Czechoslovak Mathematical Journal, first online, pp. 1-6


Fermionic Novikov algebras admitting invariant non-degenerate symmetric bilinear forms

Zhiqi Chen, Xueqing Chen, Ming Ding

Received February 21, 2019.   Published online March 31, 2020.

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.
Keywords:  Novikov algebra; fermionic Novikov algebra; invariant bilinear form
Classification MSC:  17B60, 17A30, 17D25
DOI:  10.21136/CMJ.2020.0071-19

PDF available at:  Springer   Institute of Mathematics CAS

References:
[1] A. A. Balinskii, S. P. Novikov: 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. MR 0802121 | Zbl 0606.58018
[2] D. Burde: Simple left-symmetric algebras with solvable Lie algebra. Manuscr. Math. 95 (1998), 397-411; erratum ibid. 96 (1998), 393-395. DOI 10.1007/s002290050037 | MR 1612015 | Zbl 0907.17008
[3] B. A. Dubrovin, S. P. Novikov: 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. MR 0715332 | Zbl 0553.35011
[4] 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. MR 0770656 | Zbl 0591.58012
[5] 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. DOI 10.1070/RM1975v030n05ABEH001522 | MR 0508337 | Zbl 0334.58007
[6] 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. DOI 10.1007/BF01075767 | MR 0467819 | Zbl 0347.49023
[7] I. M. Gel'fand, I. Ya. Dorfman: Hamiltonian operators and algebraic structures related to them. Funkts. Anal. Prilozh. 13 (1979), 13-30. (In Russian.) MR 0554407 | Zbl 0428.58009
[8] M. Guediri: Novikov algebras carrying an invariant Lorentzian symmetric bilinear form. J. Geom. Phys. 82 (2014), 132-144. DOI 10.1016/j.geomphys.2014.04.007 | MR 3206645 | Zbl 1361.17003
[9] B. O'Neill: Semi-Riemannian Geometry with Applications to Relativity. Pure and Applied Mathematics 103, Academic Press, New York (1983). MR 0719023 | Zbl 0531.53051
[10] 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. MR 0158414 | Zbl 0138.43301
[11] X. Xu: Hamiltonian operators and associative algebras with a derivation. Lett. Math. Phys. 33 (1995), 1-6. DOI 10.1007/BF00750806 | MR 1315250 | Zbl 0837.16034
[12] X. Xu: Hamiltonian superoperators. J. Phys. A, Math. Gen. 28 (1995), 1681-1698. DOI 10.1088/0305-4470/28/6/021 | MR 1338053 | Zbl 0852.58043
[13] X. Xu: Variational calculus of supervariables and related algebraic structures. J. Algebra 223 (2000), 396-437. DOI 10.1006/jabr.1999.8064 | MR 1735154 | Zbl 1012.37048
[14] 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. MR 0880610 | Zbl 0629.17002

Affiliations:   Zhiqi Chen, School of Mathematical Sciences and LPMC, Nankai University, Tianjin, P. R. China, e-mail: chenzhiqi@nankai.edu.cn; Xueqing Chen, Department of Mathematics, University of Wisconsin-Whitewater, 800 W. Main Street, Whitewater, WI 53190, USA, e-mail: chenx@uww.edu; Ming Ding (corresponding author), School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, P. R. China, e-mail: m-ding04@mails.tsinghua.edu.cn


 
PDF available at: