Octonions
Introduction
Octonions are a type of hypercomplex number that extends the quaternions. They form an eight-dimensional algebra over the real numbers and are one of the four normed division algebras, the others being the real numbers, complex numbers, and quaternions. Octonions are non-associative but are alternative, meaning any subalgebra generated by two elements is associative. They were discovered by John T. Graves in 1843 and are sometimes referred to as Cayley numbers after Arthur Cayley, who independently discovered them in 1845.
Algebraic Structure
Octonions can be represented as a linear combination of one real unit and seven imaginary units. The algebra of octonions is denoted by \(\mathbb{O}\) and can be written in the form: \[ O = x_0 + x_1e_1 + x_2e_2 + x_3e_3 + x_4e_4 + x_5e_5 + x_6e_6 + x_7e_7 \] where \( x_i \) are real numbers and \( e_i \) are the imaginary units. The multiplication of the imaginary units follows specific rules, summarized by the Fano plane, a mnemonic device for remembering the multiplication table of the octonions.
Properties
Octonions are non-commutative and non-associative but satisfy the weaker property of alternativity. This means that the subalgebra generated by any two elements is associative. The multiplication of octonions is neither commutative nor associative, but it is flexible, meaning: \[ (ab)c = a(bc) \] for any octonions \( a \), \( b \), and \( c \). Octonions also form a normed division algebra, which means they have a norm satisfying: \[ \|ab\| = \|a\|\|b\| \] for any octonions \( a \) and \( b \).
Conjugation and Norm
The conjugate of an octonion \( O = x_0 + \sum_{i=1}^7 x_i e_i \) is given by: \[ \overline{O} = x_0 - \sum_{i=1}^7 x_i e_i \] The norm of an octonion is defined as: \[ \|O\| = \sqrt{O \overline{O}} = \sqrt{x_0^2 + x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 + x_6^2 + x_7^2} \] This norm makes the octonions a normed division algebra.
Fano Plane and Multiplication Table
The Fano plane is a useful tool for remembering the multiplication rules of the imaginary units of octonions. It is a projective plane with seven points and seven lines (each line containing three points), and it represents the multiplication structure of the seven imaginary units. Each line in the Fano plane corresponds to a multiplication rule of the form: \[ e_i e_j = e_k \] where \( e_i \), \( e_j \), and \( e_k \) are the points on the line, with the order determined by the orientation of the line.
Applications
Octonions have applications in various fields of mathematics and theoretical physics. In particular, they appear in the study of exceptional Lie groups, string theory, and special holonomy. The exceptional Lie group \( G_2 \) can be understood as the automorphism group of the octonions, preserving the algebraic structure. In string theory, octonions are used to describe certain symmetries and dualities.
Historical Context
The discovery of octonions can be traced back to the mid-19th century. John T. Graves, a friend of William Rowan Hamilton, discovered the octonions shortly after Hamilton's discovery of quaternions. Arthur Cayley independently discovered the octonions and published his findings in 1845. The algebra was initially met with skepticism due to its non-associative nature, but it has since found its place in modern mathematics and physics.