  Gauge symmetry based on Lie algebra has a rather long history and it
successfully describes electromagnetism, weak and strong interactions in the
nature. Recently the Filippov-Nambu 3-algebras have been in the focus of
interest since they appear as gauge symmetries of new superconformal
Chern-Simons non-Abelian theories in 2 + 1 dimensions with the maximum allowed
number of N = 8 linear supersymmetries. These theories explore the low energy
dynamics of the microscopic degrees of freedom of coincident M2 branes and
constitute the boundary conformal field theories of the bulk AdS4 / S7 exact
11-dimensional supergravity backgrounds of supermembranes. These mysterious new
symmetries, the Filippov-Nambu 3-algebras represent the implementation of
non-associative algebras of coordinates of charged tensionless strings, the
boundaries of open M2 branes in antisymmetric field magnetic backgrounds of M5
branes in the M2 -M5 system. A crucial input into this construction came from
the study of the M2-M5 system in the Basu- Harvey's work where an equation
describing the Bogomol'nyi-Prasad- Sommerfeld (BPS) bound state of multiple
M2-branes ending on an M5 was formulated. The Filippov-Nambu 3-algebras are
either operator or matrix representation of the classical Nambu symmetries of
world volume preserving diffeomorphisms of M2 branes. Indeed at the classical
level the supermembrane Lagrangian, in the covariant formulation, has the world
volume preserving diffeomorphisms symmetry SDiff(M2+1). The Filippov-Nambu
3-algebras presumably correspond to the quantization of the rigid motions in
this infinite dimensional group, which describe the low energy excitation
spectrum of the M2 branes. It emphasizes the Filippov-Nambu n-algebras as the
mathematical framework for describing symmetry properties of classical and
quantum mechanical systems.
