Gelfand--Dorfman--Novikov--Poisson superalgebras and their envelopes
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 16 (2019), pp. 1843-1855.

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

We prove some result of $\mathcal{GDNP}$–algebras for $\mathcal{GDNP}$–superalgebras and embeding some $\mathcal{GDNP}$–superalgebras into $\mathcal{GDNP}$–superalgebras of vector type. Also we prove that not every $\mathcal{GDNP}$–superalgebra can be embeded into associative supercommutative superalgebra with even derivation $D$ and product given by $a\circ b=a D(b)$.
Keywords: Gelfand-Dorfman-Novikov-Poisson superalgebras, derivations of algebras, Jordan superalgebras.
@article{SEMR_2019_16_a29,
     author = {A. S. Zakharov},
     title = {Gelfand--Dorfman--Novikov--Poisson superalgebras and their envelopes},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {1843--1855},
     publisher = {mathdoc},
     volume = {16},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2019_16_a29/}
}
TY  - JOUR
AU  - A. S. Zakharov
TI  - Gelfand--Dorfman--Novikov--Poisson superalgebras and their envelopes
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2019
SP  - 1843
EP  - 1855
VL  - 16
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2019_16_a29/
LA  - ru
ID  - SEMR_2019_16_a29
ER  - 
%0 Journal Article
%A A. S. Zakharov
%T Gelfand--Dorfman--Novikov--Poisson superalgebras and their envelopes
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2019
%P 1843-1855
%V 16
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2019_16_a29/
%G ru
%F SEMR_2019_16_a29
A. S. Zakharov. Gelfand--Dorfman--Novikov--Poisson superalgebras and their envelopes. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 16 (2019), pp. 1843-1855. http://geodesic.mathdoc.fr/item/SEMR_2019_16_a29/

[1] I. M. Gel'fand, I. Ya. Dorfman, “Hamiltonian operators and algebraic structures related to them”, Functional Analysis and Its Applications, 13:4 (1979), 248–262 | DOI | MR

[2] A. A. Balinskii, S. P. Novikov, “Poisson brackets of hydrodynamic type, Frobenius algebras and Lie algebras”, Dokl. Akad. Nauk SSSR, 283:5 (1985), 1036–1039 | MR | Zbl

[3] E. I. Zelmanov, “A class of local translation-invariant Lie algebras”, Dokl. Akad. Nauk SSSR, 292:6 (1987), 1294–1297 | MR | Zbl

[4] V. T. Filippov, “A class of simple nonassociative algebras”, Mat. Zametki, 45:1 (1989), 101–105 | MR | Zbl

[5] J. M. Osborn, “Modules for Novikov algebras”, Contemp. Math., 184, 1991, 327–338 | DOI | MR

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

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

[8] X. Xu, “On Simple Novikov Algebras and Their Irreducible Modules”, J. Algebra, 185 (1996), 905–934 | DOI | MR | Zbl

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

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

[11] A. S. Dzhumadildaev, C. Lofwall, “Trees, free right-symmetric algebras, free Novikov algebras and identities”, Homology, Homotopy and Applications, 4:2 (2002), 165–190 | DOI | MR

[12] Z. Zhang, Y. Chen, L.A. Bokut, “On Free Gelfand-Dorfman-Novikov Superalgebras and a PBW Type Theorem”, International Journal of Algebra and Computation, 29:3 (2019), 481–505 | DOI | MR | Zbl

[13] A. S. Zakharov, “Embedding Novikov-Poisson algebras in Novikov-Poisson algebras of vector type”, Algebra Logika, 52:3 (2013), 236–249 | DOI | MR | Zbl

[14] A. S. Zakharov, “Novikov-Poisson algebras in low dimension”, Sib. Elektron. Mat. Izv., 12 (2015), 381–393 | MR | Zbl

[15] V. N. Zhelyabin, A. S. Tikhov, “Novikov-Poisson algebras and associative commutative derivation algebras”, Algebra Logika, 47:2 (2008), 107–117 | DOI | MR | Zbl

[16] A. S. Zakharov, “Novikov-Poisson algebras and superalgebras of Jordan brackets”, Commun. Algebra, 42:5 (2014), 2285–2298 | DOI | MR | Zbl

[17] Math. Notes, 97:3 (2015), 341–348 | DOI | DOI | MR | MR | Zbl

[18] Siberian Math. J., 32:6 (1991), 1052–1060 | DOI | MR | MR | Zbl