3.5 Jordan triple systems

Jordan triple systems extend Jordan algebras, providing a framework for ternary algebraic structures with specific symmetry properties. These systems play a crucial role in understanding symmetric spaces and quantum mechanics, offering insights into complex mathematical and physical phenomena.

The study of Jordan triple systems involves exploring their axioms, structure, and relationships to other algebraic entities. From Peirce decompositions to automorphisms and derivations, these systems reveal rich mathematical connections and find applications in diverse fields like optimization and geometry.

Definition of Jordan triple systems

• Jordan triple systems emerge as a generalization of Jordan algebras in non-associative algebra
• These systems provide a framework for studying ternary algebraic structures with specific symmetry properties
• Jordan triple systems play a crucial role in understanding the geometry of symmetric spaces and quantum mechanics

Axioms and properties

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• Jordan triple systems consist of a vector space V over a field F with a trilinear map $\{x,y,z\}$ satisfying:
  • Symmetry in the outer variables: $\{x,y,z\} = \{z,y,x\}$
  • Jordan identity: $\{a,b,\{x,y,z\}\} = \{\{a,b,x\},y,z\} - \{x,\{b,a,y\},z\} + \{x,y,\{a,b,z\}\}$
• The Jordan identity ensures a form of "partial associativity" for nested triple products
• Linearity holds in each argument of the triple product
• The triple product satisfies a fundamental identity analogous to the Jordan identity in Jordan algebras

Relation to Jordan algebras

• Jordan triple systems generalize Jordan algebras by extending the binary product to a ternary operation
• Every Jordan algebra (A, •) induces a Jordan triple system with the triple product $\{x,y,z\} = (x•y)•z + (z•y)•x - (x•z)•y$
• Not all Jordan triple systems arise from Jordan algebras, showcasing their broader algebraic scope
• The connection allows for the application of Jordan algebra techniques to study certain Jordan triple systems

Structure of Jordan triple systems

• Jordan triple systems exhibit rich algebraic structures that parallel and extend those found in Jordan algebras
• Understanding the structure of these systems is crucial for their classification and application in various mathematical fields
• The study of Jordan triple systems involves decompositions, ideals, and simplicity concepts unique to ternary algebraic structures

Peirce decomposition

• Peirce decomposition provides a way to decompose a Jordan triple system into subspaces based on idempotents
• An element e in a Jordan triple system is an idempotent if $\{e,e,e\} = e$
• The decomposition yields subspaces $V_2(e), V_1(e),$ and $V_0(e)$ satisfying:
  • $V = V_2(e) \oplus V_1(e) \oplus V_0(e)$
  • $\{V_i(e),V_j(e),V_k(e)\} \subseteq V_{i-j+k}(e)$ (with subscripts modulo 3)
• This decomposition generalizes the Peirce decomposition of Jordan algebras and provides insight into the system's structure

Ideal structure

• Ideals in Jordan triple systems are subspaces closed under the triple product with elements from the whole system
• A subspace I is an ideal if $\{I,V,V\} \subseteq I$ and $\{V,I,V\} \subseteq I$
• The study of ideals leads to concepts like simplicity and semi-simplicity in Jordan triple systems
• Minimal ideals and maximal ideals play crucial roles in understanding the overall structure of the system

Simple Jordan triple systems

• A Jordan triple system is simple if it has no non-trivial ideals and its triple product is non-zero
• Simple Jordan triple systems form the building blocks for more complex systems
• Classification of simple finite-dimensional Jordan triple systems over algebraically closed fields of characteristic 0 includes:
  • Hermitian type (rectangular matrices)
  • Symplectic type (skew-symmetric matrices)
  • Exceptional types (related to exceptional Lie algebras)

Types of Jordan triple systems

• Jordan triple systems encompass a diverse range of algebraic structures with varying properties and applications
• The classification of Jordan triple systems helps in understanding their behavior and relationships to other algebraic entities
• Different types of Jordan triple systems arise from various mathematical constructions and physical models

Hermitian Jordan triple systems

• Hermitian Jordan triple systems arise from Hermitian matrices and operators in complex Hilbert spaces
• The triple product for n×n complex Hermitian matrices A, B, C defined as $\{A,B,C\} = ABC + CBA$
• These systems play a crucial role in quantum mechanics and operator theory
• Properties of Hermitian Jordan triple systems include:
  • Positive definiteness of the quadratic form $Q(x) = \{x,x,x\}$
  • Connection to bounded symmetric domains in complex analysis

Spin factors

• Spin factors form a class of Jordan triple systems closely related to Clifford algebras and spinor spaces
• Constructed from a vector space V with a non-degenerate symmetric bilinear form (,) and an element e not in V
• The triple product defined as $\{x+λe, y+μe, z+νe\} = ((x,z)y + (y,z)x - (x,y)z) + [(λ(y,z)+μ(x,z)+ν(x,y))e]$
• Spin factors find applications in:
  • Representation theory of Lie groups
  • Geometry of symmetric spaces
  • Quantum field theory

Rectangular matrices

• Jordan triple systems of rectangular matrices generalize Hermitian Jordan triple systems
• For m×n complex matrices A, B, C, the triple product defined as $\{A,B,C\} = AB*C + CB*A$
• These systems provide a rich source of examples and play a role in:
  • Representation theory of classical Lie groups
  • Study of bounded symmetric domains
  • Optimization algorithms in linear algebra

Representation theory

• Representation theory for Jordan triple systems extends concepts from Lie algebra and Jordan algebra representations
• This theory provides tools for understanding the structure and properties of Jordan triple systems through their actions on vector spaces
• Representations of Jordan triple systems find applications in quantum mechanics and harmonic analysis

Modules over Jordan triple systems

• A module M over a Jordan triple system J consists of a vector space with a trilinear map $J × M × J → M$
• The module action satisfies axioms analogous to those of the Jordan triple product
• Types of modules include:
  • Regular modules (the Jordan triple system acting on itself)
  • Irreducible modules (having no proper submodules)
  • Composition series and Jordan-Hölder theorem for modules
• Modules provide a framework for studying linear representations of Jordan triple systems

Character theory

• Character theory for Jordan triple systems extends ideas from representation theory of Lie algebras
• The character of a finite-dimensional representation ρ defined as $χ_ρ(x,y) = Tr(ρ(\{x,y,-\}))$
• Properties of characters include:
  • Linearity and conjugation invariance
  • Orthogonality relations for irreducible representations
  • Character formulas for tensor products of representations
• Character theory aids in the classification of representations and the study of their decompositions

Automorphisms and derivations

• Automorphisms and derivations provide crucial tools for understanding the symmetries and infinitesimal transformations of Jordan triple systems
• These concepts parallel similar notions in Lie algebras and Jordan algebras, but with unique features due to the ternary nature of Jordan triple systems
• The study of automorphisms and derivations reveals important structural properties of Jordan triple systems

Automorphism groups

• An automorphism of a Jordan triple system J is a linear map $φ: J → J$ preserving the triple product
• The set of all automorphisms forms a group under composition, denoted Aut(J)
• Properties of automorphism groups include:
  • Lie group structure for finite-dimensional Jordan triple systems
  • Connection to isometry groups of associated symmetric spaces
  • Subgroups corresponding to various algebraic properties (inner automorphisms)
• Automorphism groups play a role in the classification of Jordan triple systems and their representations

Derivation algebras

• A derivation of a Jordan triple system J is a linear map $D: J → J$ satisfying the Leibniz rule: $D(\{x,y,z\}) = \{D(x),y,z\} + \{x,D(y),z\} + \{x,y,D(z)\}$
• The set of all derivations forms a Lie algebra under the commutator bracket, denoted Der(J)
• Types of derivations include:
  • Inner derivations: $D_{x,y}(z) = \{x,y,z\} - \{y,x,z\}$
  • Outer derivations: those not expressible as linear combinations of inner derivations
• The derivation algebra provides information about the infinitesimal symmetries of the Jordan triple system

Jordan triple products

• Jordan triple products form the fundamental operation in Jordan triple systems, generalizing the binary product of Jordan algebras
• These ternary operations exhibit specific symmetry and partial associativity properties
• Understanding Jordan triple products is crucial for applications in quantum mechanics and geometry

Examples of triple products

• Matrix triple product: $\{A,B,C\} = ABC + CBA$ for square matrices
• Spin factor triple product: $\{x,y,z\} = (x,z)y + (y,z)x - (x,y)z + [(x,y)z + (y,z)x + (z,x)y]e$
• Exceptional Jordan triple systems:
  • Octonion-based: $\{x,y,z\} = (x\bar{y})z + (z\bar{y})x - (x\bar{z})y$ for octonions
  • Albert algebra: $\{x,y,z\} = (x•y)•z + (z•y)•x - (x•z)•y$ for the exceptional Jordan algebra
• Geometric triple products arising from symmetric spaces and bounded domains

Triple product identities

• Fundamental identity: $\{a,b,\{x,y,z\}\} = \{\{a,b,x\},y,z\} - \{x,\{b,a,y\},z\} + \{x,y,\{a,b,z\}\}$
• Symmetry in outer variables: $\{x,y,z\} = \{z,y,x\}$
• Linearity in each argument: $\{αx+βx',y,z\} = α\{x,y,z\} + β\{x',y,z\}$ (and similarly for other arguments)
• Polarization identity: $\{x,y,z\} + \{y,x,z\} = Q(x+y,z) - Q(x,z) - Q(y,z)$ where $Q(x,y) = \{x,y,x\}$
• These identities characterize Jordan triple systems and play crucial roles in proofs and structural analysis

Applications of Jordan triple systems

• Jordan triple systems find diverse applications across mathematics and physics
• These algebraic structures provide powerful tools for understanding symmetry and geometry in various contexts
• The applications of Jordan triple systems often reveal deep connections between seemingly disparate areas of study

Quantum mechanics

• Jordan triple systems model certain aspects of quantum mechanical systems
• Applications in quantum mechanics include:
  • Description of observables and their products in quantum theory
  • Formulation of quantum logic using Jordan triple systems
  • Study of quantum entanglement through Jordan triple system structures
• The non-associative nature of Jordan triple systems aligns with the non-commutativity of quantum observables

Geometry of symmetric spaces

• Jordan triple systems provide an algebraic framework for studying symmetric spaces
• Applications in symmetric space geometry include:
  • Classification of Hermitian symmetric spaces using Jordan triple systems
  • Description of the curvature tensor of symmetric spaces in terms of triple products
  • Study of geodesics and isometries using automorphisms of associated Jordan triple systems
• The connection between Jordan triple systems and symmetric spaces reveals deep links between algebra and differential geometry

Optimization theory

• Jordan triple systems play a role in certain optimization problems and algorithms
• Applications in optimization theory include:
  • Formulation of semidefinite programming problems using Hermitian Jordan triple systems
  • Development of interior point methods for optimization over symmetric cones
  • Study of convex optimization problems related to positive semidefinite matrices
• The algebraic properties of Jordan triple systems provide insights into the geometry of optimization landscapes

Connections to other algebraic structures

• Jordan triple systems exhibit intricate relationships with various other algebraic structures
• Understanding these connections provides insights into the broader landscape of non-associative algebra
• The study of these relationships often leads to new theoretical developments and applications

Jordan triple systems vs Jordan algebras

• Jordan triple systems generalize Jordan algebras by extending the binary product to a ternary operation
• Key differences include:
  • Jordan triple systems do not require a unit element, unlike unital Jordan algebras
  • The fundamental identity in Jordan triple systems replaces the Jordan identity of Jordan algebras
  • Jordan triple systems allow for a broader class of examples, including those not derivable from Jordan algebras
• Connections between the two structures:
  • Every Jordan algebra induces a Jordan triple system via $\{x,y,z\} = (x•y)•z + (z•y)•x - (x•z)•y$
  • Some Jordan triple systems can be "polarized" to obtain Jordan pairs, which are closely related to Jordan algebras

Lie triple systems vs Jordan triple systems

• Both Lie triple systems and Jordan triple systems are ternary algebraic structures, but with different properties
• Key differences include:
  • Lie triple systems satisfy anti-symmetry in the outer variables, while Jordan triple systems are symmetric
  • The fundamental identity differs between the two structures, reflecting their distinct algebraic natures
  • Lie triple systems relate closely to Lie algebras, while Jordan triple systems connect to Jordan algebras
• Connections between the two structures:
  • Some Jordan triple systems give rise to Lie triple systems via the commutator $[x,y,z] = \{x,y,z\} - \{y,x,z\}$
  • Both structures play roles in the theory of symmetric spaces, but in different contexts

Analysis on Jordan triple systems

• Analysis on Jordan triple systems extends classical functional analysis to non-associative settings
• This field combines algebraic techniques with analytical methods to study Jordan triple systems
• The development of analysis on these structures has implications for operator theory and harmonic analysis

Spectral theory

• Spectral theory for Jordan triple systems generalizes concepts from operator theory
• Key aspects of spectral theory include:
  • Definition of spectrum for elements in Jordan triple systems
  • Spectral radius and its properties in the context of Jordan triple products
  • Resolvent identities and their applications in Jordan triple systems
• Applications of spectral theory:
  • Study of bounded symmetric domains in complex analysis
  • Analysis of operators in Hilbert spaces related to Jordan triple systems

Functional calculus

• Functional calculus extends functions of real or complex variables to elements of Jordan triple systems
• Development of functional calculus includes:
  • Definition of polynomials and rational functions for Jordan triple systems
  • Extension to more general functions using spectral theory and approximation techniques
  • Holomorphic functional calculus for certain classes of Jordan triple systems
• Applications of functional calculus:
  • Study of operator equations in Jordan triple system settings
  • Analysis of geometric properties of associated symmetric spaces

Classification of Jordan triple systems

• Classification of Jordan triple systems aims to categorize and understand the structure of these algebraic objects
• This classification extends and complements the classification of Jordan algebras
• The study reveals fundamental building blocks and relationships between different types of Jordan triple systems

Finite-dimensional classification

• Classification of simple finite-dimensional Jordan triple systems over algebraically closed fields of characteristic 0:
  • Type I: Rectangular matrix systems $M_{p,q}(F)$
  • Type II: Symmetric matrix systems $Sym(n,F)$
  • Type III: Skew-symmetric matrix systems $Skew(2n,F)$
  • Type IV: Spin factors
  • Exceptional types: Related to exceptional Lie algebras (E6, E7, E8)
• Techniques for classification include:
  • Use of Peirce decomposition
  • Analysis of automorphism groups and derivation algebras
  • Connection to classification of Hermitian symmetric spaces

Infinite-dimensional classification

• Classification of infinite-dimensional Jordan triple systems more complex and less complete
• Approaches to infinite-dimensional classification:
  • Study of JB*-triples (complex Banach Jordan triple systems with additional analytic properties)
  • Classification based on the structure of associated symmetric spaces
  • Analysis of operator algebras related to Jordan triple systems
• Challenges in infinite-dimensional classification:
  • Lack of finite-dimensionality techniques
  • Interplay between algebraic and topological properties
  • Existence of exotic examples with no finite-dimensional counterparts

Exceptional Jordan triple systems

• Exceptional Jordan triple systems arise from exceptional algebraic structures and play unique roles in mathematics and physics
• These systems often exhibit remarkable properties not found in classical Jordan triple systems
• The study of exceptional Jordan triple systems connects to exceptional Lie algebras and geometries

Octonion-based systems

• Jordan triple systems constructed using octonions, the largest of the four normed division algebras
• Properties of octonion-based Jordan triple systems:
  • Non-associativity inherited from the octonion algebra
  • Connection to exceptional Lie groups (G2, F4)
  • Role in the construction of exceptional Jordan algebras
• Examples of octonion-based Jordan triple systems:
  • The exceptional Jordan algebra (Albert algebra) viewed as a Jordan triple system
  • Freudenthal triple systems constructed using split octonions

Albert systems

• Albert systems refer to Jordan triple systems derived from or related to the exceptional Jordan algebra (Albert algebra)
• Characteristics of Albert systems:
  • 27-dimensional over their base field
  • Closely related to the exceptional Lie group E6
  • Unique structure not found in classical matrix-based Jordan triple systems
• Applications and connections of Albert systems:
  • Role in the study of exceptional geometries
  • Connection to certain quantum mechanical systems
  • Importance in the classification of simple Jordan triple systems