7.3 Principal graphs

Principal graphs are essential tools in subfactor theory, encoding crucial information about subfactor structure and properties. They provide visual representations of inclusion relationships between factors and capture the combinatorial structure of subfactor inclusions.

These graphs exhibit specific characteristics that reveal subfactor properties, aiding in classification and analysis. Understanding principal graphs is key to exploring new subfactors, computing indices, and uncovering connections between subfactor theory and other areas of mathematics.

Definition of principal graphs

• Principal graphs serve as fundamental tools in subfactor theory within von Neumann algebras
• Encode crucial information about the structure and properties of subfactors
• Provide visual representations of inclusion relationships between factors

Bipartite graphs

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• Consist of two disjoint sets of vertices with edges connecting only between sets
• Represent alternating levels of relative commutants in the subfactor tower
• Vertices correspond to irreducible bimodules or sectors in the subfactor theory
• Edges indicate tensor product decompositions or fusion rules between sectors

Connection to subfactors

• Capture the combinatorial structure of subfactor inclusions
• Reflect the standard invariant of a subfactor, a key algebraic object
• Allow for classification and analysis of subfactors based on graph properties
• Encode information about dimensions of relative commutants in the subfactor tower

Properties of principal graphs

• Principal graphs exhibit specific characteristics that reveal subfactor properties
• Analysis of these properties aids in understanding and classifying subfactors
• Provide insights into the complexity and structure of subfactor inclusions

Finite depth vs infinite depth

• Finite depth graphs terminate after a finite number of vertices
  • Correspond to subfactors with finite-dimensional standard invariants
  • Often associated with rational conformal field theories
• Infinite depth graphs continue indefinitely
  • Represent subfactors with infinite-dimensional standard invariants
  • Can arise from certain quantum group constructions or irrational conformal field theories
• Depth of a principal graph determines many important subfactor properties

Amenability and growth rate

• Amenability relates to the existence of an invariant mean on the graph
• Growth rate measures the asymptotic behavior of vertex counts at increasing depths
  • Subexponential growth rates indicate amenability of the subfactor
  • Exponential growth rates suggest non-amenability
• Connections to spectral properties of the adjacency matrix of the graph
• Impacts computational aspects and representation theory of the associated subfactor

Construction of principal graphs

• Principal graphs arise naturally from the structure of subfactor inclusions
• Construction methods provide insights into the relationship between subfactors and their graphs
• Understanding these constructions aids in exploring new subfactors and their properties

From subfactor inclusions

• Start with a subfactor inclusion $N \subset M$ of II₁ factors
• Construct the Jones tower: $N \subset M \subset M_1 \subset M_2 \subset \cdots$
• Analyze relative commutants: $N' \cap M_k$ and $M' \cap M_k$
• Vertices of the graph correspond to irreducible summands in these relative commutants
• Edges represent inclusions between successive levels in the tower

Fusion rules and graphs

• Fusion rules describe how irreducible bimodules tensor with each other
• Encode these rules as a directed graph with labeled edges
• Vertices represent irreducible bimodules, edges indicate fusion outcomes
• Apply a process of "reflection" and "reduction" to obtain the principal graph
• Resulting graph captures the essential structure of the fusion category associated with the subfactor

Classification of principal graphs

• Classification efforts aim to categorize and understand all possible principal graphs
• Provides a systematic approach to studying subfactors and their properties
• Reveals deep connections between subfactor theory and other areas of mathematics

ADE classification

• Classifies finite depth subfactors of index less than 4
• Named after the Dynkin diagrams of types A, D, and E
• An series corresponds to $SU(2)$ quantum groups at roots of unity
• Dn series relates to certain orbifold constructions
• E6, E7, and E8 graphs represent exceptional cases with unique properties
• Demonstrates a surprising connection between subfactors and Lie theory

Beyond ADE classification

• Explores subfactors with index greater than or equal to 4
• Includes infinite depth subfactors and more complex graph structures
• Haagerup subfactor with principal graph $\sqrt[3]{13}$ as a notable example
• Extended classification efforts use techniques from planar algebras and fusion categories
• Ongoing research aims to discover and classify new exotic subfactors

Planar algebra representation

• Planar algebras provide a powerful diagrammatic approach to subfactor theory
• Allow for visual representation and manipulation of algebraic structures
• Offer insights into the structure and properties of principal graphs

Jones-Wenzl projections

• Special elements in the Temperley-Lieb algebra crucial for constructing planar algebras
• Correspond to vertices in the principal graph
• Properties of Jones-Wenzl projections reflect characteristics of the associated subfactor
• Used to define higher relative commutants in the subfactor tower
• Play a key role in computing quantum dimensions and fusion rules

Temperley-Lieb algebra connection

• Temperley-Lieb algebra forms the foundation for many planar algebra constructions
• Represents the simplest non-trivial example of a planar algebra
• Encodes information about the An series of principal graphs
• Generalizations of Temperley-Lieb algebras correspond to more complex principal graphs
• Provides a bridge between subfactor theory and statistical mechanics models

Applications in subfactor theory

• Principal graphs serve as powerful tools for analyzing and understanding subfactors
• Applications extend to various aspects of operator algebras and related fields
• Demonstrate the deep connections between graph theory and von Neumann algebras

Index computation

• Principal graphs allow for efficient calculation of subfactor indices
• Index given by the square of the largest eigenvalue of the graph's adjacency matrix
• Provides a measure of the "size" or "complexity" of the subfactor inclusion
• Connects to quantum dimensions of objects in the associated fusion category
• Allows for classification of subfactors based on their index values

Standard invariants

• Principal graphs form a key component of the standard invariant of a subfactor
• Capture essential algebraic and combinatorial information about the subfactor
• Used in conjunction with other invariants (paragroups, lambda lattices) for complete classification
• Allow for reconstruction of the subfactor up to outer conjugacy in many cases
• Provide a bridge between subfactor theory and tensor category theory

Examples of principal graphs

• Concrete examples illustrate the diversity and richness of principal graphs
• Demonstrate how graph structures reflect properties of associated subfactors
• Provide benchmarks for testing and developing new theories and techniques

An and Dn series

• An series: linear graphs corresponding to $SU(2)$ quantum groups at roots of unity
  • A∞ represents the infinite depth case of the hyperfinite II₁ factor
  • Finite An graphs relate to Jones subfactors of finite depth
• Dn series: forked graphs associated with certain orbifold constructions
  • D∞ corresponds to the infinite depth $\mathbb{Z}/2\mathbb{Z}$ fixed point subfactor
  • Finite Dn graphs arise from specific finite group actions on factors

Exceptional cases

• E6, E7, and E8 graphs from the ADE classification
  • E6 related to the $G_2$ quantum group at level 1
  • E8 connected to the Haagerup subfactor and exotic fusion categories
• Haagerup subfactor with principal graph $\sqrt[3]{13}$
  • First example of an exotic subfactor beyond the ADE classification
  • Demonstrates the existence of exceptional fusion categories
• Asaeda-Haagerup subfactor and its principal graph
  • Another exotic case with no known quantum group interpretation
  • Illustrates the richness of subfactor theory beyond classical constructions

Symmetries of principal graphs

• Symmetries of principal graphs reveal important structural properties of subfactors
• Provide insights into the underlying algebraic and categorical structures
• Play a crucial role in classification and analysis of subfactors

Automorphisms

• Graph automorphisms preserve the structure of the principal graph
• Reflect symmetries in the fusion rules and bimodule category of the subfactor
• Can indicate the presence of additional structure (outer automorphisms)
• Used to define and study subfactor planar algebras with additional symmetries
• Relate to Galois theory of subfactors and intermediate subfactors

Dual principal graphs

• Represent the structure of the dual subfactor M' ⊂ M
• Often closely related to the original principal graph, but can differ in some cases
• Provide additional information about the subfactor and its standard invariant
• Studying pairs of principal and dual principal graphs aids in classification efforts
• Connections to Morita equivalence of subfactors and fusion categories

Computational aspects

• Computational methods play an increasingly important role in subfactor theory
• Allow for exploration and verification of theoretical results
• Provide tools for discovering new subfactors and analyzing their properties

Graph enumeration algorithms

• Develop systematic methods for generating potential principal graphs
• Incorporate constraints based on known properties (index bounds, fusion rules)
• Use combinatorial techniques to efficiently explore large spaces of graphs
• Implement backtracking algorithms to handle complex graph structures
• Apply graph isomorphism tests to eliminate redundant cases

Software tools for analysis

• Develop specialized software packages for subfactor and planar algebra computations
  • (Mathematica packages, GAP libraries, custom C++ programs)
• Implement algorithms for computing fusion rules from graph data
• Create visualization tools for principal graphs and related diagrammatic structures
• Utilize computer algebra systems for symbolic computations of subfactor invariants
• Develop databases of known subfactors and their principal graphs for reference and analysis

Connection to other mathematical areas

• Principal graphs and subfactor theory intersect with various branches of mathematics
• These connections provide new perspectives and tools for understanding subfactors
• Demonstrate the far-reaching implications of subfactor theory in mathematics

Representation theory

• Principal graphs encode information about representations of quantum groups and Hopf algebras
• Relate to McKay graphs of finite group representations and their generalizations
• Connect to the representation theory of affine Lie algebras and conformal field theories
• Provide a bridge between subfactor theory and the representation theory of operator algebras
• Illuminate connections between subfactors and the representation theory of fusion categories

Quantum groups

• Many principal graphs arise from quantum group constructions at roots of unity
• Quantum group representation categories often yield subfactors with interesting principal graphs
• Provide a rich source of examples and a framework for understanding subfactor structure
• Connect subfactor theory to the study of quantum symmetries and non-commutative geometry
• Allow for the application of quantum group techniques in the analysis of subfactors

Open problems and conjectures

• Ongoing research in subfactor theory continues to generate new questions and challenges
• Open problems drive the development of new techniques and insights
• Conjectures provide direction for future research and exploration

Classification challenges

• Complete classification of subfactors beyond index 4 remains an open problem
• Develop efficient methods for ruling out potential principal graphs
• Understand the relationship between principal graphs and fusion categories
• Investigate the existence of subfactors with specific graph properties
• Explore connections between subfactor classification and other classification problems in mathematics

Existence of certain graphs

• Determine whether all finite bipartite graphs can appear as principal graphs of subfactors
• Investigate the existence of subfactors with prescribed fusion rules or quantum dimensions
• Explore the possibility of exotic subfactors beyond known examples (Haagerup, Asaeda-Haagerup)
• Study the existence of subfactors with specific symmetry properties or automorphism groups
• Investigate connections between graph-theoretic properties and the existence of corresponding subfactors