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

This article is cited in 8 scientific papers (total in 8 papers)


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
\by N.~G.~Pletnev
\paper Filippov-Nambu $n$-algebras relevant to physics
\jour Sib. \`Elektron. Mat. Izv.
\yr 2009
\vol 6
\pages 272--311

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    This publication is cited in the following articles:
    1. Frank T.D., “A Fokker-Planck approach to canonical-dissipative Nambu systems: With an application to human motor control during dynamic haptic perception”, Phys. Lett. A, 374:31–32 (2010), 3136–3142  crossref  zmath  adsnasa  isi  elib
    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  crossref  adsnasa  isi
    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  crossref  isi
    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.  crossref  mathscinet  zmath  isi  elib
    5. I. B. Kaygorodov, “On $\delta$-derivations of $n$-ary algebras”, Izv. Math., 76:6 (2012), 1150–1162  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    6. Mongkolsakulvong S., Chaikhan P., Frank T.D., “Oscillatory Nonequilibrium Nambu Systems: the Canonical-Dissipative Yamaleev Oscillator”, Eur. Phys. J. B, 85:3 (2012), 90  crossref  adsnasa  isi
    7. I. B. Kaygorodov, “$(n+1)$-ary Derivations of Semisimple Filippov algebras”, Math. Notes, 96:2 (2014), 208–216  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    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  crossref  mathscinet  isi  scopus
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