Abstract
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.
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References
I. M. Gel’fand and I. Ya. Dorfman, “Hamiltonian operators and algebraic structures related to them,” Funkts. An. Pril., 13, No. 4, 13–30 (1979).
A. A. Balinskii and S. P. Novikov, “Poisson brackets of hydrodynamic type, Frobenius algebras and Lie algebras,” Dokl. Akad. Nauk SSSR, 283, No. 5, 1036–1039 (1985).
E. I. Zelmanov, “On a class of local translation invariant Lie algebras,” Dokl. Akad. Nauk SSSR, 292, No. 6, 1294–1297 (1987).
V. T. Filippov, “A class of simple nonassociative algebras,” Mat. Zametki, 45, No. 1, 101–105 (1989).
J. M. Osborn, “Modules for Novikov algebras,” in Second Int. Conf. ded. A. I. Shirshov (August 20–25, 1991, Barnaul, Russia), L. A. Bokut’ et al. (eds.), Contemp. Math., 184, Am. Math. Soc., Providence, RI (1995), pp. 327–338.
J. M. Osborn, “Novikov algebras,” Nova J. Alg. Geom., 1, No. 1, 1–13 (1992).
J. M. Osborn, “Simple Novikov algebras with an idempotent,” Commun. Alg., 20, No. 9, 2729–2753 (1992).
X. Xu, “Novikov-Poisson algebras,” J. Alg., 190, No. 2, 253–279 (1997).
X. Xu, “Classification of simple Novikov algebra and their irreducible modules of characteristic 0,” J. Alg., 246, No. 2, 673–707 (2001).
C. R. Jordan and D. A. Jordan, “The Lie structure of a commutative ring with a derivation,” J. London Math. Soc., II. Ser., 18, No. 1, 39–49 (1978).
D. A. Jordan, “Simple Lie rings of derivations of commutative rings,” J. London Math. Soc., II. Ser., 18, No. 3, 443–448 (1978).
C. R. Jordan and D. A. Jordan, “Lie rings of derivations of associative rings,” J. London Math. Soc., II. Ser., 17, No. 1, 33–41 (1978).
D. A. Jordan, “On the simplicity of Lie algebras of derivations of commutative algebras,” J. Alg., 228, No. 2, 580–585 (2000).
D. King and K. McCrimmon, “The Kantor construction of Jordan superalgebras,” Commun. Alg., 20, No. 1, 109–126 (1992).
I. P. Shestakov, “Prime alternative superalgebras of arbitrary characteristic,” Algebra Logika, 36, No. 6, 675–716 (1997).
I. P. Shestakov, “General superalgebras of vector type and (γ, δ)-superalgebras,” Resen. Inst. Mat. Estat. Univ. Sao Paulo, 5, No. 1, 85–90 (2001).
V. N. Zhelyabin and I. P. Shestakov, “Simple special Jordan superalgebras with associative even part,” Sib. Mat. Zh., 45, No. 5, 1046–1072 (2004).
A. S. Tikhov, “Novikov-Poisson algebras and Jordan algebras,” in Problems in Theoretical and Applied Mathematics, Inst. Math. Mech., UrO RAN, Yekaterinburg (2006), pp. 76–79.
X. Xu, “On simple Novikov algebras and their irreducible modules,” J. Alg., 185, No. 3, 905–934 (1996).
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Supported by RFBR (grant No. 05-01-00230), by SB RAS (Integration project No. 1.9), and by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (project NSh-344.2008.1).
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Translated from Algebra i Logika, Vol. 47, No. 2, pp. 186–202, March–April, 2008.
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Zhelyabin, V.N., Tikhov, A.S. Novikov-Poisson algebras and associative commutative derivation algebras. Algebra Logic 47, 107–117 (2008). https://doi.org/10.1007/s10469-008-9002-4
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DOI: https://doi.org/10.1007/s10469-008-9002-4