🧮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.
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 (a⋅b)⋅c=a⋅(b⋅c) for some octonions a, b, and c
Satisfy alternative laws, a weaker form of associativity
(a⋅a)⋅b=a⋅(a⋅b) and (a⋅b)⋅b=a⋅(b⋅b) for all octonions a and b
Multiplication is distributive over addition: a⋅(b+c)=(a⋅b)+(a⋅c) for all octonions a, b, and c
Noncommutative, meaning a⋅b=b⋅a for some octonions a and b
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 (G2, F4, E6, E7, and E8)
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: ∥a⋅b∥=∥a∥∥b∥ for all octonions a and b
Every non-zero octonion has a unique multiplicative inverse: for any non-zero octonion a, there exists a unique octonion a−1 such that a⋅a−1=a−1⋅a=1
Octonions are power-associative, meaning (am)n=amn for all octonions a and integers m and n
The real part of an octonion is defined as Re(a)=21(a+aˉ), where aˉ is the conjugate of a
The imaginary part of an octonion is defined as Im(a)=21(a−aˉ)
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)
Octonion multiplication is determined by the multiplication table of the imaginary units (e1 to e7)
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: e1⋅e2=e4, e2⋅e4=e1, and e4⋅e1=−e2
The conjugate of an octonion a=a0+a1e1+…+a7e7 is defined as aˉ=a0−a1e1−…−a7e7
The norm of an octonion a is defined as ∥a∥=a⋅aˉ=a02+a12+…+a72
The inverse of a non-zero octonion a is given by a−1=∥a∥2aˉ
Octonion division is defined as multiplication by the inverse: a÷b=a⋅b−1 for non-zero octonions a and b
Alternative Algebras: Definition and Examples
An alternative algebra is a nonassociative algebra satisfying the alternative laws:
Left alternative law: (a⋅a)⋅b=a⋅(a⋅b) for all elements a and b
Right alternative law: (a⋅b)⋅b=a⋅(b⋅b) for all elements a and b
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 E8, 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 K-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 (G2 and 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