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Based on the realization of three-algebras in terms of algebra of matricesand four-brackets [arXiv:0807.1570] we present the notion of u(N)-basedextended three-algebras, which for N=2 reproduces the Bagger-Lambertthree-algebra. Using these extended three-algebras we construct an su(N)\timessu(N) Chern-Simons action with explicit SO(8) invariance. The dynamical fieldsof this theory are eight complex valued bosonic and fermionic fields in thebi-fundamental representation of the su(N)\times su(N). For generic N thefermionic transformations, however, close only on a subclass of the states ofthis theory onto the 3d, {\cal N}=6 superalgebra. In this sector we deal withfour complex valued scalars and fermions, our theory is closely related to theABJM model [arXiv:0806.1218], and hence it can be viewed as the (low energyeffective) theory of N M2-branes. We discuss that our three-algebra structuresuggests a picture of open M2-brane stretched between any two pairs ofM2-branes. We also analyze the BPS configurations of our model.

A New Three-Algebra Representation for the {\cal N}=6 su(N)Xsu(N) Superconformal Chern-Simons Theory