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Chen, Zhiqi; Chen, Xueqing; Ding, Ming

Fermionic Novikov algebras admitting invariant non-degenerate symmetric bilinear forms. (English) Zbl 07285972

Czech. Math. J. 70, No. 4, 953-958 (2020).
Summary: Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic type and Hamiltonian operators in the formal variational calculus. Fermionic Novikov algebras correspond to a certain Hamiltonian superoperator in a supervariable. In this paper, we show that fermionic Novikov algebras equipped with invariant non-degenerate symmetric bilinear forms are Novikov algebras.

MSC:

17B60 Lie (super)algebras associated with other structures (associative, Jordan, etc.)
17A30 Nonassociative algebras satisfying other identities
17D25 Lie-admissible algebras

Keywords:

Novikov algebra; fermionic Novikov algebra; invariant bilinear form
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\textit{Z. Chen} et al., Czech. Math. J. 70, No. 4, 953--958 (2020; Zbl 07285972)
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References:

[1] Balinskii, A. A.; Novikov, S. P., Poisson brackets of hydrodynamic type, Frobenius algebras and Lie algebras, Sov. Math., Dokl. 32 (1985), 228-231 translated from Dokl. Akad. Nauk SSSR 283 1985 1036-1039
[2] Burde, D., Simple left-symmetric algebras with solvable Lie algebra, Manuscr. Math. 95 (1998), 397-411 erratum ibid. 96 1998 393-395
[3] Dubrovin, B. A.; Novikov, S. P., Hamiltonian formalism of one-dimensional systems of hydrodynamic type, and the Bogolyubov-Whitham averaging method, Sov. Math., Dokl. 27 (1983), 665-669 translated from Dokl. Akad. Nauk SSSR 270 1983 781-785
[4] Dubrovin, B. A.; Novikov, S. P., On Poisson brackets of hydrodynamic type, Sov. Math., Dokl. 30 (1984), 651-654 translated from Dokl. Akad. Nauk SSSR 279 1984 294-297
[5] Gel’fand, I. M.; Dikii, L. A., Asymptotic behaviour of the resolvent of Sturm-Liouville equations and the algebra of the Korteweg-de Vries equations, Russ. Math. Surv. 30 (1975), 77-113 translated from Usp. Mat. Nauk 30 1975 67-100
[6] Gel’fand, I. M.; Dikii, L. A., A Lie algebra structure in a formal variational calculation, Funct. Anal. Appl. 10 (1976), 16-22 translated from Funkts. Anal. Prilozh. 10 1976 18-25
[7] Gel’fand, I. M.; Dorfman, I. Ya., Hamiltonian operators and algebraic structures related to them, Funkts. Anal. Prilozh. Russian 13 (1979), 13-30
[8] Guediri, M., Novikov algebras carrying an invariant Lorentzian symmetric bilinear form, J. Geom. Phys. 82 (2014), 132-144
[9] O’Neill, B., Semi-Riemannian Geometry with Applications to Relativity, Pure and Applied Mathematics 103, Academic Press, New York (1983)
[10] Vinberg, E. B., The theory of convex homogeneous cones, Trans. Mosc. Math. Soc. 12 (1963), 340-403 translated from Tr. Mosk. Mat. O. 12 1963 303-358
[11] Xu, X., Hamiltonian operators and associative algebras with a derivation, Lett. Math. Phys. 33 (1995), 1-6
[12] Xu, X., Hamiltonian superoperators, J. Phys. A, Math. Gen. 28 (1995), 1681-1698
[13] Xu, X., Variational calculus of supervariables and related algebraic structures, J. Algebra 223 (2000), 396-437
[14] Zel’manov, E., On a class of local translation invariant Lie algebras, Sov. Math., Dokl. 35 (1987), 216-218 translated from Dokl. Akad. Nauk SSSR 292 1987 1294-1297
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