5.2 Cartan matrices and Dynkin diagrams

Cartan matrices and Dynkin diagrams are powerful tools for understanding root systems in Lie algebras. They encode crucial information about simple roots, their relationships, and the overall structure of semisimple Lie algebras.

These concepts help classify and visualize Lie algebras, making complex algebraic structures more accessible. By studying Cartan matrices and Dynkin diagrams, we gain insights into the fundamental properties of Lie algebras and their applications in various fields.

Cartan matrices from root systems

Construction and properties of Cartan matrices

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• Construct a Cartan matrix from a root system of a semisimple Lie algebra to encode essential information about the root system's structure
• Define the entries of a Cartan matrix as $a_{ij} = \frac{2(\alpha_i, \alpha_j)}{(\alpha_i, \alpha_i)}$, where $\alpha_i$ and $\alpha_j$ are simple roots, and $(\cdot, \cdot)$ denotes the inner product on the root space
• Diagonal entries of a Cartan matrix are always 2, while off-diagonal entries are non-positive integers (0, -1, -2, or -3)
• The Cartan matrix is symmetric if and only if the root system is simply-laced, meaning all roots have the same length (A_n, D_n, E_6, E_7, E_8)

Cartan matrix determinant and rank

• The determinant of a Cartan matrix is always positive, ensuring the linear independence of the simple roots
• The rank of a Cartan matrix equals the rank of the corresponding semisimple Lie algebra, which is the dimension of its Cartan subalgebra
• The Cartan matrix determines the angles between the simple roots and their relative lengths, which in turn determine the structure of the root system
• The entries of the Cartan matrix encode the geometrical relationships between the simple roots, such as orthogonality (0), 120-degree angle (-1), and 135-degree or 150-degree angles (-2 or -3)

Dynkin diagrams as representations

Graphical representation of Cartan matrices

• Represent a Cartan matrix graphically using a Dynkin diagram to provide a visual summary of the root system's structure
• Each node in a Dynkin diagram represents a simple root, and the edges between nodes represent the angles and relative lengths of the roots
• Connect two nodes by a single edge if the corresponding off-diagonal entry in the Cartan matrix is -1, indicating a 120-degree angle between the roots (A_2, B_2, G_2)
• Connect two nodes by a double edge, with an arrow pointing from the longer root to the shorter root, if the corresponding off-diagonal entry is -2 or -3 (B_n, C_n, F_4)
• The absence of an edge between two nodes indicates that the corresponding roots are orthogonal, forming a 90-degree angle (A_1 × A_1, B_2 × A_1)

Dynkin diagram properties

• The number of nodes in a Dynkin diagram equals the rank of the corresponding semisimple Lie algebra
• The connectedness of a Dynkin diagram reveals whether the Lie algebra is simple (connected) or a direct sum of simple Lie algebras (disconnected)
• The node labels and edge multiplicities in the Dynkin diagram determine the type of the corresponding simple Lie algebras (A_n, B_n, C_n, D_n, or exceptional)
• Dynkin diagrams provide a compact and intuitive way to visualize the structure of root systems and their associated Lie algebras

Classifying Dynkin diagrams

Connected and disconnected Dynkin diagrams

• Classify Dynkin diagrams based on their connectedness and node labels to identify the corresponding Lie algebras
• Connected Dynkin diagrams correspond to simple Lie algebras, which are the building blocks of semisimple Lie algebras (A_n, B_n, C_n, D_n, E_6, E_7, E_8, F_4, G_2)
• Disconnected Dynkin diagrams correspond to semisimple Lie algebras that are direct sums of simple Lie algebras (A_1 × A_1, A_2 × A_1, B_2 × A_1)
• The connected components of a disconnected Dynkin diagram represent the simple Lie algebras in the direct sum decomposition

Classical series and exceptional cases

• Classify the connected Dynkin diagrams into the classical series (A_n, B_n, C_n, D_n) and the exceptional cases (E_6, E_7, E_8, F_4, G_2)
• The subscript in the label of a Dynkin diagram denotes the rank of the corresponding Lie algebra
• The classical series have a regular pattern in their Dynkin diagrams, with A_n being a line of n nodes, B_n and C_n having a fork at the end, and D_n having a fork in the middle
• The exceptional Dynkin diagrams have more intricate structures and do not fit into the classical series, exhibiting unique symmetries and properties (E_8 has the largest symmetry group among exceptional cases)

Node labels and Cartan matrix reconstruction

• Use the node labels in a Dynkin diagram to determine the Cartan matrix and the corresponding Lie algebra's structure
• Assign a basis vector to each node in the Dynkin diagram, representing a simple root in the root system
• Reconstruct the Cartan matrix from the Dynkin diagram by setting the diagonal entries to 2 and the off-diagonal entries according to the edge multiplicities and arrow directions
• The Cartan matrix obtained from the Dynkin diagram uniquely determines the structure of the corresponding semisimple Lie algebra

Cartan matrices vs Lie algebras

Encoding Lie algebra structure

• The Cartan matrix and Dynkin diagram encode essential information about the structure of a semisimple Lie algebra, such as its rank, root system, and decomposition into simple Lie algebras
• The rank of a semisimple Lie algebra equals the number of nodes in its Dynkin diagram or the size of its Cartan matrix, determining the dimension of its Cartan subalgebra
• The entries of the Cartan matrix determine the commutation relations between the Chevalley generators of the Lie algebra, which in turn determine its structure constants
• The Dynkin diagram's connectedness reveals whether the Lie algebra is simple (connected) or a direct sum of simple Lie algebras (disconnected)

Classification and representation theory

• Use the Cartan matrix and Dynkin diagram to classify semisimple Lie algebras and study their representations, such as the adjoint representation and the highest weight representations
• The node labels and edge multiplicities in the Dynkin diagram determine the type of the corresponding simple Lie algebras (A_n, B_n, C_n, D_n, or exceptional), which have distinct representation theories
• The Cartan matrix and Dynkin diagram provide a framework for constructing and analyzing the representations of semisimple Lie algebras, using tools such as weight diagrams and character formulas
• The properties of the Cartan matrix and Dynkin diagram also have implications for the associated Lie group's structure and geometry, such as its maximal compact subgroup and flag manifolds (SU(n), SO(n), Sp(n))

Applications and connections

• Cartan matrices and Dynkin diagrams have applications in various areas of mathematics and physics, such as algebraic geometry, representation theory, and gauge theories
• The classification of semisimple Lie algebras using Cartan matrices and Dynkin diagrams has deep connections with the classification of regular polytopes, finite reflection groups, and root lattices
• The study of Cartan matrices and Dynkin diagrams provides a unifying framework for understanding the structure and properties of semisimple Lie algebras, revealing their intricate symmetries and interrelationships
• The techniques and insights gained from the study of Cartan matrices and Dynkin diagrams can be generalized to other algebraic structures, such as Kac-Moody algebras and quantum groups, leading to further developments in mathematics and physics