7.4 Planar algebras

Planar algebras provide a visual framework for studying von Neumann algebras and subfactors. They bridge algebraic structures with topological representations, allowing complex algebraic operations to be represented through geometric diagrams.

Developed by Vaughan Jones in the late 1990s, planar algebras emerged from his work on subfactors and index theory. They formalize the concept of planar tangles, connecting to tensor categories and quantum groups while encoding subfactor structure through diagrammatic techniques.

Definition of planar algebras

• Planar algebras provide a diagrammatic framework for studying von Neumann algebras and subfactors
• Developed by Vaughan Jones in the late 1990s as a tool to analyze subfactors and their standard invariants
• Bridges algebraic structures with topological and combinatorial representations

Historical context

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• Emerged from Jones' work on subfactors and index theory in the 1980s
• Built upon earlier diagrammatic approaches in knot theory and statistical mechanics
• Formalized the concept of planar tangles as a rigorous mathematical structure
• Influenced by the development of tensor categories and quantum groups

Relation to von Neumann algebras

• Encodes the structure of subfactors and their standard invariants
• Provides a visual representation of algebraic operations in von Neumann algebras
• Allows for the study of infinite-dimensional algebras through finite-dimensional approximations
• Connects to the theory of operator algebras through its relationship with subfactors

Fundamental concepts

• Planar algebras utilize geometric and topological structures to represent algebraic operations
• Incorporate elements of category theory, knot theory, and operator algebras
• Provide a unified framework for studying various algebraic and topological phenomena

Planar tangles

• Consist of discs with input and output points on their boundaries
• Represent operations or morphisms in the planar algebra
• Can be composed by connecting output points of one tangle to input points of another
• Subject to isotopy relations allowing continuous deformations without crossing strings

Shaded planar diagrams

• Utilize alternating shaded and unshaded regions to represent different types of spaces
• Shading encodes additional structure and constraints on the algebra
• Preserve shading when composing tangles
• Play a crucial role in defining subfactor planar algebras

Planar operad

• Generalizes the notion of planar tangles to a categorical setting
• Defines a collection of operations with multiple inputs and one output
• Satisfies associativity and unit axioms for composition
• Provides a formal framework for studying planar algebra structures

Structure of planar algebras

• Planar algebras consist of a family of vector spaces with associated operations
• Incorporate both algebraic and topological structures
• Allow for the study of infinite-dimensional algebras through finite-dimensional approximations

Vector spaces and operations

• Each planar algebra contains a sequence of vector spaces indexed by natural numbers
• Vector spaces correspond to diagrams with a specific number of boundary points
• Operations defined by planar tangles act on these vector spaces
• Include multiplication, trace, and various other algebraic operations

Composition of tangles

• Tangles can be composed by connecting output points of one tangle to input points of another
• Composition is associative and preserves the planar structure
• Allows for the construction of complex diagrams from simpler building blocks
• Corresponds to composition of operators in the associated von Neumann algebra

Isotopy invariance

• Planar algebra operations are invariant under continuous deformations of tangles
• Allows for simplification and manipulation of diagrams without changing their algebraic meaning
• Connects planar algebras to topological invariants and knot theory
• Provides a powerful tool for proving identities and simplifying calculations

Types of planar algebras

• Various types of planar algebras exist, each with specific properties and applications
• Different types capture different aspects of algebraic and topological structures
• Allow for the study of diverse mathematical phenomena within a unified framework

Subfactor planar algebras

• Encode the standard invariant of a subfactor
• Incorporate additional structure related to the Jones index
• Include a distinguished "generating" tangle corresponding to the Jones projection
• Satisfy positivity and duality conditions reflecting properties of von Neumann algebras

Temperley-Lieb algebra

• Fundamental example of a planar algebra
• Consists of non-crossing diagrams with a fixed number of boundary points
• Plays a crucial role in statistical mechanics and knot theory
• Serves as a building block for more complex planar algebras

Jones-Wenzl projections

• Special elements in the Temperley-Lieb algebra with important properties
• Satisfy recursion relations and orthogonality conditions
• Play a key role in the construction of quantum invariants of knots and 3-manifolds
• Connect planar algebras to representation theory of quantum groups

Properties and characteristics

• Planar algebras possess various algebraic and topological properties
• These properties allow for powerful techniques in analysis and computation
• Connect planar algebras to other areas of mathematics and physics

Modularity

• Some planar algebras exhibit modular properties related to conformal field theory
• Involves the existence of certain special elements and relations
• Connects planar algebras to modular tensor categories
• Allows for the construction of 3-manifold invariants

Duality

• Planar algebras often possess a natural duality structure
• Relates to the existence of adjoint operations in von Neumann algebras
• Allows for the definition of inner products and traces
• Plays a crucial role in the theory of subfactors and their standard invariants

Trace and inner product

• Planar algebras often come equipped with a trace functional
• Trace defined using a specific "cup" tangle
• Allows for the definition of an inner product on the vector spaces
• Connects to the trace on von Neumann algebras and statistical mechanical partition functions

Applications in mathematics

• Planar algebras find applications in various areas of mathematics and theoretical physics
• Provide a unifying framework for studying diverse phenomena
• Allow for the transfer of techniques between different fields

Knot theory connections

• Planar algebras provide a natural setting for studying knot and link invariants
• Allow for the construction of quantum invariants (Jones polynomial)
• Connect to the theory of braids and tangles
• Provide diagrammatic techniques for simplifying knot calculations

Quantum groups relationship

• Planar algebras encode representation theory of quantum groups
• Allow for the study of quantum symmetries in a diagrammatic setting
• Connect to the theory of tensor categories
• Provide a bridge between algebraic and topological aspects of quantum groups

Topological quantum field theory

• Planar algebras provide a framework for constructing and studying TQFTs
• Allow for the definition of invariants of 3-manifolds and cobordisms
• Connect to conformal field theory and statistical mechanics
• Provide insights into the structure of quantum field theories

Planar algebra techniques

• Various techniques have been developed for working with planar algebras
• These techniques combine algebraic, topological, and combinatorial methods
• Allow for efficient calculations and proofs in planar algebra theory

Skein theory

• Provides a method for simplifying planar diagrams using local relations
• Allows for the definition of invariants through skein relations
• Connects planar algebras to knot theory and quantum invariants
• Provides a powerful tool for computations in planar algebras

Diagrammatic calculus

• Utilizes graphical representations to perform algebraic calculations
• Allows for intuitive manipulation of complex algebraic expressions
• Provides visual proofs of identities and relations
• Connects to other diagrammatic methods in mathematics and physics

Planar algebra manipulations

• Involve operations such as rotation, reflection, and contraction of tangles
• Allow for the simplification and normalization of planar diagrams
• Provide techniques for proving identities and solving equations in planar algebras
• Connect to algebraic manipulations in von Neumann algebras and subfactors

Advanced topics

• Various advanced topics in planar algebra theory extend and generalize the basic concepts
• These topics connect planar algebras to deeper aspects of operator algebras and category theory
• Provide new tools and insights for studying subfactors and related structures

Planar algebra subfactors

• Construct subfactors directly from planar algebra data
• Allow for the classification of subfactors through planar algebraic methods
• Provide a connection between combinatorial and operator algebraic aspects of subfactors
• Include techniques for computing invariants and analyzing structural properties

Principal graphs

• Encode important information about the structure of subfactors and planar algebras
• Consist of bipartite graphs representing the inclusion structure of certain algebras
• Play a crucial role in the classification of subfactors
• Connect to representation theory and quantum symmetries

Fusion rules in planar algebras

• Describe the decomposition of tensor products of irreducible objects
• Encoded diagrammatically in planar algebra structure
• Connect to fusion categories and quantum groups
• Provide important invariants for classifying and studying planar algebras

Computational aspects

• Various computational tools and techniques have been developed for working with planar algebras
• These tools allow for efficient calculations and exploration of planar algebraic structures
• Provide important resources for research and applications in planar algebra theory

Planar algebra software

• Specialized software packages for manipulating and analyzing planar diagrams
• Include tools for computing invariants and performing diagrammatic calculations
• Utilize computer algebra systems and graphical interfaces
• Facilitate exploration and discovery in planar algebra research

Diagrammatic algorithms

• Algorithmic approaches to manipulating and simplifying planar diagrams
• Include methods for normalizing tangles and computing traces
• Utilize combinatorial and graph-theoretic techniques
• Connect to algorithmic aspects of knot theory and tensor networks

Current research directions

• Planar algebra theory continues to be an active area of research
• New developments connect planar algebras to various areas of mathematics and physics
• Open problems drive further exploration and generalization of planar algebraic concepts

Open problems

• Classification of subfactor planar algebras beyond small index
• Connections between planar algebras and conformal field theory
• Generalization of planar algebra techniques to higher dimensions
• Applications of planar algebras in quantum information theory

Recent developments

• Extension of planar algebra techniques to study fusion categories and tensor networks
• Connections between planar algebras and categorification in representation theory
• Applications of planar algebras in topological phases of matter and anyonic systems
• Development of new invariants and classification results using planar algebraic methods