quantifies the size of within von Neumann algebras. Introduced by in 1983, it revolutionized operator algebras and connected them to knot theory and .
The index takes values in a specific set, including some discrete points and a continuous range. It provides insights into subfactor structure, influences statistical dimensions in physics, and has applications in and knot invariants.
Definition of Jones index
Jones index quantifies the relative size of a subfactor within a larger von Neumann algebra
Introduced by Vaughan Jones in 1983, revolutionizing the study of operator algebras and their applications
Plays a crucial role in understanding the structure of von Neumann algebras and their subfactors
Historical context
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Homer Nodded: Von Neumann’s Surprising Oversight | Foundations of Physics View original
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Von-Neumann Finiteness and Reversibility in some Classes of Non-Associative Algebras | SpringerLink View original
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Local Nets of Von Neumann Algebras in the Sine–Gordon Model | SpringerLink View original
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Homer Nodded: Von Neumann’s Surprising Oversight | Foundations of Physics View original
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Von-Neumann Finiteness and Reversibility in some Classes of Non-Associative Algebras | SpringerLink View original
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Top images from around the web for Historical context
Homer Nodded: Von Neumann’s Surprising Oversight | Foundations of Physics View original
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Von-Neumann Finiteness and Reversibility in some Classes of Non-Associative Algebras | SpringerLink View original
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Local Nets of Von Neumann Algebras in the Sine–Gordon Model | SpringerLink View original
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Homer Nodded: Von Neumann’s Surprising Oversight | Foundations of Physics View original
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Von-Neumann Finiteness and Reversibility in some Classes of Non-Associative Algebras | SpringerLink View original
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Emerged from Vaughan Jones' work on subfactors in the early 1980s
Built upon previous research in operator algebras by Murray and von Neumann
Resulted from attempts to classify II₁ factors and their subfactors
Led to unexpected connections with knot theory and statistical mechanics
Motivation and significance
Addresses the fundamental question of how "large" a subfactor is within its parent factor
Provides a numerical invariant for classifying subfactors
Bridges gap between operator algebras and other areas of mathematics and physics
Sparked renewed interest in von Neumann algebras and their applications
Properties of Jones index
Serves as a measure of the "size" or "complexity" of a subfactor inclusion
Connects abstract algebraic structures to concrete numerical values
Exhibits surprising restrictions and patterns, leading to new mathematical insights
Fundamental properties
Always takes values in the set 4cos2(π/n):n=3,4,5,...∪[4,∞]
Finite index implies both factor and subfactor are II₁ factors
Satisfies multiplicativity: [M:N]=[M:P][P:N] for intermediate subfactor N ⊂ P ⊂ M
Invariant under isomorphisms of subfactor inclusions
Lower bound: Jones index is always greater than or equal to 1
Relation to subfactors
Measures the "relative dimension" of a subfactor within its parent factor
Smaller index indicates a "tighter" inclusion of subfactors
Provides information about the structure of intermediate subfactors
Relates to the minimal index in subfactor theory
Influences the possible values of statistical dimensions in conformal field theory
Calculation methods
Involve techniques from various areas of mathematics, including linear algebra, representation theory, and functional analysis
Require understanding of trace properties in von Neumann algebras
Often utilize diagrammatic methods developed by Jones and others
Basic techniques
Use of and Jones projections
Computation via Pimsner-Popa basis
Trace method: [M:N]=trM(eN)−1, where eN is the Jones projection
Calculation through statistical dimensions in certain cases
Application of Temperley-Lieb algebra for some subfactors
Advanced approaches
Utilization of planar algebra techniques
Employment of Ocneanu's paragroup theory
Analysis of principal graphs and their growth rates
Use of subfactor homology and cohomology theories
Application of free probability methods in certain cases
Jones index theorem
Establishes a surprising restriction on possible values of the Jones index
Connects abstract algebraic structures to concrete numerical values
Has far-reaching implications in operator algebras and related fields
Statement of theorem
For a II₁ factor M and a subfactor N, the Jones index [M : N] takes values in the set:
4cos2(\π/n):n=3,4,5,...∪[4,∞]
Known as the "Jones set" or "Jones' discrete series"
Excludes all values between 1 and 4 except for specific algebraic numbers
Proof involves intricate analysis of Temperley-Lieb algebras and representation theory
Implications and consequences
Reveals unexpected structure in subfactor theory
Leads to classification of subfactors with small index
Connects to representation theory of quantum groups
Influences development of quantum invariants in knot theory
Provides insights into conformal field theory and statistical mechanics models
Applications in mathematics
Demonstrates the far-reaching impact of Jones index theory beyond operator algebras
Illustrates unexpected connections between different areas of mathematics
Provides new tools and perspectives for solving long-standing problems
Knot theory
Jones polynomial derived from Jones index theory
Subfactors associated with knots and links provide topological invariants
Planar algebra techniques used in both subfactor theory and knot diagrams
Connection to quantum groups and their representations in knot invariants
Application in the study of 3-manifolds and topological quantum field theories
Operator algebras
Classification of subfactors with small index (≤ 4)
Study of automorphism groups of factors
Investigation of amenability and property T for II₁ factors
Development of fusion categories and tensor categories
Analysis of infinite-dimensional Lie algebras and their representations
Jones index in physics
Demonstrates the relevance of abstract mathematical concepts in physical theories
Provides a bridge between operator algebraic methods and physical phenomena
Offers new perspectives on quantum systems and statistical models
Quantum field theory
Describes statistical dimensions of superselection sectors in algebraic quantum field theory
Relates to fusion rules and operator product expansions in conformal field theory
Connects to anyonic statistics and topological order in condensed matter physics
Influences the study of boundary conditions in quantum field theories
Applies to the analysis of defects and interfaces in topological phases
Statistical mechanics
Describes critical behavior in lattice models (Potts model, Ising model)
Relates to transfer matrices and partition functions in exactly solvable models
Connects to integrable systems and Yang-Baxter equations
Applies to the study of phase transitions and critical phenomena
Influences the analysis of entanglement entropy in quantum many-body systems
Generalizations and extensions
Expands the applicability of Jones index theory to broader contexts
Addresses limitations of the original theory in certain situations
Provides new tools for analyzing more complex algebraic structures
Higher dimensional cases
Generalization to higher-dimensional subfactors and planar algebras
Development of higher-dimensional quantum invariants
Study of subfactors in type III von Neumann algebras
Investigation of index theory for inclusions of C*-algebras
Application to higher-dimensional conformal field theories and topological phases
Non-factor extensions
Extension of index theory to inclusions of von Neumann algebras that are not factors
Development of relative entropy techniques for general von Neumann algebra inclusions
Study of index for inclusions of W*-categories and 2-categories
Investigation of index theory in the context of Hopf algebra actions
Application to quantum groupoids and weak Hopf algebras
Relation to other concepts
Illustrates connections between Jones index and other important mathematical and physical quantities
Provides different perspectives on the meaning and significance of the Jones index
Offers new avenues for applying index theory in various contexts
Jones index vs coupling constant
Jones index as a generalization of coupling constants in physics
Relationship to Kac-Moody algebras and their levels
Connection to central charge in conformal field theory
Influence on the study of quantum groups at roots of unity
Application in the analysis of integrable systems and exactly solvable models
Connection to entropy
Relation between Jones index and various notions of entropy in operator algebras
Connection to entanglement entropy in quantum systems
Application in the study of quantum information theory
Influence on the development of free entropy dimension
Relationship to amenability and hyperfiniteness in von Neumann algebras
Open problems and research
Highlights ongoing areas of investigation in Jones index theory and related fields
Identifies key challenges and potential directions for future research
Demonstrates the continued relevance and vitality of the subject
Current challenges
Complete classification of subfactors beyond index 5
Understanding the structure of subfactors with infinite index
Developing effective computational methods for Jones index in complex cases
Clarifying the relationship between Jones index and quantum dimensions in general
Extending index theory to more general algebraic structures (e.g., tensor categories)
Future directions
Application of Jones index theory to quantum computing and quantum error correction
Investigation of Jones index in the context of non-commutative geometry
Development of index theory for inclusions of vertex operator algebras
Exploration of connections between Jones index and geometric group theory
Study of Jones index in relation to higher-dimensional topology and geometry