mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, an Albert algebra is a 27-dimensional exceptional

Jordan algebra In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms: # xy = yx (commutative law) # (xy)(xx) = x(y(xx)) (). The product of two elements ''x'' and ''y'' in a Jordan al ...

. They are named after

Abraham Adrian Albert Abraham Adrian Albert (November 9, 1905 – June 6, 1972) was an American mathematician. In 1939, he received the American Mathematical Society's Cole Prize in Algebra for his work on Riemann matrices. He is best known for his work on the Al ...

, who pioneered the study of

non-associative algebras A non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative. That is, an algebraic structure ''A'' is a non-associative algebra over a field ''K'' i ...

, usually working over the

real numbers In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...

. Over the real numbers, there are three such Jordan algebras

up to Two Mathematical object, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' wi ...

isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...

.Springer & Veldkamp (2000) 5.8, p.153 One of them, which was first mentioned by and studied by , is the set of 3×3

self-adjoint In mathematics, and more specifically in abstract algebra, an element ''x'' of a *-algebra is self-adjoint if x^*=x. A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold. A collection ''C'' of elements of a sta ...

matrices over the

octonions In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have e ...

, equipped with the binary operation :

x \circ y = \frac12 (x \cdot y + y \cdot x),

where

\cdot

denotes matrix multiplication. Another is defined the same way, but using split octonions instead of octonions. The final is constructed from the non-split octonions using a different standard involution. Over any

algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...

, there is just one Albert algebra, and its automorphism group ''G'' is the simple split group of type F₄.Springer & Veldkamp (2000) 7.2 (For example, the

complexification In mathematics, the complexification of a vector space over the field of real numbers (a "real vector space") yields a vector space over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include t ...

s 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 In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group ''G'' associated to a field extension ''L''/''K'' acts in a nat ...

group H¹(''F'',''G'').Knus et al (1998) p.517 The

Kantor–Koecher–Tits construction In algebra, the Kantor–Koecher–Tits construction is a method of constructing a Lie algebra from a Jordan algebra In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms ...

applied to an Albert algebra gives a form of the

E7 Lie algebra In mathematics, E7 is the name of several closely related Lie groups, linear algebraic groups or their Lie algebras e7, all of which have dimension 133; the same notation E7 is used for the corresponding root lattice, which has rank 7. The ...

. The split Albert algebra is used in a construction of a 56-dimensional structurable algebra whose automorphism group has identity component the simply-connected algebraic group of type E₆. The space of

cohomological invariant In mathematics, a cohomological invariant of an algebraic group ''G'' over a field is an invariant of forms of ''G'' taking values in a Galois cohomology group. Definition Suppose that ''G'' is an algebraic group defined over a field ''K'', and c ...

s of Albert algebras a field ''F'' (of characteristic not 2) with coefficients in Z/2Z is a

free module In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in t ...

over the cohomology ring of ''F'' with a basis 1, ''f''₃, ''f''₅, of degrees 0, 3, 5.Garibaldi, Merkurjev, Serre (2003), p.50 The cohomological invariants with 3-torsion coefficients have a basis 1, ''g''₃ of degrees 0, 3.Garibaldi (2009), p.20 The invariants ''f''₃ and ''g''₃ are the primary components of the

Rost invariant In mathematics, the Rost invariant is a cohomological invariant of an absolutely simple simply connected algebraic group ''G'' over a field ''k'', which associates an element of the Galois cohomology group H3(''k'', Q/Z(2)) to a principal homogeneo ...

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See also

Notes

References

Further reading