Dynkin Diagrams

Introduction

Dynkin diagrams are a type of graph used in various branches of mathematics, particularly in the study of Lie algebras, algebraic groups, and root systems. These diagrams are named after the Russian mathematician Eugene Dynkin, who introduced them in the context of classifying semisimple Lie algebras. Dynkin diagrams provide a visual representation of the relationships between the simple roots of a root system, capturing the essential structure of the corresponding Lie algebra.

Definition and Construction

Dynkin diagrams are constructed from the root system of a semisimple Lie algebra. A root system is a finite set of vectors in a Euclidean space that satisfies specific axioms. Each root in the system corresponds to a node in the Dynkin diagram. The edges between nodes represent the angles between the corresponding roots.

Nodes and Edges

In a Dynkin diagram:

Each node represents a simple root.
An edge between two nodes indicates that the corresponding roots are not orthogonal.
The number of edges (single, double, or triple) and their orientation (if any) encode the angle between the roots.

For example, a single edge represents an angle of 120 degrees, a double edge (with an arrow) represents an angle of 135 degrees, and a triple edge (with an arrow) represents an angle of 150 degrees.

Classification of Simple Lie Algebras

Dynkin diagrams play a crucial role in the classification of simple Lie algebras. The classification is based on the structure of the root system, which can be represented by a Dynkin diagram. There are four infinite families of simple Lie algebras, denoted by \(A_n\), \(B_n\), \(C_n\), and \(D_n\), and five exceptional Lie algebras, denoted by \(E_6\), \(E_7\), \(E_8\), \(F_4\), and \(G_2\).

Infinite Families

**\(A_n\)**: Corresponds to the special linear Lie algebra \(\mathfrak{sl}(n+1)\). The Dynkin diagram is a simple chain of \(n\) nodes.
**\(B_n\)**: Corresponds to the orthogonal Lie algebra \(\mathfrak{so}(2n+1)\). The Dynkin diagram consists of a chain of \(n\) nodes with a double edge at the end.
**\(C_n\)**: Corresponds to the symplectic Lie algebra \(\mathfrak{sp}(2n)\). The Dynkin diagram is similar to \(B_n\) but with a different orientation of the double edge.
**\(D_n\)**: Corresponds to the orthogonal Lie algebra \(\mathfrak{so}(2n)\). The Dynkin diagram is a chain of \(n-1\) nodes with a forked end.

Exceptional Lie Algebras

**\(E_6\)**: The Dynkin diagram consists of six nodes forming a T-shape.
**\(E_7\)**: The Dynkin diagram extends \(E_6\) with an additional node.
**\(E_8\)**: The Dynkin diagram extends \(E_7\) with another node, forming a long chain with a branch.
**\(F_4\)**: The Dynkin diagram consists of four nodes with a unique structure.
**\(G_2\)**: The Dynkin diagram consists of two nodes connected by a triple edge.

Applications in Lie Theory

Dynkin diagrams are instrumental in the study of Lie algebras and their representations. They provide a compact way to encode the structure of the root system and facilitate the classification of semisimple Lie algebras. The diagrams also help in understanding the relationships between different Lie algebras and their subalgebras.

Root Systems and Weyl Groups

The root system of a Lie algebra can be decomposed into simple roots, which form the basis for the Dynkin diagram. The Weyl group, associated with the root system, acts on the roots and preserves the structure of the Dynkin diagram. This action provides insights into the symmetries and automorphisms of the Lie algebra.

Representation Theory

In representation theory, Dynkin diagrams help classify the irreducible representations of semisimple Lie algebras. The highest weight of a representation can be expressed in terms of the simple roots, and the Dynkin diagram encodes the possible weights and their multiplicities.

Generalizations and Extensions

Dynkin diagrams have been generalized to other mathematical structures, such as Kac-Moody algebras and quivers. These generalizations extend the applicability of Dynkin diagrams beyond the realm of semisimple Lie algebras.

Kac-Moody Algebras

Kac-Moody algebras are a generalization of semisimple Lie algebras that can be infinite-dimensional. The Dynkin diagrams for Kac-Moody algebras include additional nodes and edges, representing more complex relationships between the roots.

Quivers

In representation theory, quivers are directed graphs that generalize Dynkin diagrams. Quivers are used to study the representations of algebras and have applications in areas such as homological algebra and representation theory.

References

Humphreys, James E. "Introduction to Lie Algebras and Representation Theory." Springer, 1972.
Carter, Roger. "Lie Algebras of Finite and Affine Type." Cambridge University Press, 2005.
Kac, Victor G. "Infinite-Dimensional Lie Algebras." Cambridge University Press, 1990.