Non-associative Algebra

🧮Non-associative Algebra Unit 4 – Octonions & Alternative Algebras

Octonions are an 8-dimensional algebra extending complex numbers and quaternions. They're non-associative and non-commutative, but satisfy alternative laws. Discovered in the 19th century, octonions have unique properties that set them apart from other number systems. Alternative algebras, including octonions, have applications in physics and mathematics. They're used in string theory, M-theory, and exceptional Lie groups. Despite challenges due to non-associativity, researchers continue exploring octonions' properties and potential applications in various fields.

What Are Octonions?

  • Octonions are an 8-dimensional normed division algebra over the real numbers, extending the complex numbers and quaternions
  • Consist of 8 basis elements: one real unit element and 7 imaginary unit elements, each squaring to -1
  • Non-associative, meaning (ab)ca(bc)(a \cdot b) \cdot c \neq a \cdot (b \cdot c) for some octonions aa, bb, and cc
  • Satisfy alternative laws, a weaker form of associativity
    • (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 octonions aa and bb
  • Multiplication is distributive over addition: a(b+c)=(ab)+(ac)a \cdot (b + c) = (a \cdot b) + (a \cdot c) for all octonions aa, bb, and cc
  • Noncommutative, meaning abbaa \cdot b \neq b \cdot a for some octonions aa and bb
  • Can be represented using the Cayley-Dickson construction, which doubles the dimension of the algebra at each step (real numbers → complex numbers → quaternions → octonions)

Historical Background

  • Octonions were discovered independently by John T. Graves and Arthur Cayley in the mid-19th century
  • Graves discovered them in 1843 while studying hypercomplex number systems, extending William Rowan Hamilton's work on quaternions
  • Cayley published a paper on octonions in 1845, providing a more comprehensive treatment of their properties
  • Initially called "Cayley numbers" or "Cayley's octonions" in recognition of Cayley's contributions
  • Later work by Adolf Hurwitz and others explored the connections between octonions and other mathematical structures, such as Lie groups and spinors
  • In the 20th century, octonions found applications in various areas of mathematics and physics, including:
    • Exceptional Lie groups (G2G_2, F4F_4, E6E_6, E7E_7, and E8E_8)
    • String theory and M-theory
    • Quantum mechanics and the standard model of particle physics

Fundamental Properties

  • Octonions form a normed division algebra, meaning they have a multiplicative inverse and a norm satisfying certain properties
  • The octonion norm is multiplicative: ab=ab\lVert a \cdot b \rVert = \lVert a \rVert \lVert b \rVert for all octonions aa and bb
  • Every non-zero octonion has a unique multiplicative inverse: for any non-zero octonion aa, there exists a unique octonion a1a^{-1} such that aa1=a1a=1a \cdot a^{-1} = a^{-1} \cdot a = 1
  • Octonions are power-associative, meaning (am)n=amn(a^m)^n = a^{mn} for all octonions aa and integers mm and nn
  • The real part of an octonion is defined as Re(a)=12(a+aˉ)\text{Re}(a) = \frac{1}{2}(a + \bar{a}), where aˉ\bar{a} is the conjugate of aa
  • The imaginary part of an octonion is defined as Im(a)=12(aaˉ)\text{Im}(a) = \frac{1}{2}(a - \bar{a})
  • Octonions satisfy the Moufang identities, a set of weaker associativity conditions named after Ruth Moufang

Octonion Arithmetic

  • Octonion addition is component-wise: (a0,a1,,a7)+(b0,b1,,b7)=(a0+b0,a1+b1,,a7+b7)(a_0, a_1, \ldots, a_7) + (b_0, b_1, \ldots, b_7) = (a_0 + b_0, a_1 + b_1, \ldots, a_7 + b_7)
  • Octonion multiplication is determined by the multiplication table of the imaginary units (e1e_1 to e7e_7)
    • The product of any two distinct imaginary units is another imaginary unit, with a sign determined by the orientation of the arrows in the Fano plane
    • Example: e1e2=e4e_1 \cdot e_2 = e_4, e2e4=e1e_2 \cdot e_4 = e_1, and e4e1=e2e_4 \cdot e_1 = -e_2
  • The conjugate of an octonion a=a0+a1e1++a7e7a = a_0 + a_1e_1 + \ldots + a_7e_7 is defined as aˉ=a0a1e1a7e7\bar{a} = a_0 - a_1e_1 - \ldots - a_7e_7
  • The norm of an octonion aa is defined as a=aaˉ=a02+a12++a72\lVert a \rVert = \sqrt{a \cdot \bar{a}} = \sqrt{a_0^2 + a_1^2 + \ldots + a_7^2}
  • The inverse of a non-zero octonion aa is given by a1=aˉa2a^{-1} = \frac{\bar{a}}{\lVert a \rVert^2}
  • Octonion division is defined as multiplication by the inverse: a÷b=ab1a \div b = a \cdot b^{-1} for non-zero octonions aa and bb

Alternative Algebras: Definition and Examples

  • An alternative algebra is a nonassociative algebra satisfying the alternative laws:
    • Left alternative law: (aa)b=a(ab)(a \cdot a) \cdot b = a \cdot (a \cdot b) for all elements aa and bb
    • Right alternative law: (ab)b=a(bb)(a \cdot b) \cdot b = a \cdot (b \cdot b) for all elements aa and bb
  • The octonions form the most well-known example of an alternative algebra
  • Other examples of alternative algebras include:
    • The split-octonions (or hyperbolic octonions), a non-division algebra obtained by modifying the multiplication rules of the octonions
    • The sedenions, a 16-dimensional extension of the octonions that is neither alternative nor a division algebra
    • Cayley-Dickson algebras, a class of algebras obtained by iterating the Cayley-Dickson construction, which includes the octonions and sedenions as special cases
  • Alternative algebras have connections to various areas of mathematics, such as:
    • Lie algebras and Malcev algebras
    • Jordan algebras and exceptional Jordan algebras
    • Composition algebras and triality

Comparison with Other Number Systems

  • Octonions extend the complex numbers and quaternions, forming the third and largest normed division algebra over the real numbers
  • Like the complex numbers and quaternions, octonions are a non-commutative algebra
  • Unlike the complex numbers and quaternions, octonions are non-associative
  • Octonions share some properties with the real numbers, complex numbers, and quaternions, such as:
    • Existence of a multiplicative identity element (1)
    • Existence of multiplicative inverses for non-zero elements
    • Distributivity of multiplication over addition
  • Octonions differ from the real numbers, complex numbers, and quaternions in several ways:
    • Non-associativity of multiplication
    • Non-commutativity of multiplication (shared with quaternions)
    • Lack of a matrix representation that respects the octonion multiplication (unlike the complex numbers and quaternions)
  • The Cayley-Dickson construction provides a unified framework for understanding the relationships between the real numbers, complex numbers, quaternions, and octonions

Applications in Physics and Mathematics

  • Octonions have found various applications in theoretical physics, particularly in the context of string theory and M-theory
    • In M-theory, the exceptional Lie group E8E_8, which is closely related to the octonions, plays a crucial role in describing the symmetries of the theory
    • Octonions have been used to construct explicit solutions for certain classes of supergravity theories, which are low-energy limits of string theory
  • In mathematics, octonions have connections to a wide range of topics, including:
    • Exceptional Lie groups and their associated Lie algebras
    • Spinors and Clifford algebras
    • Algebraic topology and KK-theory
    • Bott periodicity and the Freudenthal magic square
  • Octonions have been used to construct interesting examples and counterexamples in various areas of mathematics, such as:
    • Non-associative division algebras
    • Projective planes and other geometric structures
    • Algebraic varieties and manifolds with exceptional holonomy groups (G2G_2 and Spin(7)\text{Spin}(7))

Challenges and Open Problems

  • Despite their fascinating properties and applications, octonions remain less well-understood than their lower-dimensional counterparts (real numbers, complex numbers, and quaternions)
  • Some open problems and challenges related to octonions include:
    • Developing a more intuitive geometric understanding of octonion multiplication and its non-associativity
    • Exploring the connections between octonions and other areas of mathematics, such as number theory, algebraic geometry, and representation theory
    • Investigating the potential applications of octonions in physics beyond string theory and M-theory, such as in the study of spacetime, quantum mechanics, and the standard model of particle physics
  • The non-associativity of octonions poses challenges in generalizing certain mathematical concepts and results that rely on associativity, such as:
    • Tensor products and modules over octonions
    • Exponential and logarithmic functions of octonions
    • Octonion analysis and the development of a consistent theory of octonion-valued functions
  • Researchers continue to explore the mathematical properties of octonions and their connections to other areas of mathematics and physics, seeking to uncover new insights and applications of these fascinating objects


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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.