|XXIX Workshop on Geometric Methods in Physics||27.06-03.07.2010|
Graded algebras generalizing the octonions
The algebra of quaternions is not commutative. However, viewed as a graded algebra over (Z_2)^2 or (Z_2)^3, the algebra of quaternions becomes graded-commutative! More generally, any Clifford algebra is an associative graded-commutative algebra. We will show that this property completely characterizes (simple) Clifford algebras.
The classical algebra of octonions is neither commutative nor associative,but it also becomes graded-commutative and graded-associative over (Z_2)^3. We will introduce a series of algebras generalizing the octonions in the same way as
Clifford algebras generalize the quaternions. We will discuss the main properties of these algebras and mention numerous applications. In particular, we will obtain explicit solutions for the Hurwitz-Radon sum of squares problem. (Joint work with V.Ovsienko)