Login
Login

Albert algebra

In mathematics, an Albert algebra is a 27-dimensional exceptional Jordan algebra. They are named after Abraham Adrian Albert, who pioneered the study of non-associative algebras, usually working over the real numbers. Over the real numbers, there are three such Jordan algebras up to isomorphism.Springer & Veldkamp (2000) 5.8, p.153 One of them, which was first mentioned by {{harvs|txt|| last1 = Jordan | first1 = Pascual | author1-link = Pascual Jordan | last2 = Neumann | first2 = John von| last3 = Wigner | first3 = Eugene | author2-link = John von Neumann | author3-link = Eugene Wigner | year=1934}} and studied by {{harvtxt|Albert|1934}}, is the set of 3×3 self-adjoint matrices over the octonions, 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, there is just one Albert algebra, and its automorphism group G is the simple split group of type F4.Springer & Veldkamp (2000) 7.2{{cite journal |author=Chevalley C, Schafer RD |title=The Exceptional Simple Lie Algebras F(4) and E(6) |journal=Proc. Natl. Acad. Sci. U.S.A. |volume=36 |issue=2 |pages=137–41 |date=February 1950 |pmid=16588959 |pmc=1063148 |doi= 10.1073/pnas.36.2.137|bibcode=1950PNAS...36..137C|doi-access=free }}Garibaldi, Petersson, Racine (2024), p. 577 (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).Knus et al (1998) p.517Garibaldi, Petersson, Racine (2024), pp. 599, 600

The Kantor–Koecher–Tits construction applied to an Albert algebra gives a form of the E7 Lie algebra. 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.{{cite journal|title=Structurable Algebras and Groups of Type E_6 and E_7|author=Skip Garibaldi|author-link=Skip Garibaldi|journal=Journal of Algebra|volume=236|year=2001|issue=2|pages=651–691|doi=10.1006/jabr.2000.8514|arxiv=math/9811035}}

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.Garibaldi, Merkurjev, Serre (2003), p.50 The cohomological invariants with 3-torsion coefficients have a basis 1, g3 of degrees 0, 3.Garibaldi (2009), p.20 The invariants f3 and g3 are the primary components of the Rost invariant.

See also

Notes

{{Reflist}}

References

  • {{Citation | last1=Albert | first1=A. Adrian | author-link=Abraham Adrian Albert | title=On a Certain Algebra of Quantum Mechanics | series=Second Series | year=1934 | journal=Annals of Mathematics | issn=0003-486X | volume=35 | issue=1 | pages=65–73 | jstor=1968118 | doi=10.2307/1968118}}
  • {{citation|mr=1999383 |last1= Garibaldi|first1= Skip|author-link1=Skip Garibaldi |last2= Merkurjev|first2= Alexander|last3= Serre|first3= Jean-Pierre|author-link3=Jean-Pierre Serre | title= Cohomological invariants in Galois cohomology|series=University Lecture Series|volume= 28|publisher= American Mathematical Society|place= Providence, RI|year= 2003|isbn= 978-0-8218-3287-5 }}
  • {{cite book|last=Garibaldi|first=Skip|author-link=Skip Garibaldi | title=Cohomological invariants: exceptional groups and Spin groups|series=Memoirs of the American Mathematical Society |year=2009|volume=200|issue=937|isbn=978-0-8218-4404-5|doi=10.1090/memo/0937}}
  • {{cite book|last1=Garibaldi|first1=Skip|author-link1=Skip Garibaldi|last2=Petersson|first2=Holger P.|last3=Racine|first3=Michel L.|title=Albert algebras over commutative rings|series=New Mathematical Monographs|year=2024|volume=48|publisher=Cambridge University Press|isbn=978-1-0094-2685-5|doi=10.1017/9781009426862}}
  • {{citation | doi = 10.2307/1968117 | last1 = Jordan | first1 = Pascual | author1-link = Pascual Jordan | last2 = Neumann | first2 = John von| last3 = Wigner | first3 = Eugene | author2-link = John von Neumann | author3-link = Eugene Wigner | year = 1934 | title = On an Algebraic Generalization of the Quantum Mechanical Formalism | journal = Annals of Mathematics | volume = 35 | issue = 1 | pages = 29–64 | jstor = 1968117}}
  • {{citation | last1=Knus | first1=Max-Albert | last2=Merkurjev | first2=Alexander | author2-link=Alexander Merkurjev | last3=Rost | first3=Markus | author3-link=Markus Rost | last4=Tignol | first4=Jean-Pierre | author-link4=Jean-Pierre Tignol | title=The book of involutions | others=With a preface by J. Tits | zbl=0955.16001 | series=Colloquium Publications | publisher=American Mathematical Society | volume=44 | location=Providence, RI | year=1998 | isbn=978-0-8218-0904-4 }}
  • {{Citation | last1=McCrimmon | first1=Kevin | title=A taste of Jordan algebras |authorlink=Kevin McCrimmon | url=https://books.google.com/books?isbn=0387954473 | publisher=Springer-Verlag | location=Berlin, New York | series=Universitext | isbn=978-0-387-95447-9 | doi=10.1007/b97489 | year=2004 | mr=2014924}}
  • {{Citation | last1=Springer | first1=Tonny A. | author1-link=T. A. Springer | last2=Veldkamp | first2=Ferdinand D. | title=Octonions, Jordan algebras and exceptional groups | orig-year=1963 | url=https://books.google.com/books?isbn=3540663371 | publisher=Springer-Verlag | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-540-66337-9 | year=2000 | mr=1763974}}

Further reading

Category:Non-associative algebras