7.3 Jordan rings

Jordan rings are non-associative algebraic structures that extend concepts from associative algebra. They provide a framework for studying symmetric elements in associative algebras with involution and are crucial in understanding quantum mechanics and projective geometry.

Introduced in 1934 by Jordan, von Neumann, and Wigner, Jordan rings formalize the algebra of observables in quantum mechanics. They exhibit unique decomposition properties and play a key role in understanding symmetries and transformations in various mathematical systems.

Definition of Jordan rings

• Non-associative algebraic structures extending concepts from associative algebra to non-associative settings
• Provide a framework for studying symmetric elements in associative algebras with involution
• Crucial in understanding quantum mechanics and projective geometry

Axioms of Jordan rings

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• Commutative binary operation denoted by $\circ$ (Jordan product)
• Satisfy the Jordan identity: $(a \circ b) \circ (a \circ a) = a \circ (b \circ (a \circ a))$ for all elements $a$ and $b$
• Distributive property: $a \circ (b + c) = a \circ b + a \circ c$
• Scalar multiplication compatibility: $\alpha(a \circ b) = (\alpha a) \circ b = a \circ (\alpha b)$ for scalar $\alpha$

Historical context

• Introduced by Pascual Jordan, John von Neumann, and Eugene Wigner in 1934
• Developed to formalize the algebra of observables in quantum mechanics
• Initially studied as part of the quest to understand the mathematical foundations of quantum theory
• Gained prominence in algebraic research due to connections with exceptional Lie algebras

Structure of Jordan rings

• Fundamental building blocks in non-associative algebra, bridging associative and non-associative structures
• Exhibit unique decomposition properties not found in associative rings
• Play a crucial role in understanding symmetries and transformations in various mathematical systems

Idempotents in Jordan rings

• Elements $e$ satisfying $e \circ e = e$
• Form the backbone of the structural theory of Jordan rings
• Classify into primitive idempotents (cannot be decomposed further)
• Orthogonal idempotents: $e_i \circ e_j = 0$ for $i \neq j$
• Complete system of orthogonal idempotents sums to the identity element

Peirce decomposition

• Decomposes a Jordan ring into subspaces based on idempotents
• For an idempotent $e$, the Peirce decomposition is $J = J_0 \oplus J_{\frac{1}{2}} \oplus J_1$
• $J_0 = \{x \in J : e \circ x = 0\}$, $J_1 = \{x \in J : e \circ x = x\}$
• $J_{\frac{1}{2}} = \{x \in J : e \circ x = \frac{1}{2}x\}$
• Multiplication rules between Peirce subspaces determine the overall structure

Special Jordan rings

• Subclass of Jordan rings with additional properties or derived from other algebraic structures
• Bridge the gap between associative and non-associative algebra
• Provide concrete examples for studying general Jordan ring theory

Jordan algebras from associative algebras

• Construct Jordan algebras from associative algebras using the Jordan product
• Define $a \circ b = \frac{1}{2}(ab + ba)$ for elements $a$ and $b$ in an associative algebra
• Resulting structure satisfies all Jordan ring axioms
• Examples include:
  • Symmetric matrices under the Jordan product
  • Self-adjoint operators on a Hilbert space

Quadratic Jordan algebras

• Generalization of Jordan algebras using quadratic operators
• Define $U_a(b) = 2a \circ (a \circ b) - (a \circ a) \circ b$ as the U-operator
• Satisfy the fundamental formula: $U_{U_a(b)} = U_a U_b U_a$
• Include all special Jordan algebras and some exceptional Jordan algebras
• Examples:
  • Spin factors in quantum mechanics
  • Jordan algebras of symmetric bilinear forms

Jordan triple systems

• Ternary algebraic structures generalizing Jordan rings
• Provide a unified framework for studying various non-associative structures
• Important in the classification of symmetric spaces and bounded symmetric domains

Connection to Jordan rings

• Every Jordan ring gives rise to a Jordan triple system
• Define the triple product $\{x,y,z\} = (x \circ y) \circ z + (z \circ y) \circ x - (x \circ z) \circ y$
• Jordan triple systems satisfy the Jordan triple identity: $\{a,b,\{x,y,z\}\} = \{\{a,b,x\},y,z\} - \{x,\{b,a,y\},z\} + \{x,y,\{a,b,z\}\}$
• Not every Jordan triple system comes from a Jordan ring

Examples of Jordan triple systems

• Rectangular matrices with triple product $\{X,Y,Z\} = XY^*Z + ZY^*X$
• Spin factors with triple product $\{x,y,z\} = (x \cdot y)z + (z \cdot y)x - (x \cdot z)y$
• Hermitian operators on a Hilbert space with $\{A,B,C\} = ABC + CBA$

Representation theory

• Studies how Jordan rings act on vector spaces (modules)
• Provides insights into the structure and properties of Jordan rings
• Connects Jordan ring theory to other areas of mathematics and physics

Modules over Jordan rings

• Vector spaces $M$ equipped with a bilinear map $J \times M \to M$
• Satisfy the Jordan module identity: $(a \circ b) \cdot m = a \cdot (b \cdot m) + b \cdot (a \cdot m) - (a \circ b) \cdot m$
• Types of modules:
  • Unital modules (preserve the identity element)
  • Faithful modules (only the zero element acts as the zero operator)
• Homomorphisms between modules preserve the Jordan ring action

Irreducible representations

• Modules with no proper non-zero submodules
• Fundamental building blocks of representation theory
• Schur's lemma applies to irreducible representations of Jordan rings
• Classification of irreducible representations:
  • For finite-dimensional simple Jordan algebras over algebraically closed fields
  • Involves the structure theory and Peirce decomposition

Exceptional Jordan algebras

• Non-special Jordan algebras not derived from associative algebras
• Closely related to exceptional Lie algebras
• Play a crucial role in the classification of simple Jordan algebras

Albert algebra

• 27-dimensional exceptional Jordan algebra over real or complex numbers
• Constructed using 3x3 Hermitian matrices over octonions
• Elements of the form: \alpha & a & \bar{b} \\ \bar{a} & \beta & c \\ b & \bar{c} & \gamma \end{pmatrix}$$ where $\alpha, \beta, \gamma$ are real and $a, b, c$ are octonions
• Jordan product defined as $A \circ B = \frac{1}{2}(AB + BA)$

27-dimensional exceptional Jordan algebra

• Unique up to isomorphism over algebraically closed fields
• Connected to the exceptional Lie group $E_6$
• Automorphism group is the compact real form of $E_6$
• Applications:
  • Quantum mechanics of supersymmetric particles
  • M-theory and string theory

Applications of Jordan rings

• Extend beyond pure mathematics to various scientific disciplines
• Provide mathematical frameworks for describing physical phenomena
• Connect abstract algebra to concrete real-world problems

Quantum mechanics

• Jordan algebras model observables in quantum systems
• Self-adjoint operators on Hilbert spaces form Jordan algebras
• Jordan-von Neumann-Wigner theorem classifies finite-dimensional formally real Jordan algebras
• Applications:
  • Describe spin systems
  • Model quantum information and quantum computation

Projective geometry

• Jordan algebras of symmetric matrices relate to projective spaces
• Projective planes over composition algebras correspond to certain Jordan algebras
• Applications:
  • Describe symmetries of projective spaces
  • Study exceptional geometries (Cayley plane)

Identities in Jordan rings

• Fundamental equations governing the behavior of elements in Jordan rings
• Distinguish Jordan rings from other algebraic structures
• Provide tools for proving theorems and analyzing Jordan ring properties

Fundamental formulas

• Power associativity: $a^m \circ a^n = a^{m+n}$ for all positive integers $m$ and $n$
• Linearized Jordan identity: $(a \circ b) \circ (c \circ d) + (a \circ d) \circ (b \circ c) = (a \circ (b \circ c)) \circ d + (a \circ (b \circ d)) \circ c$
• MacDonald identity: $(a^2 \circ b) \circ a = a^2 \circ (b \circ a)$
• Shirshov-Cohn theorem: Any Jordan algebra generated by two elements is special

Jordan identity vs associativity

• Jordan identity weaker than associativity but stronger than commutativity
• Associative algebras satisfy Jordan identity but not vice versa
• Differences:
  • Jordan rings allow for non-associative multiplication
  • Associative rings have a richer ideal structure
• Consequences:
  • Jordan rings have a more limited representation theory
  • Some classical ring theory results do not hold for Jordan rings

Classification of Jordan rings

• Organizes Jordan rings into categories based on their properties
• Provides a systematic way to study and understand different types of Jordan rings
• Reveals connections between seemingly disparate algebraic structures

Simple Jordan rings

• Jordan rings with no non-trivial ideals
• Classification theorem (Jordan-von Neumann-Wigner):
  • Four infinite families of simple Jordan algebras
  • One exceptional case (Albert algebra)
• Finite-dimensional simple Jordan algebras over algebraically closed fields:
  • $H_n(F)$: n x n Hermitian matrices over $F$
  • $J(V,f)$: Jordan algebra of a non-degenerate symmetric bilinear form
  • $J(S,*)$: Symmetrized algebra of an associative algebra with involution
  • 27-dimensional exceptional Jordan algebra

Artinian Jordan rings

• Jordan rings satisfying the descending chain condition on ideals
• Structure theorem:
  • Decompose into a direct sum of simple Artinian Jordan rings
  • Each simple Artinian Jordan ring is isomorphic to a matrix ring over a division Jordan ring
• Properties:
  • Finite-dimensional Jordan algebras are Artinian
  • Artinian Jordan rings have a finite number of minimal ideals

Derivations and automorphisms

• Study symmetries and transformations of Jordan rings
• Provide insights into the structure and properties of Jordan rings
• Connect Jordan ring theory to Lie algebra theory

Derivation algebra

• Linear maps $D: J \to J$ satisfying $D(a \circ b) = D(a) \circ b + a \circ D(b)$
• Form a Lie algebra under the commutator bracket
• Types of derivations:
  • Inner derivations: $D_{a,b}(x) = [L_a, L_b](x) = (a \circ b) \circ x - a \circ (b \circ x)$
  • Outer derivations: Not expressible as linear combinations of inner derivations
• Derivation algebra structure theorem for simple finite-dimensional Jordan algebras

Inner automorphisms

• Automorphisms of Jordan rings preserving the Jordan product
• Generated by exponentials of inner derivations
• Properties:
  • Form a normal subgroup of the full automorphism group
  • Connect to the structure group of the Jordan ring
• Examples:
  • For matrix Jordan algebras, conjugation by invertible matrices
  • For Albert algebra, certain transformations involving octonion multiplication

Jordan ring extensions

• Construct larger Jordan rings from smaller ones
• Provide tools for studying the structure of Jordan rings
• Analogous to field extensions in commutative algebra

Split null extension

• Construct a Jordan ring from a Jordan triple system
• Given a Jordan triple system $(V, \{\cdot,\cdot,\cdot\})$, define $J = F \oplus V$
• Jordan product: $(α,x) \circ (β,y) = (αβ, αy + βx)$
• Properties:
  • Contains the original Jordan triple system as a subspace
  • Useful for studying representations of Jordan triple systems

Isotopes of Jordan rings

• Deformations of Jordan rings preserving their structure
• For an invertible element $a$, define new product $x \circ_a y = (x \circ a) \circ (a^{-1} \circ y)$
• Properties:
  • Isotopes are isomorphic to the original Jordan ring
  • Useful for studying automorphisms and derivations
• Applications:
  • Classify simple Jordan rings
  • Study structure preserving maps between Jordan rings

Relation to other algebraic structures

• Places Jordan rings in the broader context of algebra
• Reveals connections and differences between various algebraic systems
• Provides insights into the nature of non-associative structures

Jordan rings vs Lie algebras

• Both are non-associative algebraic structures
• Differences:
  • Jordan rings are commutative, Lie algebras are anti-commutative
  • Jordan identity vs Jacobi identity
• Connections:
  • Derivation algebra of a Jordan ring is a Lie algebra
  • Certain constructions (TKK construction) relate Jordan and Lie algebras
• Applications:
  • Study exceptional Lie algebras using Jordan algebras
  • Analyze symmetries in physical systems

Jordan rings vs associative rings

• Jordan rings generalize certain aspects of associative rings
• Differences:
  • Jordan rings lack full associativity
  • Ideal structure is more complex in associative rings
• Connections:
  • Every associative ring gives rise to a Jordan ring
  • Special Jordan rings embed into associative rings
• Consequences:
  • Some classical ring theory results do not hold for Jordan rings
  • Jordan rings provide insights into symmetric elements of associative rings with involution