n-Lie algebra, where n ≥ 2, is a generalization of the notion of Lie algebra. An integer n indicates the number of elements of algebra necessary to form a Lie bracket. Thus, a Lie bracket of n-Lie algebra g is an n-ary multilinear mapping g×g×…×g (n times) → g, which is skew-symmetric and satisﬁes the Filippov-Jacobi identity (also called fundamental identity).

A notion of n-Liealgebra was proposed by V. T. Filippov in[1]. Earlier Y. Nambu in [2] proposed a generalization of Hamiltonian mechanics by means of a ternary (or, more generally, n-ary) bracket of functions determined on a phasespace. This n-ary bracket can be regarded as an analog of the Poisson bracket in Hamiltonian mechanics and later this n-ary bracket was called the Nambu bracket. The dynamics of this generalization of Hamiltonian mechanics is based on the Nambu-Hamilton equation of motion that contains n−1 Hamilton functions. It is worth to mention that Y. Nambu proposed and developed his generalization of Hamiltonian mechanics based on a notion of triple bracket with the goal to apply this approach to quarks model, where baryons are combinations of three quarks. Later it was shown that the Nambu bracket satisﬁed the Filippov-Jacobi identity and thus, the space of functions endowed with the n-ary Nambu bracket can be regarded as the example of n-Lie algebra. Further development this direction has received in the paper [5], where the authorintroduced a notion of Nambu-Poisson manifold, which can be regarded as an analog of the notion of Poisson manifold in Hamiltonian mechanics. In the same paper the author also treated the problem of quantization of Nambu bracket and constructed the explicit representation of Nambu-Heisenberg commutation relations by means of a primitive cubic root of unity.

Another area of theoretical physics, where the 3-Lie algebras are applied, is a ﬁeld theory. Particularly the authors of the paper [3] proposes a generalization of the Nahm’s equation by means of quantum Nambu 4-bracket and show that their generalization of the Nahm’s equation describes M2 branes ending on M5 branes.

References

[1] V.T. Filippov, n-Lie algebras, Siberian Math. J. **26** (1985), 879–891.

[2] Y. Nambu, Generalized Hamiltonian mechanics, Phys. Rev. D, **7** (1973), 2405–2412.

[3] A. Basu, J.A. Harvey, The M2-M5 brane system and a generalized Nahm’s equation, Nucl. Phys. B **713** (2005), 136–155.

[4] N.Cantarini,V.Kac, Classiﬁcation of simple linearly compact n-Lie superalgebras, Commun. Math. Phys. **298**(2010), 833–853.

[5] L. Takhtajan, On foundation of generalized Nambu mechanics, Comm. Math. Phys. **160**(2) (1994), 295 – 315.

Contact: Viktor Abramov, viktor.abramov@ut.ee

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