octonions

octonions. In mathematics, the octonions are a normed division algebra over the real numbers, a generalization of the complex numbers and quaternions. They form an 8-dimensional real vector space and are the largest of the four normed division algebras, as stated by the Hurwitz theorem. Their nonassociative algebraic structure makes them a key object of study in abstract algebra and theoretical physics.

Definition and basic properties

The octonions can be constructed from the quaternions via the Cayley–Dickson construction, doubling the dimension to eight. A standard basis is denoted $\backslash \{1,\; i,\; j,\; k,\; l,\; il,\; jl,\; kl\backslash \}$, where the imaginary units satisfy specific multiplication rules. Each octonion is a linear combination of these basis elements with real coefficients. The algebra is equipped with a conjugate operation and a multiplicative norm satisfying $\backslash |xy\backslash |\; =\; \backslash |x\backslash |\backslash |y\backslash |$, making it a composition algebra. Unlike the quaternions, multiplication is neither commutative nor associative, but it is alternative, meaning any two elements generate an associative subalgebra.

History and discovery

The discovery of the octonions followed shortly after William Rowan Hamilton's 1843 breakthrough with the quaternions. In late 1843, John T. Graves, a friend of Hamilton, communicated his discovery of an 8-dimensional algebra to Hamilton in a letter. Independently, the mathematician Arthur Cayley published his findings in 1845, leading to the octonions sometimes being called **Cayley numbers**. Their early study was sporadic, with significant later contributions from figures like Ruth Moufang, who explored their Moufang identities, and Élie Cartan, who connected them to Lie group theory. The full classification of normed division algebras by Adolf Hurwitz in 1898 cemented their unique status.

Algebraic structure

The octonions form a non-associative algebra that is a division algebra, meaning every nonzero element has a multiplicative inverse. The multiplication table is encoded by the Fano plane, a finite projective plane of order 2, where points correspond to imaginary units. This structure highlights the alternativity and permits the definition of an automorphism group, which is the exceptional Lie group G₂. The lack of associativity complicates analysis, but the algebra satisfies the Moufang identities, which provide a weaker form of associativity crucial for projective geometry. The Jacobson identity does not hold, distinguishing them from Lie algebras.

Applications

While less directly applied than complex numbers or quaternions, octonions have found important roles in several advanced theoretical domains. In theoretical physics, they appear in string theory, particularly in connections to supersymmetry and M-theory, as explored by researchers like John C. Baez. They are also used in the construction of the exceptional Lie groups, such as E₈, which feature in grand unified theory models. In geometry, they relate to the study of Rosenfeld projective planes and symmetric spaces. Some approaches to quantum mechanics, such as those by Pascual Jordan, have utilized Jordan algebras derived from octonions.

Related mathematical structures

The octonions are the third member of the sequence of Cayley–Dickson algebras, preceded by the real numbers, complex numbers, and quaternions, and succeeded by the sedenions, which are not a division algebra. They are intimately connected to exceptional structures in Lie theory, including the exceptional Jordan algebra or Albert algebra, and the five exceptional simple Lie groups: G₂, F₄, E₆, E₇, and E₈. Their spinor representations link them to Clifford algebras and triality in 8 dimensions. The Pfister form theory in quadratic forms also utilizes their multiplicative properties.

Category:Hypercomplex numbers Category:Division algebras Category:Non-associative algebras