7.4 Planar algebras
7 min read•august 21, 2024
provide a visual framework for studying von Neumann algebras and . They bridge algebraic structures with topological representations, allowing complex algebraic operations to be represented through geometric diagrams.
Developed by in the late 1990s, planar algebras emerged from his work on subfactors and index theory. They formalize the concept of , connecting to and while encoding subfactor structure through .
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
- 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 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
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 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, , 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 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 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 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
Key Terms to Review (27)
Composition of tangles: The composition of tangles refers to the operation of combining two or more tangles into a new tangle, following specific rules about how the strands and crossings interact. This operation is central to understanding planar algebras as it allows for the construction of more complex structures from simpler ones, revealing deeper algebraic properties.
Diagrammatic calculus: Diagrammatic calculus is a graphical representation used to simplify and manipulate algebraic expressions, particularly in the study of planar algebras. This method allows complex relationships and operations to be visualized through diagrams, making it easier to understand the underlying mathematical structures. It facilitates calculations by providing a visual framework that can express intricate relationships in a more accessible way.
Diagrammatic techniques: Diagrammatic techniques are visual representations used to understand and manipulate algebraic structures, particularly in the context of planar algebras. They help simplify complex relationships and operations by translating them into graphical forms, making it easier to analyze and communicate intricate concepts in mathematics. These techniques often involve drawing diagrams that encode algebraic information, which allows for a more intuitive grasp of the underlying mathematical principles.
Duality: Duality refers to a fundamental principle that describes a relationship between two distinct yet interconnected concepts or structures. In planar algebras, duality often manifests in the way planar diagrams can represent both algebraic structures and their dual counterparts, revealing deep insights into their properties and interactions. This connection not only enhances the understanding of the original structures but also provides a framework for exploring their relationships and transformations.
Fusion rules in planar algebras: Fusion rules in planar algebras refer to the algebraic relationships that determine how different simple objects (or 'labels') combine or 'fuse' together to form new objects within the framework of planar algebras. These rules are crucial for understanding the structure of a planar algebra, as they dictate how different components can interact and form larger, more complex entities, establishing connections to other algebraic structures and categories.
Inner product: An inner product is a mathematical operation that takes two vectors from a vector space and returns a scalar, providing a way to define geometric concepts like length and angle in the space. This operation is crucial in understanding the structure of Hilbert spaces, where it enables the concept of orthogonality and helps in defining the notions of convergence and completeness. Inner products also play a significant role in the GNS construction, where they are used to represent states as vectors in a Hilbert space, and in planar algebras, where they help define the relationships between different elements and their interactions.
Isotopy invariance: Isotopy invariance refers to a property of certain mathematical structures, where two structures are considered equivalent if one can be continuously transformed into the other without cutting or gluing. This concept is crucial in various areas of mathematics, particularly in the study of planar algebras, as it allows for the classification and comparison of different configurations based on their underlying structure rather than their specific representation.
Jones-Wenzl Projections: Jones-Wenzl projections are special types of projections used in the study of planar algebras and categorification of knot invariants. They play a key role in defining the structure of certain categories associated with these algebras, particularly in the context of quantum groups and representations of the symmetric group.
Knot theory connections: Knot theory connections refer to the mathematical study of knots, which are closed loops in three-dimensional space, and their properties and classifications. This area of study involves understanding how different knots can be transformed into one another through various manipulations and has significant implications in topology and related fields. Knot theory connections can also be applied in other disciplines, such as biology and chemistry, where the study of molecular structures benefits from understanding knot-like configurations.
Modularity: Modularity refers to a property of a von Neumann algebra that captures the notion of 'independence' between certain subalgebras. It indicates that the relative position of the subalgebras allows for a well-defined structure of conditional expectations and states, leading to a richer understanding of the algebra's decomposition. This concept plays an important role in understanding how different parts of an algebra interact, especially in the context of certain theoretical frameworks.
Operator Algebras: Operator algebras are mathematical structures that study collections of bounded linear operators on a Hilbert space, focusing on their algebraic and topological properties. They serve as a bridge between functional analysis and quantum mechanics, encapsulating concepts like states, weights, and noncommutative geometry, which have applications in various fields including statistical mechanics and quantum field theory.
Planar algebra manipulations: Planar algebra manipulations refer to the operations and techniques used to work with planar algebras, which are mathematical structures that allow for the representation of algebraic relations in a two-dimensional format. These manipulations involve the addition, multiplication, and transformation of diagrams or strings in a planar manner, leading to a better understanding of the underlying algebraic properties. By manipulating these diagrams, one can visualize complex relationships and deduce new results within the framework of planar algebras.
Planar algebras: Planar algebras are algebraic structures that allow for the manipulation of planar diagrams representing algebraic operations, providing a way to encode and study properties of certain types of operator algebras and knot theory. They emphasize the role of planar diagrams in defining algebraic relations, and they connect deeply with topics like conformal nets, which describe quantum field theories and their associated symmetries.
Planar tangles: Planar tangles are graphical structures that consist of a collection of strands connected at junctions, arranged in a way that can be drawn on a plane without any crossings between strands. These tangles play a critical role in the study of planar algebras, as they provide a visual and combinatorial approach to understanding the algebraic properties and operations within this framework. Their configurations can represent various algebraic identities and relationships, making them essential for exploring the interactions of different algebraic elements.
Principal graphs: Principal graphs are graphical representations that capture the structure of subfactors in von Neumann algebras, acting as essential tools for understanding their relationships and properties. These graphs help visualize the connections between different objects in a subfactor and allow for a clearer analysis of their hierarchy and modularity. They play a significant role in various areas, including the study of planar algebras and the classification of subfactors, providing insight into the behavior of these mathematical structures.
Quantum groups: Quantum groups are mathematical structures that generalize the concept of groups to a noncommutative framework, playing a significant role in the study of symmetries in quantum mechanics. They provide a rich algebraic structure that can be utilized in various fields, including noncommutative geometry, representation theory, and statistical mechanics. Quantum groups serve as a bridge connecting classical algebraic concepts to the complexities of quantum theory and other advanced mathematical constructs.
Quantum groups relationship: The quantum groups relationship refers to the study of structures that arise in the context of noncommutative geometry and quantum algebra, where classical group concepts are generalized to accommodate quantum phenomena. This concept connects with planar algebras through their shared foundations in operator algebras and their applications in mathematical physics, particularly in the study of symmetries and invariants under quantum transformations.
Skein theory: Skein theory is a mathematical framework used to study knots and links through the use of algebraic structures known as skeins. It connects knot theory to other areas of mathematics, such as topology and representation theory, by allowing the transformation of knot diagrams into algebraic expressions. This theory provides tools for calculating knot invariants and understanding the relationships between different knots.
Standard invariants: Standard invariants are mathematical structures associated with a subfactor, which provide essential information about its representation theory and modular properties. They serve as a bridge between the algebraic aspects of von Neumann algebras and their geometric interpretations, particularly in planar algebras and Jones-Wassermann subfactors. Understanding standard invariants is crucial for analyzing the behavior of subfactors and their connections to various areas in operator algebras.
Subfactor Planar Algebras: Subfactor planar algebras are a specific type of planar algebra that arises from the study of subfactors, which are inclusions of von Neumann algebras. These algebras provide a framework for understanding the relationships between different subfactors and enable the visualization of various algebraic structures through planar diagrams. This connection highlights the interplay between geometry and algebra in the analysis of operator algebras.
Subfactors: Subfactors are inclusions of a von Neumann algebra into a larger von Neumann algebra, forming a new algebra with certain properties. This concept allows for the analysis of the structure of algebras and their relationships, leading to insights into topics such as the classification of factors and the understanding of modular theory. Subfactors also play a critical role in determining the relative positions and indices of algebras, highlighting their significance in the study of operator algebras.
Temperley-Lieb Algebra: The Temperley-Lieb algebra is a specific type of algebraic structure that arises in the study of planar diagrams and is connected to knot theory and statistical mechanics. It consists of linear combinations of diagrams that represent certain relations, making it an important tool for understanding the connections between algebra, geometry, and topology, particularly in the context of planar algebras.
Tensor Categories: Tensor categories are a type of mathematical structure that combine the concepts of category theory and tensor products, providing a framework to study objects and morphisms in a way that respects both their algebraic and categorical properties. They allow for the definition of a tensor product between objects, a notion of duality, and coherence conditions that facilitate the interaction between these structures, making them useful in areas like quantum physics and representation theory.
Topological quantum field theory: Topological quantum field theory (TQFT) is a branch of theoretical physics and mathematics that studies quantum field theories which are invariant under continuous deformations of spacetime. In TQFT, physical phenomena are expressed in terms of topological properties rather than geometric ones, focusing on the structure of spacetime rather than the specific metric. This leads to a rich interplay between topology, algebra, and quantum mechanics, often finding applications in both physics and mathematics.
Trace: In the context of operator algebras, a trace is a linear functional defined on a von Neumann algebra that satisfies specific properties, including positivity and normalization. Traces play a crucial role in understanding the structure of factors and provide insights into the representation theory of algebras, making them essential for various applications in mathematics and physics.
Vaughan Jones: Vaughan Jones is a prominent mathematician known for his groundbreaking work in the field of von Neumann algebras, particularly his introduction of the Jones index and planar algebras. His contributions have significantly influenced the study of subfactors and their interconnections with other areas in mathematics, including knot theory and operator algebras.
Vector Spaces: Vector spaces are mathematical structures formed by a collection of vectors, which can be added together and multiplied by scalars. They serve as fundamental frameworks in linear algebra and have applications across various areas, including geometry and functional analysis. The properties of vector spaces, such as closure, associativity, and the existence of a zero vector, are essential for understanding more complex mathematical concepts.