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 Sib. Èlektron. Mat. Izv., 2009, Volume 6, Pages 272–311 (Mi semr68)

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Filippov-Nambu $n$-algebras relevant to physics

N. G. Pletnev

Sobolev Institute of Mathematics, Novosibirsk, Russia

Abstract: 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 $\mathcal N=8$ linear supersymmetries. These theories explore the low energy dynamics of the microscopic degrees of freedom of coincident $\mathrm M2$ branes and constitute the boundary conformal field theories of the bulk $AdS_4\times S_7$ 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 $\mathrm M5$ branes in the $\mathrm M2$-$\mathrm M5$ system. A crucial input into this construction came from the study of the $\mathrm M2$-$\mathrm M5$ system in the Basu–Harvey's work where an equation describing the Bogomol'nyi–Prasad–Sommerfield (BPS) bound state of multiple $\mathrm M2$-branes ending on an $\mathrm M5$ was formulated. The Filippov–Nambu $3$-algebras are either operator or matrix representation of the classical Nambu symmetries of world volume preserving diffeomorphisms of $\mathrm M2$ branes. Indeed at the classical level the supermembrane Lagrangian, in the covariant formulation, has the world volume preserving diffeomorphisms symmetry $SDiff(M_{2+1})$. 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 $\mathrm M2$ branes. It emphasizes the Filippov–Nambu $n$-algebras as the mathematical framework for describing symmetry properties of classical and quantum mechanical systems.

Keywords: Filippov $n$-algebra, Nambu bracket, supersymmetry, super $p$-branes.

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Document Type: Article
UDC: 512.5
MSC: 13A99
Received July 8, 2009, published October 16, 2009
Language: English

Citation: N. G. Pletnev, “Filippov-Nambu $n$-algebras relevant to physics”, Sib. Èlektron. Mat. Izv., 6 (2009), 272–311

Citation in format AMSBIB
\Bibitem{Ple09} \by N.~G.~Pletnev \paper Filippov-Nambu $n$-algebras relevant to physics \jour Sib. \Elektron. Mat. Izv. \yr 2009 \vol 6 \pages 272--311 \mathnet{http://mi.mathnet.ru/semr68} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2586691} `

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2. Frank T.D., “Active systems with Nambu dynamics: with applications to rod wielding for haptic length perception and self-propagating systems on two-spheres”, Eur. Phys. J. B, 74:2 (2010), 195–203
3. Frank T.D., “Unifying mass-action kinetics and Newtonian mechanics by means of Nambu brackets”, Journal of Biological Physics, 37:4 (2011), 375–385
4. Frank T.D., “Nambu bracket formulation of nonlinear biochemical reactions beyond elementary mass action kinetics”, J. Nonlinear Math. Phys., 19:1 (2012), 1250007, 17 pp.
5. I. B. Kaygorodov, “On $\delta$-derivations of $n$-ary algebras”, Izv. Math., 76:6 (2012), 1150–1162
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7. I. B. Kaygorodov, “$(n+1)$-ary Derivations of Semisimple Filippov algebras”, Math. Notes, 96:2 (2014), 208–216
8. Frank T.D., “Active and Purely Dissipative Nambu Systems in General Thermostatistical Settings Described By Nonlinear Partial Differential Equations Involving Generalized Entropy Measures”, Entropy, 19:1 (2017), 8
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