Geodesic
On $\delta$-derivations of $n$-ary algebras
Izvestiya. Mathematics , Tome 76 (2012) no. 6, pp. 1150-1162.

Voir la notice de l'article provenant de la source Math-Net.Ru

We give a description of $\delta$-derivations of $(n+1)$-dimensional $n$-ary Filippov algebras and, as a consequence, of simple finite-dimensional Filippov algebras over an algebraically closed field of characteristic zero. We also give new examples of non-trivial $\delta$-derivations of Filippov algebras and show that there are no non-trivial $\delta$-derivations of the simple ternary Mal'tsev algebra $M_8$.
Keywords: $\delta$-derivation, Filippov algebra, ternary Mal'tsev algebra. 
@article{IM2_2012_76_6_a4,
     author = {I. B. Kaygorodov},
     title = {On $\delta$-derivations of $n$-ary algebras},
     journal = {Izvestiya. Mathematics },
     pages = {1150--1162},
     publisher = {mathdoc},
     volume = {76},
     number = {6},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2012_76_6_a4/}
}
TY  - JOUR
AU  - I. B. Kaygorodov
TI  - On $\delta$-derivations of $n$-ary algebras
JO  - Izvestiya. Mathematics 
PY  - 2012
SP  - 1150
EP  - 1162
VL  - 76
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2012_76_6_a4/
LA  - en
ID  - IM2_2012_76_6_a4
ER  - 
%0 Journal Article
%A I. B. Kaygorodov
%T On $\delta$-derivations of $n$-ary algebras
%J Izvestiya. Mathematics 
%D 2012
%P 1150-1162
%V 76
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2012_76_6_a4/
%G en
%F IM2_2012_76_6_a4
I. B. Kaygorodov. On $\delta$-derivations of $n$-ary algebras. Izvestiya. Mathematics , Tome 76 (2012) no. 6, pp. 1150-1162. http://geodesic.mathdoc.fr/item/IM2_2012_76_6_a4/

[1] N. C. Hopkins, “Generalized derivations of nonassociative algebras”, Nova J. Math. Game Theory Algebra, 5:3 (1996), 215–224 | MR | Zbl

[2] V. T. Filippov, “Lie algebras satisfying identities of degree 5”, Algebra and Logic, 34:6 (1995), 379–394 | DOI | MR | Zbl

[3] V. T. Filippov, “On $\delta$-derivations of Lie algebras”, Siberian Math. J., 39:6 (1998), 1218–1230 | DOI | MR | Zbl

[4] V. T. Filippov, “$\delta$-derivations of prime Lie algebras”, Siberian Math. J., 40:1 (1999), 174–184 | DOI | MR | Zbl

[5] V. T. Filippov, “$\delta$-derivations of prime alternative and Mal'tsev algebras”, Algebra and Logic, 39:5 (2000), 354–358 | DOI | MR | Zbl

[6] N. B. Kaygorodov, “$\delta$-Derivations of simple finite-dimensional Jordan superalgebras”, Algebra and Logic, 46:5 (2007), 318–329 | DOI | MR | Zbl

[7] N. B. Kaygorodov, “$\delta$-derivations of classical Lie superalgebras”, Siberian Math. J., 50:3 (2009), 434–449 | DOI | MR | Zbl

[8] N. B. Kaygorodov, “$\delta$-Superderivations of simple finite-dimensional Jordan and Lie superalgebras”, Algebra and Logic, 49:2 (2010), 130–144 | DOI | MR | Zbl

[9] P. Zusmanovich, “On $\delta$-derivations of Lie algebras and superalgebras”, J. Algebra, 324:12 (2010), 3470–3486 | DOI | MR | Zbl

[10] V. N. Zhelyabin, N. B. Kaygorodov, “On $\delta$-superderivations of simple superalgebras of Jordan brackets”, St. Petersburg Math. J., 23:4 (2012), 665–677 | DOI | MR

[11] N. B. Kaygorodov, “$\delta$-Superderivations of semisimple finite-dimensional Jordan superalgebras”, Math. Notes, 91:2 (2012), 187–197 | DOI

[12] Y. Nambu, “Generalized Hamiltonian dynamics”, Phys. Rev. D (3), 7:8 (1973), 2405–2412 | DOI | MR | Zbl

[13] N. G. Pletnev, “Filippov–Nambu $n$-algebras relevant to physics”, Sib. elektron. matem. izv., 6 (2009), 272–311 | MR

[14] J. A. de Azcárraga, J. M. Izquierdo, “$n$-ary algebras: a review with applications”, J. Phys. A, 43:29 (2010), ID 293001 | DOI | Zbl

[15] J. Bagger, N. Lambert, “Three-algebras and $N=6$ Chern–Simons gauge theories”, Phys. Rev. D, 79:2 (2009), 025002 | DOI | MR | Zbl

[16] J. Bagger, G. Bruhn, Three-algebras in $N=5,6$ superconformal Chern–Simons theories: representations and relations, arXiv: 1006.0040

[17] D. Gaiotto, E. Witten, “Janus configurations, Chern–Simons couplings, and the $\theta$-angle in $N=4$ super Yang–Mills theory”, J. High Energy Phys., 2010, no. 6, 97 | DOI | MR

[18] N. Cantarini, V. G. Kac, “Classification of linearly compact simple algebraic $N=6$ 3-algebras”, Transform. Groups, 16:3 (2011), 649–671 | DOI | MR | Zbl

[19] P. de Medeiros, J. Figueroa-O'Farrill, E. Méndez-Escobar, P. Ritter, “On the Lie-algebraic origin of metric $3$-algebras”, Comm. Math. Phys., 290:3 (2009), 871–902 | DOI | MR | Zbl

[20] V. T. Filippov, “$n$-Lie algebras”, Siberian Math. J., 26:6 (1985), 879–891 | DOI | MR | Zbl | Zbl

[21] W. Ling, On the structure of $n$-Lie algebras, Thesis, Siegen University-GHS-Siegen, 1993 | Zbl

[22] A. P. Pozhidaev, “$n$-ary Mal'tsev algebras”, Algebra and Logic, 40:3 (2001), 170–182 | DOI | MR | Zbl

[23] A. P. Pozhidaev, P. Saraiva, “On derivations of the ternary Malcev algebra $M_8$”, Comm. Algebra, 34:10 (2006), 3593–3608 | DOI | MR | Zbl

[24] A. P. Pozhidaev, “Root decomposition of the ternary Malcev algebra $M_8$”, Siberian Math. J., 46:4 (2005), 717–721 | DOI | MR | Zbl

[25] K. A. Zhevlakov, A. M. Slinko, I. P. Shestakov, A. I. Shirshov, Koltsa, blizkie k assotsiativnym, Nauka, M., 1978 | MR | Zbl

[26] I. B. Kaygorodov, “$(n+1)$-Ary derivations of simple $n$-ary algebras”, Algebra and Logic, 50:5 (2011), 470–471 | DOI | MR

[27] I. Kaygorodov, “$\delta$-derivations of algebras and superalgebras”, Proceedings of the International Conference on Algebra 2010, Advances in Algebraic Structures (Gadjah Mada University, Indonesia, 2010), World Scientific, Hackensack, NJ, 2012, 374–380