ 2): The odd orthogonal Lie algebra so(2n+1, C), consisting of (2n+1) × (2n+1) skew-symmetric matrices
• Cn (n ≥ 3): The symplectic Lie algebra sp(2n, C), consisting of 2n × 2n matrices that preserve a non-degenerate skew-symmetric bilinear form
• Dn (n ≥ 4): The even orthogonal Lie algebra so(2n, C), consisting of 2n × 2n skew-symmetric matrices

The Five Exceptional Simple Lie Algebras

• G2: The smallest exceptional Lie algebra with dimension 14
• F4: An exceptional Lie algebra with dimension 52
• E6: An exceptional Lie algebra with dimension 78
• E7: An exceptional Lie algebra with dimension 133
• E8: The largest exceptional Lie algebra with dimension 248
  • Each exceptional Lie algebra has unique properties and can be characterized by its root system and Dynkin diagram

Dynkin Diagrams for Lie Algebras

Dynkin Diagrams as Graphical Representations

• A Dynkin diagram is a graphical representation of the root system of a semisimple Lie algebra, encoding its structure and properties
  • Each node represents a simple root of the Lie algebra
  • Edges between nodes represent the angles between the corresponding roots
• The Dynkin diagram uniquely determines the semisimple Lie algebra up to isomorphism, and vice versa

Dynkin Diagrams for Infinite Families and Exceptional Lie Algebras

• Infinite families:
  • An: A chain of n nodes, each connected to its neighbors by a single edge
  • Bn: A chain of n nodes, with a double edge connecting the last two nodes
  • Cn: A chain of n nodes, with a double edge connecting the first two nodes
  • Dn: A chain of n-2 nodes, with two additional nodes connected to the (n-2)th node, forming a fork
• Exceptional Lie algebras have more complex Dynkin diagrams involving multiple edges and different edge orientations

Applications of Lie Algebra Classification

Determining Invariants and Representation Theory

• The classification theorem helps determine dimension, rank, and other invariants of a semisimple Lie algebra based on its decomposition into simple Lie algebras
• The root system and Dynkin diagram provide information about the representation theory, including dimensions and characters of irreducible representations

Studying Algebraic Structures and Their Relations

• The classification enables the study of subalgebras, ideals, and automorphisms of semisimple Lie algebras
• It also helps understand the relations between semisimple Lie algebras and other algebraic structures (Lie groups, symmetric spaces)

Applications in Mathematics and Physics

• Exceptional Lie algebras have applications in various areas:
  • Study of exceptional groups
  • Classification of simple singularities
  • Construction of certain gauge theories and supergravity models

Semisimple Lie Algebras over Fields of Positive Characteristic

• The classification of semisimple Lie algebras over fields of positive characteristic is more complex, involving additional families and exceptional cases
• Basic principles and techniques from the complex case can often be adapted to this setting