Geodesic
Novikov--Poisson algebras and associative commutative derivation algebras
Algebra i logika, Tome 47 (2008) no. 2, pp. 186-202.

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

We describe Novikov–Poisson algebras in which a Novikov algebra is not simple while its corresponding associative commutative derivation algebra is differentially simple. In particular, it is proved that a Novikov algebra is simple over a field of characteristic not 2 iff its associative commutative derivation algebra is differentially simple. The relationship is established between Novikov–Poisson algebras and Jordan superalgebras.
Keywords: Novikov algebra, Lie algebra, derivation algebra, Jordan superalgebra. 
@article{AL_2008_47_2_a3,
     author = {V. N. Zhelyabin and A. S. Tikhov},
     title = {Novikov--Poisson algebras and associative commutative derivation algebras},
     journal = {Algebra i logika},
     pages = {186--202},
     publisher = {mathdoc},
     volume = {47},
     number = {2},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2008_47_2_a3/}
}
TY  - JOUR
AU  - V. N. Zhelyabin
AU  - A. S. Tikhov
TI  - Novikov--Poisson algebras and associative commutative derivation algebras
JO  - Algebra i logika
PY  - 2008
SP  - 186
EP  - 202
VL  - 47
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AL_2008_47_2_a3/
LA  - ru
ID  - AL_2008_47_2_a3
ER  - 
%0 Journal Article
%A V. N. Zhelyabin
%A A. S. Tikhov
%T Novikov--Poisson algebras and associative commutative derivation algebras
%J Algebra i logika
%D 2008
%P 186-202
%V 47
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AL_2008_47_2_a3/
%G ru
%F AL_2008_47_2_a3
V. N. Zhelyabin; A. S. Tikhov. Novikov--Poisson algebras and associative commutative derivation algebras. Algebra i logika, Tome 47 (2008) no. 2, pp. 186-202. http://geodesic.mathdoc.fr/item/AL_2008_47_2_a3/

[1] I. M. Gelfand, I. Ya. Dorfman, “Gamiltonovy operatory i svyazannye s nimi algebraicheskie struktury”, Funkts. analiz pril., 13:4 (1979), 13–30 | MR | Zbl

[2] A. A. Balinskii, S. P. Novikov, “Skobki Puassona gidrodinamicheskogo tipa, frobeniusovy algebry i algebry Li”, DAN SSSR, 283:5 (1985), 1036–1039 | MR | Zbl

[3] E. I. Zelmanov, “Ob odnom klasse lokalnykh translyatsionno invariantnykh algebr Li”, DAN SSSR, 292:6 (1987), 1294–1297 | MR

[4] V. T. Filippov, “Ob odnom klasse prostykh neassotsiativnykh algebr”, Matem. zametki, 45:1 (1989), 101–105 | MR | Zbl

[5] J. M. Osborn, “Modules for Novikov algebras”, Second int. conf. algebra ded. mem. A. I. Shirshov, Proc. conf. algebra (August 20–25, 1991, Barnaul, Russia), Contemp. Math., 184, eds. L. A. Bokut' et al., Am. Math. Soc., Providence, RI, 1995, 327–338 | MR | Zbl

[6] J. M. Osborn, “Novikov algebras”, Nova J. Algebra Geom., 1:1 (1992), 1–13 | MR | Zbl

[7] J. M. Osborn, “Simple Novikov algebras with an idempotent”, Commun. Algebra, 20:9 (1992), 2729–2753 | DOI | MR | Zbl

[8] X. Xu, “Novikov–Poisson algebras”, J. Algebra, 190:2 (1997), 253–279 | DOI | MR | Zbl

[9] X. Xu, “Classification of simple Novikov algebra and their irreducible modules of characteristic 0”, J. Algebra, 246:2 (2001), 673–707 | DOI | MR | Zbl

[10] C. R. Jordan, D. A. Jordan, “The Lie structure of a commutative ring with a derivation”, J. Lond. Math. Soc. (2), 18:1 (1978), 39–49 | DOI | MR | Zbl

[11] D. A. Jordan, “Simple Lie rings of derivations of commutative rings”, J. Lond. Math. Soc. (2), 18:3 (1978), 443–448 | DOI | MR | Zbl

[12] C. R. Jordan, D. A. Jordan, “Lie rings of derivations of associative rings”, J. Lond. Math. Soc. (2), 17:1 (1978), 33–41 | DOI | MR | Zbl

[13] D. A. Jordan, “On the simplicity of Lie algebras of derivations of commutative algebras”, J. Algebra, 228:2 (2000), 580–585 | DOI | MR | Zbl

[14] D. King, K. McCrimmon, “The Kantor construction of Jordan superalgebras”, Commun. Algebra, 20:1 (1992), 109–126 | DOI | MR | Zbl

[15] I. P. Shestakov, “Pervichnye alternativnye superalgebry proizvolnoi kharakteristiki”, Algebra i logika, 36:6 (1997), 675–716 | MR | Zbl

[16] I. P. Shestakov, “General superalgebras of vector type and $(\gamma,\delta)$-superalgebras”, Resen. Inst. Mat. Estat. Univ. Sao Paulo, 5:1 (2001), 85–90 | MR | Zbl

[17] V. N. Zhelyabin, I. P. Shestakov, “Prostye spetsialnye iordanovy superalgebry s assotsiativnoi chëtnoi chastyu”, Sib. matem. zh., 45:5 (2004), 1046–1072 | MR | Zbl

[18] A. S. Tikhov, “Algebry Novikova-Puassona i iordanovy superalgebry”, Problemy teoreticheskoi i prikladnoi matematiki, Tr. 37-oi region. molod. konf. (30 yanvarya – 3 fevralya 2006 g.), In-t matem. mekh. UrO RAN, Ekaterinburg, 2006, 76–79

[19] X. Xu, “On simple Novikov algebras and their irreducible modules”, J. Algebra, 185:3 (1996), 905–934 | DOI | MR | Zbl