🧮Non-associative Algebra Unit 7 – Non-associative Rings

Key Concepts and Definitions

• Non-associative rings generalize the concept of associative rings by removing the requirement of associativity for multiplication
• A non-associative ring $(R, +, \cdot)$ consists of a set $R$ with two binary operations, addition $(+)$ and multiplication $(\cdot)$, satisfying certain axioms
  • Addition is associative: $a + (b + c) = (a + b) + c$ for all $a, b, c \in R$
  • Addition is commutative: $a + b = b + a$ for all $a, b \in R$
  • Additive identity: There exists an element $0 \in R$ such that $a + 0 = a$ for all $a \in R$
  • Additive inverses: For each $a \in R$, there exists an element $-a \in R$ such that $a + (-a) = 0$
• Multiplication is distributive over addition: $a \cdot (b + c) = a \cdot b + a \cdot c$ and $(a + b) \cdot c = a \cdot c + b \cdot c$ for all $a, b, c \in R$
• Examples of non-associative rings include Lie algebras, Jordan algebras, and octonions
• The center of a non-associative ring $R$ is the set of elements that associate and commute with all other elements: $Z(R) = \{a \in R : (a \cdot b) \cdot c = a \cdot (b \cdot c), a \cdot b = b \cdot a \text{ for all } b, c \in R\}$

Historical Context and Development

• Non-associative rings emerged in the early 20th century as mathematicians explored generalizations of classical algebraic structures
• The study of non-associative algebras gained momentum with the development of Lie algebras by Sophus Lie in the late 19th century
  • Lie algebras were introduced to study the symmetries of differential equations and have since found applications in various areas of mathematics and physics
• In the 1930s, Pascual Jordan introduced Jordan algebras as a generalization of the observables in quantum mechanics
  • Jordan algebras satisfy the Jordan identity: $(a \cdot b) \cdot a^2 = a \cdot (b \cdot a^2)$ for all $a, b \in R$
• The octonions, discovered by John T. Graves and Arthur Cayley in the mid-19th century, are a non-associative division algebra that extends the quaternions
• The development of non-associative rings has been driven by their connections to various areas of mathematics, including geometry, topology, and mathematical physics
• Non-associative rings have also found applications in coding theory, cryptography, and the study of non-classical logics

Types of Non-associative Rings

• Alternative rings: Rings satisfying the alternative laws $(a \cdot a) \cdot b = a \cdot (a \cdot b)$ and $(a \cdot b) \cdot b = a \cdot (b \cdot b)$ for all $a, b \in R$
  • Examples include the octonions and the sedenions
• Flexible rings: Rings satisfying $(a \cdot b) \cdot a = a \cdot (b \cdot a)$ for all $a, b \in R$
• Power-associative rings: Rings in which every subring generated by a single element is associative
• Lie rings: Non-associative rings satisfying the Jacobi identity $[a, [b, c]] + [b, [c, a]] + [c, [a, b]] = 0$ for all $a, b, c \in R$, where $[a, b] = a \cdot b - b \cdot a$ is the Lie bracket
• Jordan rings: Commutative non-associative rings satisfying the Jordan identity $(a \cdot b) \cdot a^2 = a \cdot (b \cdot a^2)$ for all $a, b \in R$
• Division rings: Non-associative rings in which every non-zero element has a multiplicative inverse

Properties and Characteristics

• Non-associative rings may have zero divisors: elements $a, b \in R$ such that $a \neq 0, b \neq 0$, but $a \cdot b = 0$
• The center of a non-associative ring is always an associative subring
• The associator $[a, b, c] = (a \cdot b) \cdot c - a \cdot (b \cdot c)$ measures the failure of associativity
  • In an associative ring, the associator is always zero
• The Kleinfeld function $K(a, b, c) = (a \cdot b) \cdot c + (b \cdot c) \cdot a + (c \cdot a) \cdot b - a \cdot (b \cdot c) - b \cdot (c \cdot a) - c \cdot (a \cdot b)$ is used to study the structure of non-associative rings
• The nucleus of a non-associative ring $R$ is the set of elements that associate with all other elements: $N(R) = \{a \in R : (a \cdot b) \cdot c = a \cdot (b \cdot c) \text{ for all } b, c \in R\}$
  • The nucleus is always an associative subring containing the center
• Non-associative rings may have idempotent elements: elements $e \in R$ such that $e \cdot e = e$
  • Idempotents can be used to decompose non-associative rings into direct sums of subrings

Important Theorems and Proofs

• Artin's theorem: In an alternative ring, the subalgebra generated by any two elements is associative
  • This theorem highlights the close relationship between alternative rings and associative rings
• Zorn's vector matrix algebra construction: Every simple non-associative algebra over a field can be realized as a subalgebra of the algebra of vector matrices
• Schafer's classification of alternative division rings: Every alternative division ring is either associative or a Cayley-Dickson algebra over its center
• The Peirce decomposition theorem for Jordan algebras: Every Jordan algebra can be decomposed into a direct sum of subspaces determined by idempotent elements
• The structure theorem for alternative algebras: Every alternative algebra over a field of characteristic not 2 is the direct sum of a nilpotent ideal and a semisimple subalgebra
• The Wedderburn principal theorem for alternative algebras: Every semisimple alternative algebra over a field is a direct sum of simple alternative algebras

Applications and Examples

• Lie algebras are used in the study of symmetries in differential equations, quantum mechanics, and particle physics
  • The Lie algebra $\mathfrak{su}(2)$ describes the angular momentum operators in quantum mechanics
• Jordan algebras have applications in quantum mechanics, where they describe the observables of a quantum system
  • The Jordan algebra of self-adjoint operators on a Hilbert space is a key example
• The octonions have applications in geometry, topology, and mathematical physics
  • The exceptional Lie groups $G_2$, $F_4$, and $E_8$ can be constructed using the octonions
• Non-associative rings are used in the study of non-classical logics, such as non-associative Lambek calculus and substructural logics
• In coding theory, non-associative rings are used to construct error-correcting codes with improved parameters compared to codes over associative rings
• Non-associative rings have potential applications in cryptography, where the lack of associativity can provide additional security against certain types of attacks

Connections to Other Algebraic Structures

• Non-associative rings generalize the concept of associative rings by removing the associativity requirement for multiplication
• Lie algebras are a special case of non-associative rings satisfying the Jacobi identity and anticommutativity of the Lie bracket
• Jordan algebras are commutative non-associative rings satisfying the Jordan identity
• Alternative algebras, including the octonions, are a class of non-associative rings satisfying the alternative laws
• Non-associative division rings, such as the octonions and the sedenions, extend the concept of associative division rings (fields)
• The study of non-associative rings has connections to the theory of quasigroups and loops in combinatorics
  • Quasigroups and loops are non-associative algebraic structures with a single binary operation and unique division
• Non-associative rings have links to the theory of operads, which provide a framework for studying algebraic structures with multiple operations

Challenges and Open Problems

• Classifying all finite-dimensional simple non-associative algebras over algebraically closed fields
  • The classification is complete for alternative algebras and Jordan algebras, but remains open for more general non-associative algebras
• Developing a structure theory for non-associative rings analogous to the Wedderburn-Artin theorem for associative rings
• Exploring the connections between non-associative rings and non-classical logics, such as substructural logics and non-associative Lambek calculus
• Investigating the potential applications of non-associative rings in cryptography and coding theory
  • Designing efficient algorithms for arithmetic in non-associative rings and analyzing their security properties
• Studying the representation theory of non-associative rings and its connections to the representation theory of associated Lie algebras and Jordan algebras
• Exploring the role of non-associative rings in mathematical physics, particularly in the context of quantum gravity and string theory
• Developing computational tools and algorithms for working with non-associative rings, such as Gröbner basis methods and computer algebra systems
• Investigating the connections between non-associative rings and other areas of mathematics, such as topology, geometry, and number theory