Non-associative Algebra

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

Non-associative rings expand on traditional ring theory by removing the requirement of associativity for multiplication. These structures include Lie algebras, Jordan algebras, and octonions, each with unique properties and applications in mathematics and physics. The study of non-associative rings began in the early 20th century, driven by the need to understand symmetries in differential equations and quantum mechanics. Today, these structures find applications in diverse fields, from coding theory to non-classical logics.

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,+,)(R, +, \cdot) consists of a set RR with two binary operations, addition (+)(+) and multiplication ()(\cdot), satisfying certain axioms
    • Addition is associative: a+(b+c)=(a+b)+ca + (b + c) = (a + b) + c for all a,b,cRa, b, c \in R
    • Addition is commutative: a+b=b+aa + b = b + a for all a,bRa, b \in R
    • Additive identity: There exists an element 0R0 \in R such that a+0=aa + 0 = a for all aRa \in R
    • Additive inverses: For each aRa \in R, there exists an element aR-a \in R such that a+(a)=0a + (-a) = 0
  • Multiplication is distributive over addition: a(b+c)=ab+aca \cdot (b + c) = a \cdot b + a \cdot c and (a+b)c=ac+bc(a + b) \cdot c = a \cdot c + b \cdot c for all a,b,cRa, b, c \in R
  • Examples of non-associative rings include Lie algebras, Jordan algebras, and octonions
  • The center of a non-associative ring RR is the set of elements that associate and commute with all other elements: Z(R)={aR:(ab)c=a(bc),ab=ba for all b,cR}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: (ab)a2=a(ba2)(a \cdot b) \cdot a^2 = a \cdot (b \cdot a^2) for all a,bRa, 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 (aa)b=a(ab)(a \cdot a) \cdot b = a \cdot (a \cdot b) and (ab)b=a(bb)(a \cdot b) \cdot b = a \cdot (b \cdot b) for all a,bRa, b \in R
    • Examples include the octonions and the sedenions
  • Flexible rings: Rings satisfying (ab)a=a(ba)(a \cdot b) \cdot a = a \cdot (b \cdot a) for all a,bRa, 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[a, [b, c]] + [b, [c, a]] + [c, [a, b]] = 0 for all a,b,cRa, b, c \in R, where [a,b]=abba[a, b] = a \cdot b - b \cdot a is the Lie bracket
  • Jordan rings: Commutative non-associative rings satisfying the Jordan identity (ab)a2=a(ba2)(a \cdot b) \cdot a^2 = a \cdot (b \cdot a^2) for all a,bRa, 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,bRa, b \in R such that a0,b0a \neq 0, b \neq 0, but ab=0a \cdot b = 0
  • The center of a non-associative ring is always an associative subring
  • The associator [a,b,c]=(ab)ca(bc)[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)=(ab)c+(bc)a+(ca)ba(bc)b(ca)c(ab)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 RR is the set of elements that associate with all other elements: N(R)={aR:(ab)c=a(bc) for all b,cR}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 eRe \in R such that ee=ee \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 su(2)\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 G2G_2, F4F_4, and E8E_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


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© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.