7.6 Bisch-Haagerup subfactors

Bisch-Haagerup subfactors are a key class in von Neumann algebra theory, bridging group theory and operator algebras. They're constructed from finite groups acting on factors, offering insights into subfactor structure and classification.

These subfactors have unique properties, including specific index values and characteristic principal graphs. Their study involves sophisticated construction methods, planar algebra representations, and connections to fusion categories, contributing to broader subfactor theory and classification efforts.

Definition of Bisch-Haagerup subfactors

• Bisch-Haagerup subfactors represent a significant class of subfactors in von Neumann algebra theory
• These subfactors bridge group theory and operator algebras, providing insights into both fields
• Understanding Bisch-Haagerup subfactors enhances our comprehension of subfactor structure and classification

Historical context

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• Introduced in the late 1990s by Dietmar Bisch and Uffe Haagerup
• Emerged from the study of subfactors associated with finite groups and their subgroups
• Built upon Jones' groundbreaking work on subfactors and index theory
• Addressed questions about intermediate subfactors and their properties

Key characteristics

• Constructed from two finite groups G and H with an outer action on a II₁ factor
• Index values determined by the orders of the groups: $[M:N] = |G||H|/(|G \cap H|)$
• Principal graphs often exhibit a characteristic "diamond" shape
• Possess rich combinatorial and algebraic structures
• Serve as examples of subfactors with nontrivial intermediate subfactors

Construction methods

• Construction of Bisch-Haagerup subfactors involves sophisticated techniques from operator algebras
• These methods provide a bridge between group theory and von Neumann algebras
• Understanding construction techniques is crucial for analyzing subfactor properties

Tensor product approach

• Utilizes the tensor product of two group-subgroup subfactors
• Starts with a II₁ factor M and two finite groups G and H
• Constructs the subfactor as $(M^G \otimes M^H) \subset (M \otimes M)$
• Requires careful analysis of the tensor product structure
• Allows for the study of interactions between the two group actions

Group-subgroup method

• Based on the inclusion of a subgroup K in the intersection of G and H
• Constructs the subfactor as $(M^K)^{G/K} \subset M^K$
• Involves analyzing the group actions and their fixed-point algebras
• Provides a direct link to the group-theoretic properties of G and H
• Enables the study of intermediate subfactors corresponding to subgroups

Properties of Bisch-Haagerup subfactors

• Bisch-Haagerup subfactors exhibit unique characteristics that set them apart in subfactor theory
• These properties provide valuable insights into the structure of von Neumann algebras
• Understanding these properties is essential for classification and application of Bisch-Haagerup subfactors

Index values

• Always take the form of integer products: $[M:N] = |G||H|/(|G \cap H|)$
• Can be rational numbers, unlike Jones subfactors which have index ≥ 4 or of the form 4cos²(π/n)
• Provide information about the relative sizes of the groups G and H
• Help in distinguishing between different Bisch-Haagerup subfactors
• Can be used to study the lattice of intermediate subfactors

Principal graphs

• Often exhibit a characteristic "diamond" shape
• Reflect the group-theoretic structure of G and H
• Contain information about the fusion rules of bimodules
• Can be computed using representation theory of the groups involved
• Serve as a powerful invariant for classifying Bisch-Haagerup subfactors

Classification theory

• Classification of Bisch-Haagerup subfactors remains an active area of research in operator algebras
• Contributes to the broader classification program for subfactors
• Helps in understanding the landscape of subfactors and their properties

Known examples

• Include subfactors arising from dihedral groups (D₂n)
• Encompass subfactors derived from symmetric groups (Sn)
• Feature subfactors constructed from alternating groups (An)
• Contain examples based on finite simple groups (PSL(2,q))
• Include subfactors arising from product groups and their subgroups

Open problems

• Complete classification of all Bisch-Haagerup subfactors remains unresolved
• Determining which groups can give rise to Bisch-Haagerup subfactors
• Understanding the relationship between group properties and subfactor invariants
• Classifying Bisch-Haagerup subfactors with specific index values or principal graphs
• Exploring connections between Bisch-Haagerup subfactors and other mathematical structures (quantum groups)

Planar algebra representation

• Planar algebras provide a powerful visual and algebraic tool for studying Bisch-Haagerup subfactors
• This representation connects subfactor theory to other areas of mathematics (knot theory)
• Understanding planar algebras enhances our ability to analyze and classify Bisch-Haagerup subfactors

Generators and relations

• Planar algebras for Bisch-Haagerup subfactors typically have two generators
• Generators correspond to the groups G and H in the construction
• Relations reflect the group-theoretic properties of G and H
• Include specific relations for the intersection of G and H
• Can be used to derive other properties of the subfactor (principal graphs)

Diagrammatic calculus

• Utilizes a system of diagrams to represent elements and operations in the subfactor
• Includes specific symbols or colors for generators corresponding to G and H
• Incorporates rules for manipulating diagrams based on group properties
• Allows for visual computation of subfactor invariants (traces)
• Provides intuitive understanding of the subfactor structure and properties

Applications in operator algebras

• Bisch-Haagerup subfactors have significant implications across various areas of operator algebras
• These subfactors provide concrete examples for testing general theories
• Studying their applications enhances our understanding of von Neumann algebras and related fields

Connections to fusion categories

• Bisch-Haagerup subfactors give rise to specific fusion categories
• These categories reflect the group-theoretic structure of G and H
• Provide examples of fusion categories with non-trivial associators
• Allow for the study of categorical properties (modularity)
• Contribute to the classification of fusion categories

Role in subfactor theory

• Serve as important examples in the classification of subfactors
• Provide insights into the structure of intermediate subfactors
• Help in understanding the relationship between group theory and operator algebras
• Contribute to the development of new techniques in subfactor analysis
• Offer a testing ground for conjectures in subfactor theory

Generalizations and extensions

• Bisch-Haagerup construction has inspired various generalizations in subfactor theory
• These extensions broaden the scope of applications and connections to other areas
• Studying generalizations enhances our understanding of subfactor structure and classification

Higher-rank cases

• Extend the construction to involve more than two groups
• Include subfactors arising from multiple group-subgroup inclusions
• Require more complex analysis of group interactions and fixed-point algebras
• Offer richer structure and potentially new invariants
• Provide connections to higher-category theory

Quantum group analogues

• Replace finite groups with quantum groups in the construction
• Utilize Hopf algebra techniques in the analysis
• Offer examples of subfactors with non-integer index values
• Provide connections to quantum symmetries and non-commutative geometry
• Allow for the study of deformation theories in subfactor context

Computational aspects

• Computational methods play a crucial role in the study of Bisch-Haagerup subfactors
• These techniques enable the exploration of complex structures and invariants
• Understanding computational aspects is essential for advancing research in this field

Algorithms for principal graphs

• Utilize group-theoretic data to construct principal graphs
• Employ representation theory techniques for computing fusion rules
• Implement graph-theoretic algorithms for analyzing graph structure
• Use combinatorial methods to determine graph growth and periodicity
• Incorporate techniques from computer algebra systems for efficient computation

Software tools

• Include specialized packages for subfactor analysis (Knotted)
• Utilize general-purpose mathematical software (SageMath)
• Implement custom algorithms for specific Bisch-Haagerup constructions
• Provide visualization tools for planar algebra representations
• Offer databases of known Bisch-Haagerup subfactors and their properties

Related subfactor families

• Bisch-Haagerup subfactors form part of a larger landscape of subfactor constructions
• Understanding relationships between different subfactor families enhances our overall comprehension
• Comparing and contrasting these families provides insights into subfactor structure and classification

Bisch-Haagerup vs Jones subfactors

• Bisch-Haagerup subfactors allow for rational index values, unlike Jones subfactors
• Jones subfactors arise from representations of Temperley-Lieb algebras
• Bisch-Haagerup construction directly incorporates group theory
• Jones subfactors have a more restricted set of possible principal graphs
• Both families contribute to the understanding of subfactor classification and structure

Connections to GHJ construction

• GHJ (Goodman-de la Harpe-Jones) construction uses graphs to build subfactors
• Bisch-Haagerup subfactors can sometimes be realized as GHJ subfactors
• Both constructions provide insights into the relationship between graphs and subfactors
• GHJ construction offers a more general framework for certain subfactor families
• Comparing the two approaches enhances our understanding of subfactor structure

Research directions

• Bisch-Haagerup subfactors continue to be an active area of research in operator algebras
• Current and future research aims to deepen our understanding and expand applications
• Exploring new directions contributes to the broader field of von Neumann algebras

Recent developments

• Exploration of Bisch-Haagerup subfactors with exceptional groups
• Connections to conformal field theory and boundary conditions
• Applications in quantum information theory and entanglement
• Development of new invariants for distinguishing Bisch-Haagerup subfactors
• Investigations into the structure of planar algebras associated with these subfactors

Future prospects

• Complete classification of Bisch-Haagerup subfactors for specific group families
• Exploration of higher-rank generalizations and their properties
• Applications in topological quantum computing and anyonic systems
• Connections to other areas of mathematics (representation theory)
• Development of new computational tools for analyzing complex subfactor structures