Albert algebra

In

${\displaystyle x\circ y={\frac {1}{2}}(x\cdot y+y\cdot x),}$

where ${\displaystyle \cdot }$ denotes matrix multiplication. Another is defined the same way, but using

Over any algebraically closed field, there is just one Albert algebra, and its automorphism group G is the simple split group of type F₄.^[2][3] (For example, the complexifications of the three Albert algebras over the real numbers are isomorphic Albert algebras over the complex numbers.) Because of this, for a general field F, the Albert algebras are classified by the Galois cohomology group H¹(F,G).^[4]

The

The space of cohomological invariants of Albert algebras a field F (of characteristic not 2) with coefficients in Z/2Z is a free module over the cohomology ring of F with a basis 1, f₃, f₅, of degrees 0, 3, 5.^[6] The cohomological invariants with 3-torsion coefficients have a basis 1, g₃ of degrees 0, 3.^[7] The invariants f₃ and g₃ are the primary components of the Rost invariant.

See also

• Euclidean Jordan algebra
  for the Jordan algebras considered by Jordan, von Neumann and Wigner
• Euclidean Hurwitz algebra for details of the construction of the Albert algebra for the octonions

Notes