7.5 Subfactor lattices

Subfactor lattices are a key concept in von Neumann algebras, providing a structured way to analyze nested subalgebras. They offer insights into algebraic and combinatorial properties of operator algebras, bridging abstract algebra and functional analysis.

These lattices organize intermediate subfactors between factors N and M, with elements corresponding to subalgebras. The meet and join operations represent intersection and generation of subalgebras, respectively. Lattice diagrams visually depict inclusion relations between subfactors.

Definition of subfactor lattices

• Subfactor lattices form a crucial component in the study of Von Neumann Algebras, providing a structured approach to analyze nested subalgebras
• These lattices offer insights into the algebraic and combinatorial properties of operator algebras, bridging abstract algebra and functional analysis

Subfactors vs factors

Top images from around the web for Subfactors vs factors

• 

  Infinite Measures on von Neumann Algebras | SpringerLink View original

  Is this image relevant?

• 

  Von-Neumann Finiteness and Reversibility in some Classes of Non-Associative Algebras | SpringerLink View original

  Is this image relevant?

• 

  Some aspects of quantum sufficiency for finite and full von Neumann algebras | SpringerLink View original

  Is this image relevant?

• 

  Infinite Measures on von Neumann Algebras | SpringerLink View original

  Is this image relevant?

• 

  Von-Neumann Finiteness and Reversibility in some Classes of Non-Associative Algebras | SpringerLink View original

  Is this image relevant?

1 of 3

Top images from around the web for Subfactors vs factors

• 

  Infinite Measures on von Neumann Algebras | SpringerLink View original

  Is this image relevant?

• 

  Von-Neumann Finiteness and Reversibility in some Classes of Non-Associative Algebras | SpringerLink View original

  Is this image relevant?

• 

  Some aspects of quantum sufficiency for finite and full von Neumann algebras | SpringerLink View original

  Is this image relevant?

• 

  Infinite Measures on von Neumann Algebras | SpringerLink View original

  Is this image relevant?

• 

  Von-Neumann Finiteness and Reversibility in some Classes of Non-Associative Algebras | SpringerLink View original

  Is this image relevant?

1 of 3

• Factors represent irreducible von Neumann algebras with trivial centers
• Subfactors denote inclusions of one factor within another, written as $N \subset M$
• Subfactor theory investigates the relative position and properties of these nested algebras
• Jones index $[M:N]$ quantifies the "size" of the subfactor, measuring how much larger M is compared to N

Lattice structure basics

• Subfactor lattices organize the intermediate subfactors between N and M
• Lattice elements correspond to von Neumann subalgebras between N and M
• Meet operation (∧) represents the intersection of two subalgebras
• Join operation (∨) denotes the von Neumann algebra generated by two subalgebras
• Lattice diagrams visually represent the inclusion relations between subfactors

Properties of subfactor lattices

• Subfactor lattices exhibit unique structural properties that differentiate them from general lattices
• These properties play a crucial role in the classification and analysis of von Neumann algebras

Finite vs infinite index

• Finite index subfactors have Jones index $[M:N] < \infty$
• Infinite index subfactors occur when $[M:N] = \infty$
• Finite index subfactors possess richer structure and are more amenable to classification
• Hyperfinite II₁ factor serves as a prototype for studying finite index subfactors
• Infinite index subfactors often arise in the context of group actions and crossed products

Irreducibility and extremality

• Irreducible subfactors satisfy $N' \cap M = \mathbb{C}$, where N' denotes the commutant of N in M
• Extremal subfactors minimize the Jones index among all subfactors in their conjugacy class
• Irreducibility ensures the absence of non-trivial intermediate subfactors
• Extremality provides a canonical choice of subfactor within its conjugacy class
• These properties simplify the analysis and classification of subfactor lattices

Depth and amenability

• Depth measures the complexity of the subfactor lattice structure
• Finite depth subfactors have a periodic structure in their standard invariant
• Amenable subfactors satisfy certain approximation properties
• Depth relates to the number of iterations needed to reach the relative commutant
• Amenability ensures the existence of a unique trace-preserving conditional expectation

Jones index theory

• Jones index theory revolutionized the study of subfactors and von Neumann algebras
• This theory provides powerful tools for classifying and analyzing subfactor lattices

Jones index definition

• Jones index $[M:N]$ measures the relative size of factor M compared to subfactor N
• For finite index subfactors, $[M:N] = \dim_N(L^2(M))$, where $L^2(M)$ is the GNS Hilbert space
• Jones' celebrated result: $[M:N] \in \{4\cos^2(\pi/n) : n \geq 3\} \cup [4,\infty]$
• Index values below 4 occur only at discrete points (Jones' quadratic series)
• Continuous spectrum of index values exists for indices greater than or equal to 4

Principal graph

• Principal graph encodes the fusion rules of bimodules in the subfactor
• Represents the Bratteli diagram of higher relative commutants
• Vertices correspond to irreducible N-N and N-M bimodules
• Edges represent tensor product decompositions of bimodules
• Principal graph serves as a powerful invariant for classifying subfactors

Standard invariant

• Standard invariant captures the essential algebraic structure of a subfactor
• Consists of towers of higher relative commutants: $N' \cap N \subset N' \cap M \subset N' \cap M_1 \subset \cdots$
• Includes the principal graph, dual principal graph, and planar algebra structure
• Governs the reconstruction of the subfactor up to outer conjugacy
• Plays a crucial role in the classification of subfactors, especially for small index values

Classification of subfactor lattices

• Classification of subfactor lattices represents a major goal in operator algebra research
• This area combines techniques from various mathematical disciplines to categorize and understand subfactor structures

Small index classification

• Complete classification achieved for subfactors with index less than 4
• Index values 1, 2, and 3 correspond to well-understood subfactors
• $A_n$ series subfactors arise from inclusions of finite cyclic groups
• $D_n$ series subfactors relate to dihedral group inclusions
• $E_6$, $E_7$, and $E_8$ exceptional subfactors discovered through classification efforts

Haagerup's list

• Haagerup classified all subfactors with index in the interval $(4,3+\sqrt{3})$
• List includes the Haagerup subfactor with index $\frac{5+\sqrt{13}}{2}$
• Asaeda-Haagerup subfactor appears on this list with index $\frac{5+\sqrt{17}}{2}$
• These exotic subfactors have no known group-theoretical or quantum group construction
• Haagerup's classification utilized advanced techniques in operator algebras and combinatorics

Asaeda-Haagerup subfactors

• Asaeda-Haagerup subfactors represent a family of exotic subfactors
• Discovered as part of the classification of subfactors with small index
• No known construction from familiar mathematical objects (groups, quantum groups)
• Possess unique principal graphs and fusion rules
• Serve as important examples in the study of fusion categories and conformal field theory

Planar algebra approach

• Planar algebra approach provides a powerful graphical calculus for studying subfactors
• This framework unifies various aspects of subfactor theory and enables new computational techniques

Jones' planar algebra

• Planar algebras encode the standard invariant of a subfactor
• Consist of vector spaces $P_{n,\pm}$ with planar operations (composition, tensor product, trace)
• Diagrams represent elements of the planar algebra, with strings corresponding to bimodules
• Planar isotopy invariance captures key algebraic relations in the subfactor
• Jones' construction allows reconstruction of the subfactor from its planar algebra

Temperley-Lieb algebra

• Temperley-Lieb algebra TL(δ) forms the simplest non-trivial example of a planar algebra
• Generated by "cup" and "cap" diagrams with parameter δ (loop value)
• Corresponds to the $A_n$ series of subfactors when $\delta = 2\cos(\pi/(n+1))$
• Plays a fundamental role in statistical mechanics and knot theory
• Serves as a building block for more complex planar algebras

Bisch-Jones planar algebras

• Bisch-Jones planar algebras generalize Temperley-Lieb algebras to include additional generators
• Correspond to subfactors arising from finite group actions and their subgroups
• Introduce "coloring" of strings to represent different bimodules
• Provide a unified framework for studying group-subgroup and crossed product subfactors
• Enable combinatorial analysis of fusion rules and principal graphs for these subfactors

Applications of subfactor lattices

• Subfactor lattices find applications across various areas of mathematics and physics
• These structures provide insights into quantum symmetries and topological phenomena

Quantum groups

• Subfactors naturally arise from quantum group representations
• Jones-Wassermann subfactors correspond to loop group representations
• Quantum double construction relates to Drinfeld centers of fusion categories
• Subfactor theory provides tools for studying quantum symmetries in low dimensions
• Applications include classification of quantum group categories and modular tensor categories

Conformal field theory

• Subfactors model chiral algebras and operator product expansions in CFT
• Fusion categories derived from subfactors describe fusion rules of primary fields
• Modular invariance in CFT relates to modularity of subfactor fusion categories
• Subfactor planar algebras encode correlation functions and conformal blocks
• Applications include classification of rational conformal field theories and boundary CFTs

Topological quantum computation

• Anyonic systems in topological quantum computation modeled by subfactor fusion categories
• Braiding statistics of anyons described by subfactor planar algebras
• Subfactor invariants (principal graphs, fusion rules) determine computational power of anyonic systems
• Jones representations of braid groups arise from subfactor theory
• Applications include design and analysis of topological quantum error-correcting codes

Subfactor lattice constructions

• Various construction methods exist for creating subfactor lattices
• These techniques allow for the systematic study and generation of new subfactors

Group-subgroup construction

• Subfactors arise from inclusion of fixed-point algebras $M^H \subset M^G$ for $H \subset G$
• Jones index equals [G:H], the index of the subgroup
• Principal graph relates to the representation theory of G and H
• Intermediate subfactors correspond to intermediate subgroups
• Examples include cyclic group inclusions ($A_n$ series) and dihedral group inclusions ($D_n$ series)

Crossed product construction

• Subfactors formed by $N \subset N \rtimes G$ for a group G acting on factor N
• Jones index equals |G|, the order of the group
• Principal graph encodes the group action and its fixed points
• Dual construction $N \subset (N \rtimes G)^G$ yields the fixed-point subfactor
• Applications in studying group actions on von Neumann algebras and ergodic theory

Composition of subfactors

• New subfactors created by composing existing subfactors: $P \subset N \subset M$
• Jones index multiplicativity: $[M:P] = [M:N][N:P]$
• Principal graphs of composed subfactors relate to fiber products of original graphs
• Intermediate subfactors of the composition form a sublattice
• Technique used to construct exotic subfactors from simpler building blocks

Advanced topics in subfactor theory

• Subfactor theory connects to various advanced mathematical concepts
• These topics represent active areas of research in operator algebras and category theory

Fusion categories

• Fusion categories arise as categories of N-N bimodules in finite-depth subfactors
• Encode tensor product and duality structures of bimodules
• Classification of fusion categories closely related to subfactor classification
• Drinfeld center of a fusion category yields a modular tensor category
• Applications in topological quantum field theory and anyonic systems

Subfactor planar algebras

• Planar algebras provide a graphical calculus for subfactor invariants
• Skein theory of planar algebras relates to knot invariants and TQFTs
• Subfactor planar algebras generalize Temperley-Lieb and Hecke algebras
• Reconstruction theorems allow recovery of subfactors from planar algebras
• Tools include graph planar algebras, jellyfish algorithms, and subfactor arithmetic

Bisch-Haagerup subfactors

• Subfactors arising from finite-depth inclusions of II₁ factors
• Generalize group-subgroup and crossed product constructions
• Principal graphs exhibit periodic structure with finite "core"
• Classification program for Bisch-Haagerup subfactors ongoing
• Connections to free probability theory and random matrices

Computational aspects

• Computational techniques play an increasingly important role in subfactor theory
• These methods enable exploration and verification of theoretical results

Computing Jones index

• Numerical methods for approximating Jones index in infinite-depth cases
• Exact computation possible for finite-depth subfactors using fusion rules
• Trace methods on finite-dimensional approximants of factors
• Statistical approaches using Monte Carlo simulations on spin models
• Applications in verifying index restrictions and classifying small-index subfactors

Generating principal graphs

• Algorithms for enumerating possible principal graphs up to a given depth
• Ocneanu's paragroup theory provides constraints on admissible graphs
• Fusion rule algebra computations determine graph structure
• Graph-theoretic approaches (e.g., depth reduction) for efficient enumeration
• Applications in small-index classification and discovery of exotic subfactors

Software tools for subfactors

• Specialized software packages for subfactor computations (Knotted Cord, Subfactor Atlas)
• Computer algebra systems adapted for planar algebra calculations
• Visualization tools for principal graphs and planar algebra elements
• Database systems for storing and querying known subfactors and their invariants
• Machine learning approaches for pattern recognition in subfactor data