Abstract
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A.A. Balinskii, S. P. Novikov: Poisson brackets of hydrodynamic type, Frobenius alge- bras and Lie algebras. Sov. Math., Dokl. 32 (1985), 228–231; translated from Dokl. Akad. Nauk SSSR 283 (1985), 1036-1039.
D. Burde: Simple left-symmetric algebras with solvable Lie algebra. Manuscr. Math. 95 (1998), 397–411; erratum ibid. 96 (1998), 393-395.
B.A. Dubrovin, S. P. Novikov: Hamiltonian formalism of one-dimensional systems of hy- drodynamic type, and the Bogolyubov-Whitham averaging method. Sov. Math., Dokl. 27 (1983), 665–669; translated from Dokl. Akad. Nauk SSSR 270 (1983), 781-785.
B.A. Dubrovin, S. P. Novikov: On Poisson brackets of hydrodynamic type. Sov. Math., Dokl. 30 (1984), 651–654; translated from Dokl. Akad. Nauk SSSR 279 (1984), 294-297.
I. M. Gel'fand, L.A. Dikii: 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.
I. M. Gel'fand, L.A. Dikii: 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.
I. M. Gel'fand, I.Ya. Dorfman: Hamiltonian operators and algebraic structures related to them. Funkts. Anal. Prilozh. 13 (1979), 13–30. (In Russian.)
M. Guediri: Novikov algebras carrying an invariant Lorentzian symmetric bilinear form. J. Geom. Phys. 82 (2014), 132–144.
B. O'Neill: Semi-Riemannian Geometry with Applications to Relativity. Pure and Ap- plied Mathematics 103, Academic Press, New York, 1983.
E.B. Vinberg: The theory of convex homogeneous cones. Trans. Mosc. Math. Soc. 12 (1963), 340–403; translated from Tr. Mosk. Mat. O. 12 (1963), 303-358.
X. Xu: Hamiltonian operators and associative algebras with a derivation. Lett. Math. Phys. 33 (1995), 1–6.
X. Xu: Hamiltonian superoperators. J. Phys. A. Math. Gen. 28 (1995), 1681–1698.
X. Xu: Variational calculus of supervariables and related algebraic structures. J. Algebra 223 (2000), 396–437.
E. Zel'manov: 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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chen, Z., Chen, X. & Ding, M. Fermionic Novikov Algebras Admitting Invariant Non-Degenerate Symmetric Bilinear Forms. Czech Math J 70, 953–958 (2020). https://doi.org/10.21136/CMJ.2020.0071-19
Received:
Published:
Issue Date:
DOI: https://doi.org/10.21136/CMJ.2020.0071-19