The algebraic and geometric classification of $3$-Poisson algebras

$3$-Poisson (ternary Nambu--Poisson) algebras are the algebras of observables of Nambu mechanics, the generalization of Hamiltonian mechanics in which the time evolution $\dot f=\{f,H_1,H_2\}$ is governed by two Hamiltonians and preserves the phase volume; their $3$-Lie part also gives the gauge algebra of the Bagger--Lambert--Gustavsson model of multiple M2-branes. We give the algebraic classification of complex $4$-dimensional $3$-Poisson algebras (seven algebras) and of complex $5$-dimensional nilpotent $3$-Poisson algebras ($38$ algebras and one one-parameter family) with both multiplications non-zero. We also describe the algebraic counterpart of the Inönü--Wigner contractions of physics: the variety of complex $4$-dimensional $3$-Poisson algebras has dimension $16$ and seven irreducible components, one of which is the orbit closure of the simple $3$-Lie algebra of the M2-brane model, and the variety of complex $5$-dimensional nilpotent $3$-Poisson algebras has dimension $20$ and five irreducible components.

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