×
Zakharov, A. S.

A class of generalized derivations. (English. Russian original) Zbl 07786474

Algebra Logic 61, No. 6, 466-480 (2023); translation from Algebra Logika 61, No. 6, 87-705 (2022).
Summary: We consider a class of generalized derivations that arise in connection with the problem of adjoining unity to an algebra with generalized derivation, and of searching envelopes for Novikov-Poisson algebras. Conditions for the existence of the localization of an algebra with ternary derivation are specified, as well as conditions under which given an algebra with ternary derivation, we can construct a Novikov-Poisson algebra and a Jordan superalgebra. Finally, we show how the simplicity of an algebra with Brešar generalized derivation is connected with simplicity of the appropriate Novikov algebra.

Cited in 1 Document

MSC:

03-XX Mathematical logic and foundations
20-XX Group theory and generalizations

Keywords:

differential algebra; ternary derivation; generalized derivation; Novikov-Poisson algebra; Jordan superalgebra
Cite Review PDF
Full Text: DOI

References:

[1] R. D. Schafer, An Introduction to Nonassociative Algebras, Pure Appl. Math., 22, Academic Press, New York (1966). · Zbl 0145.25601
[2] Jimenez-Gestal, C.; Perez-Izquierdo, JM, Ternary derivations of generalized Cayley-Dickson algebras, Comm. Alg., 31, 10, 5071-5094 (2003) · Zbl 1036.17001 · doi:10.1081/AGB-120023148
[3] Filippov, VT, On δ-derivations of Lie algebras, Sib. Math. J., 39, 6, 1218-1230 (1998) · Zbl 0936.17020 · doi:10.1007/BF02674132
[4] Brešar, M., On the distance of the composition of two derivations to the generalized derivations, Glasg. Math. J., 33, 1, 89-93 (1991) · Zbl 0731.47037 · doi:10.1017/S0017089500008077
[5] Leger, GF; Luks, EM, Generalized derivations of Lie algebras, J. Alg., 228, 1, 165-203 (2000) · Zbl 0961.17010 · doi:10.1006/jabr.1999.8250
[6] I. M. Gel’fand and I. Ya. Dorfman, “Hamiltonian operators and algebraic structures related to them,” Funkts. Anal. Prilozhen., 13, No. 4, 13-30 (1979). · Zbl 0428.58009
[7] Balinskii, AA; Novikov, SP, Poisson brackets of hydrodynamic type, Frobenius algebras and Lie algebras, Dokl. Akad. Nauk SSSR, 283, 5, 1036-1039 (1985) · Zbl 0606.58018
[8] Zelmanov, EI, On a class of local translation invariant Lie algebras, Dokl. Akad. Nauk SSSR, 292, 6, 1294-1297 (1987) · Zbl 0629.17002
[9] Filippov, VT, A class of simple nonassociative algebras, Mat. Zametki, 45, 1, 101-105 (1989) · Zbl 0659.17003
[10] J. M. Osborn, “Modules for Novikov algebras,” in: Contemp. Math., 184, Am. Math. Soc., Providence, RI (1995), pp. 327-338. · Zbl 0842.17002
[11] Osborn, JM, Novikov algebras, Nova J. Alg. Geom., 1, 1, 1-13 (1992) · Zbl 0876.17005
[12] Osborn, JM, Simple Novikov algebras with an idempotent, Comm. Alg., 20, 9, 2729-2753 (1992) · Zbl 0772.17001 · doi:10.1080/00927879208824486
[13] Xu, X., On simple Novikov algebras and their irreducible modules, J. Alg., 185, 3, 905-934 (1996) · Zbl 0863.17003 · doi:10.1006/jabr.1996.0356
[14] Xu, X., Novikov-Poisson algebras, J. Alg., 190, 2, 253-279 (1997) · Zbl 0872.17030 · doi:10.1006/jabr.1996.6911
[15] A. Dzhumadil’daev and C. Löfwall, “Trees, free right-symmetric algebras, free Novikov algebras and identities,” Homol. Homotopy Appl., 4, No. 2(1), 165-190 (2002). · Zbl 1029.17001
[16] Zhang, Z.; Chen, Y.; Bokut, LA, Free Gelfand-Dorfman-Novikov superalgebras and a Poincaré-Birkhoff-Witt type theorem, Int. J. Alg. Comput., 29, 3, 481-505 (2019) · Zbl 1444.17001 · doi:10.1142/S0218196719500115
[17] A. S. Zakharov, “Gelfand-Dorfman-Novikov-Poisson superalgebras and their envelopes,” Sib. El. Mat. Izv., 16, 1843-1855 (2019); http://semr.math.nsc.ru/v16/p1843-1855.pdf. · Zbl 1472.17017
[18] Zhelyabin, VN; Tikhov, AS, Novikov-Poisson algebras and associative commutative derivation algebras, Algebra and Logic, 47, 2, 107-117 (2008) · Zbl 1164.17001 · doi:10.1007/s10469-008-9002-4
[19] Zakharov, AS, Novikov-Poisson algebras and superalgebras of Jordan brackets, Comm. Alg., 42, 5, 2285-2298 (2014) · Zbl 1301.17004 · doi:10.1080/00927872.2012.758269
[20] Zakharov, AS, Embedding Novikov-Poisson algebras in Novikov-Poisson algebras of vector type, Algebra and Logic, 52, 3, 236-249 (2013) · Zbl 1317.17001 · doi:10.1007/s10469-013-9237-6
[21] A. S. Zakharov, “Novikov-Poisson algebras in low dimension,” Sib. El. Mat. Izv., 12, 381-393 (2015); http://semr.math.nsc.ru/v12/p381-393.pdf · Zbl 1361.17029
[22] Posner, EC, Differentiable simple rings, Proc. Am. Math. Soc., 11, 3, 337-343 (1960) · Zbl 0103.26802 · doi:10.1090/S0002-9939-1960-0113908-6
[23] Kantor, IL, “Jordan and Lie superalgebras defined by the Poisson algebra”, in Algebra and Analysis, 55-80 (1989), Tomsk: Tomsk State Univ, Tomsk · Zbl 0803.17012
[24] King, D.; McCrimmon, K., The Kantor construction of Jordan superalgebras, Comm. Alg., 20, 1, 109-126 (1992) · Zbl 0743.17030 · doi:10.1080/00927879208824334
[25] I. B. Kaigorodov, “Generalized Kantor double,” Vest. SamGU. Estestv.-Nauch. Ser., No. 4(78), 42-50 (2010).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.