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We use superalgebras to realize the 3-algebras used to construct N=6, 8Chern-Simons-matter (CSM) theories. We demonstrate that the superalgebrarealization of the 3-algebras provides a unified framework for classifying thegauge groups of the N \geq 5 theories based on 3-algebras. Using thisrealization, we rederive the ordinary Lie algebra construction of the generalN=6 CSM theory from its 3-algebra counterpart, and reproduce all known examplesas well. In particular, we explicitly construct the Nambu 3-bracket in terms ofa double graded commutator of PSU(2|2). The N = 8 theory of Bagger, Lambert andGustavsson (BLG) with SO(4) gauge group is constructed by using severaldifferent ways. A quantization scheme for the 3-brackets is proposed bypromoting the double graded commutators as quantum mechanical double gradedcommutators.

Superalgebra Realization of the 3-algebras in N=6, 8 Chern-Simons-matter Theories