Free Access

On the solvability of graded Novikov algebras

Ualbai Umirbaev

Department of Mathematics, Wayne State University, Detroit, MI 48202, USA

Department of Mathematics, Al-Farabi Kazakh National University, Almaty 050040, Kazakhstan

Institute of Mathematics of the SB of RAS, Novosibirsk 630090, Russia

Institute of Mathematics and Mathematical Modeling, Almaty 050010, Kazakhstan

E-mail Address: umirbaev@wayne.edu

Corresponding author.

Search for more papers by this author

and

Viktor Zhelyabin

Institute of Mathematics of the SB of RAS, Novosibirsk 630090, Russia

E-mail Address: vicnic@math.nsc.ru

Search for more papers by this author

https://doi.org/10.1142/S0218196721500491Cited by:8 (Source: Crossref)

Abstract

We show that the right ideal of a Novikov algebra generated by the square of a right nilpotent subalgebra is nilpotent. We also prove that a $G$ $G$ -graded Novikov algebra $N$ $N$ over a field $K$ $K$ with solvable $0$ $0$ -component $N_{0}$ $N_{0}$ is solvable, where $G$ $G$ is a finite additive abelean group and the characteristic of $K$ $K$ does not divide the order of the group $G$ $G$ . We also show that any Novikov algebra $N$ $N$ with a finite solvable group of automorphisms $G$ $G$ is solvable if the algebra of invariants $N^{G}$ $N^{G}$ is solvable.

Communicated by I. Shestakov

Keywords:

AMSC: 17D25, 17B30, 17B70

References

1. I. M. Balinskii and S. P. Novikov , Poisson brackets of hydrodynamic type, Frobenius algebras and Lie algebras, (Russian) Dokl. Akad. Nauk SSSR 283(5) (1985) 1036–1039. Google Scholar
2. G. M. Bergman and I. M. Isaacs , Rings with fixed-point-free group actions, Proc. London Math. Soc. (3) 27 (1973) 69–87. Crossref, Web of Science, Google Scholar
3. L. A. Bokut, Y. Chen and Z. Zhang , On free Gelfand–Dorfman–Novikov–Poisson algebras and a PBW theorem, J. Algebra 500 (2018) 153–170. Crossref, Web of Science, Google Scholar
4. G. V. Dorofeev , An instance of a solvable, though nonnilpotent, alternative ring, (Russian) Uspehi Mat. Nauk 153(93) (1960) 147–150. Google Scholar
5. A. S. Dzhumadil’daev and K. M. Tulenbaev , Engel theorem for Novikov algebras, Comm. Algebra 34(3) (2006) 883–888. Crossref, Web of Science, Google Scholar
6. V. T. Filippov , A class of simple nonassociative algebras (Russian) Mat. Zametki 45(1) (1989) 101–105; translation in Math. Notes 45(1–2) (1989) 68. Google Scholar
7. V. T. Filippov , On right-symmetric and Novikov nil algebras of bounded index, (Russian) Mat. Zametki 70(2) (2001) 289–295; translation in Math. Notes 70(1–2) (2001) 258–263. Google Scholar
8. I. M. Gel’fand and I. Y. Dorfman , Hamiltonian operators and algebraic structures related to them, (Russian) Funktsional. Anal. i Prilozhen 13(4) (1979) 3–30. Google Scholar
9. M. Goncharov , On solvable $ℤ_{3}$ -graded alternative algebras, Algebra Discrete Math. 20(2) (2015) 203–216. Web of Science, Google Scholar
10. G. Higman , Groups and rings which have automorphisms without non-trivial fixed elements, J. London Math. Soc. 32(2) (1957) 321–334. Crossref, Google Scholar
11. V. K. Kharchenko , Galois extensions and quotient rings, Algebra Logic 13(4) (1974) 265–281. Crossref, Google Scholar
12. V. A. Kreknin and A. I. Kostrikin , Lie algebras with a regular automorphism, Sov. Math. Dokl. 4 (1963) 355–358. Google Scholar
13. V. A. Kreknin , Solvability of a Lie algebra containing a regular automorphism, Sib. Math. J. 8 (1967) 536–537. Google Scholar
14. N. Y. Makarenko , A nilpotent ideal in the Lie rings with automorphism of prime order, Sib. Mat. Zh. 46(6) (2005) 1360–1373. Crossref, Web of Science, Google Scholar
15. L. Makar-Limanov and U. Umirbaev , The Freiheitssatz for Novikov algebras, TWMS J. Pure Appl. Math. 2(2) (2011) 228–235. Google Scholar
16. W. S. Martindale and S. Montgomery , Fixed elements of Jordan automorphisms of associative rings, Pac. J. Math. 72(1) (1977) 181–196. Crossref, Web of Science, Google Scholar
17. J. M. Osborn , Novikov algebras, Nova J. Algebra Geom. 1(1) (1992) 1–13. Google Scholar
18. J. M. Osborn , Simple Novikov algebras with an idempotent, Comm. Algebra 20(9) (1992) 2729–2753. Crossref, Web of Science, Google Scholar
19. J. M. Osborn , Infinite-dimensional Novikov algebras of characteristic $0$ , J. Algebra 167(1) (1994) 146–167. Crossref, Web of Science, Google Scholar
20. A. P. Semenov , Subrings of invariants of a finite group of automorphisms of a Jordan ring, Sib. Math. J. 32(1) (1991) 169–172. Crossref, Web of Science, Google Scholar
21. I. Shestakov and Z. Zhang , Solvability and nilpotency of Novikov algebras, Comm. Algebra 48(12) (2020) 5412–5420. Crossref, Web of Science, Google Scholar
22. O. N. Smirnov , Solvability of alternative $ℤ_{2}$ -graded algebras and alternative superalgebras, Sib. Math. J. 32(6) (1991) 1030–1034. Crossref, Web of Science, Google Scholar
23. X. Xu , On simple Novikov algebras and their irreducible modules, J. Algebra 185(3) (1996) 905–934. Crossref, Web of Science, Google Scholar
24. X. Xu , Classification of simple Novikov algebras and their irreducible modules of characteristic 0, J. Algebra 246(2) (2001) 673–707. Crossref, Web of Science, Google Scholar
25. E. I. Zelmanov , A class of local translation-invariant Lie algebras, Dokl. Akad. Nauk SSSR 292(6) (1987) 1294–1297. Google Scholar
26. E. I. Zelmanov , On solvability of Jordan nil-algebras, Sib. Adv. Math. 1 (1991) 185–203; translation from Tr. Inst. Mat. 16 (1989) 37–54. Google Scholar
27. V. N. Zhelyabin , Jordan superalgebras with a solvable even part, Algebra Logic 34(1) (1995) 25–34. Crossref, Google Scholar
28. V. Zhelyabin and U. Umirbaev , On the solvability of $ℤ_{3}$ -graded Novikov algebras, Symmetry 312(2) (2021) 13. Google Scholar
29. K. A. Zhevlakov , Solvability of alternative nil-rings, Sib. Math. J. 3 (1962) 368–377. Google Scholar
30. K. A. Zhevlakov, A. M. Slinko, I. P. Shestakov and A. I. Shirshov , Rings that are Nearly Associative (Academic Press, New York, 1982). Google Scholar