Superconformal M2-branes and generalized Jordan triple systems

Three-dimensional conformal theories with six supersymmetries and SU(4) R-symmetry describing stacks of M2-branes are here proposed to be related to generalized Jordan triple systems. Writing the four-index structure constants in an appropriate form, the Chern-Simons part of the action immediately suggests a connection to such triple systems. In contrast to the previously considered three-algebras, the additional structure of a generalized Jordan triple system is associated to a graded Lie algebra, which corresponds to an extension of the gauge group. In this note we show that the whole theory with six manifest supersymmetries can be naturally expressed in terms of such a graded Lie algebra. Also the BLG theory with eight supersymmetries is included as a special case.