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

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 $(a \cdot b) \cdot c \neq a \cdot (b \cdot c)$ for some octonions $a$, $b$, and $c$
• Satisfy alternative laws, a weaker form of associativity
  • $(a \cdot a) \cdot b = a \cdot (a \cdot b)$ and $(a \cdot b) \cdot b = a \cdot (b \cdot b)$ for all octonions $a$ and $b$
• Multiplication is distributive over addition: $a \cdot (b + c) = (a \cdot b) + (a \cdot c)$ for all octonions $a$, $b$, and $c$
• Noncommutative, meaning $a \cdot b \neq b \cdot a$ 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 ($G_2$, $F_4$, $E_6$, $E_7$, and $E_8$)
  • 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: $\lVert a \cdot b \rVert = \lVert a \rVert \lVert b \rVert$ 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 $a \cdot a^{-1} = a^{-1} \cdot a = 1$
• Octonions are power-associative, meaning $(a^m)^n = a^{mn}$ for all octonions $a$ and integers $m$ and $n$
• The real part of an octonion is defined as $\text{Re}(a) = \frac{1}{2}(a + \bar{a})$, where $\bar{a}$ is the conjugate of $a$
• The imaginary part of an octonion is defined as $\text{Im}(a) = \frac{1}{2}(a - \bar{a})$
• Octonions satisfy the Moufang identities, a set of weaker associativity conditions named after Ruth Moufang

Octonion Arithmetic

• Octonion addition is component-wise: $(a_0, a_1, \ldots, a_7) + (b_0, b_1, \ldots, b_7) = (a_0 + b_0, a_1 + b_1, \ldots, a_7 + b_7)$
• Octonion multiplication is determined by the multiplication table of the imaginary units ($e_1$ to $e_7$)
  • 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: $e_1 \cdot e_2 = e_4$, $e_2 \cdot e_4 = e_1$, and $e_4 \cdot e_1 = -e_2$
• The conjugate of an octonion $a = a_0 + a_1e_1 + \ldots + a_7e_7$ is defined as $\bar{a} = a_0 - a_1e_1 - \ldots - a_7e_7$
• The norm of an octonion $a$ is defined as $\lVert a \rVert = \sqrt{a \cdot \bar{a}} = \sqrt{a_0^2 + a_1^2 + \ldots + a_7^2}$
• The inverse of a non-zero octonion $a$ is given by $a^{-1} = \frac{\bar{a}}{\lVert a \rVert^2}$
• Octonion division is defined as multiplication by the inverse: $a \div b = a \cdot b^{-1}$ 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: $(a \cdot a) \cdot b = a \cdot (a \cdot b)$ for all elements $a$ and $b$
  • Right alternative law: $(a \cdot b) \cdot b = a \cdot (b \cdot b)$ 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 $E_8$, 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 $K$-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 ($G_2$ and $\text{Spin}(7)$)

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