Abstract
The generalized supersymmetries admitting abelian bosonic tensorial central charges are classified in accordance with their division algebra structure (over
R
,
C
,
H
or
O
). It is shown in particular that in D=11 dimensions, the
M
-superalgebra admits a consistent octonionic formulation, involving 52 real bosonic generators (in place of the 528 of the standard
M
-superalgebra). The octonionic
M5
(super-5-brane) sector coincides with the octonionic
M1
and
M2
sectors, while in the standard formulation these sectors are all independent. The octonionic conformal and superconformal
M
-algebras are explicitly constructed. They are respectively given by the
Sp(8|O)
(
OSp(1,8|O)
) (super)algebra of octonionic-valued (super)matrices, whose bosonic subalgebra consists of 232 (and respectively 239) generators.