Nonassociative, division, and Jordan algebras

Core idea

Nonassociative algebra studies vector spaces with bilinear multiplication without assuming the associative law. This section develops the structures needed for the octonions, Jordan algebras, and their exceptional symmetry groups, from general definitions through the Albert algebra and Peirce theory.

Foundations

• Nonassociative algebra
• Alternative algebra
• Composition algebra
• Real normed division algebra
• Hurwitz's theorem

The octonions

• Octonion algebra
• Octonion conjugation, norm, and inner product
• Determinant of a two-by-two octonionic Hermitian matrix
• Octonions as complex vectors
• Conjugated cross product on $\mathbb C^3$
• $SU(3)$ stabilizer of a complex octonion subalgebra

Exceptional structures

The automorphism group of the octonions is the compact exceptional Lie group $G_2$, with Lie algebra $\mathfrak g_2$. Octonions also supply the entries of the exceptional Jordan algebra.

Jordan-algebra foundations

• Jordan algebra
• Jordan algebra homomorphism
• Jordan subalgebra
• Special and exceptional Jordan algebras
• Euclidean Jordan algebra
• Simple Euclidean Jordan algebra
• Trace form of a Euclidean Jordan algebra
• Automorphism group of a Jordan algebra
• Ideal in a Jordan algebra
• Power-associative algebra
• Derivation of a Jordan algebra
• Spectral theorem for Euclidean Jordan algebras
• Automorphism transitivity on Jordan frames
• Hermitian Jordan triple system
• Tripotent in a Jordan triple system

Hermitian matrix and spin-factor examples

• Hermitian matrix Jordan algebra
• Complex-qubit Jordan algebra
• Complex-qutrit Jordan algebra
• Spin-factor Jordan algebra
• Octonionic spin factor
• Exceptional Jordan algebra

Idempotents, frames, and Peirce theory

• Jordan idempotent
• Orthogonal Jordan idempotents
• Primitive Jordan idempotent
• Jordan frame
• Peirce decomposition of a Jordan algebra
• Peirce-one Jordan corner
• Frame decomposition of Hermitian Jordan algebras

Exceptional corners, decompositions, and stabilizers

• Unique octonionic spin-factor corner
• Complex-subalgebra decomposition in the Albert algebra
• $\operatorname{Spin}(8)$ stabilizer of an Albert-algebra frame
• $\operatorname{Spin}(9)$ stabilizer of an octonionic spin factor
• Complex-qutrit stabilizer in $F_4$
• $F_4$ transitivity on complex-qutrit subalgebras
• $F_4$ transitivity on compatible Jordan-subalgebra pairs

Paper-guided expansion

• Complete E7 and exceptional Jordan-algebra paper index
• Complete quaternionic PSH, convexity, and octonionic paper index