We prove that the variety of Lie algebras arising from splicing operation coincides with the variety CM of centreby-metabelian Lie algebras. Using these Lie algebras we find the minimal dimension algebras generated the variety CM and the variety of its associative envelope algebras. We study the splicing n-ary operation. We show that all n-ary (n > 2) commutator algebras arising from this operation are nilpotent of index 3. We investigate the generalization of the splicing n-ary operation, and we formulate a series of open problems.
@article{bwmeta1.element.doi-10_2478_s11533-014-0438-6,
author = {Sergei Sverchkov},
title = {Commutator algebras arising from splicing operations},
journal = {Open Mathematics},
volume = {12},
year = {2014},
pages = {1687-1699},
zbl = {06335865},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-014-0438-6}
}
Sergei Sverchkov. Commutator algebras arising from splicing operations. Open Mathematics, Tome 12 (2014) pp. 1687-1699. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-014-0438-6/
[1] De Azcárraga J., Izquierdo J., n-ary algebras: a review with applications, J. Phys. A., 2010, 43(293001) | Zbl 1202.81187
[2] Birkhoff G., Lattice theory, Amer. Math. Soc., Providence, vol. 25, 1967
[3] Bremner M.R., Jordan algebras arising from intermolecular recombination, ACM SIGSAM Bull., 2005, 39(4), 106–117 http://dx.doi.org/10.1145/1140378.1140380 | Zbl 1226.17026
[4] Bremner M., Polynomial identities for ternary intermolecular recombination, Discrete Contin. Dyn. Syst., Ser. S, 2011, 4(6), 1387–1399 http://dx.doi.org/10.3934/dcdss.2011.4.1387 | Zbl 1256.17001
[5] Filippov V., n-Lie algebras. Sib. Mat. Zh, 1985, 26(6), 126–140 | Zbl 0585.17002
[6] Goze N., Remm E., Dimension theorem for free ternary partially associative algebras and applications, J. Algebra, 2011, 348(1), 14–36 http://dx.doi.org/10.1016/j.jalgebra.2011.09.011 | Zbl 1275.17008
[7] De Graaf W., Classification of solvable Lie Algebras, Experiment. Math., 2005, 14(1), 15–25 http://dx.doi.org/10.1080/10586458.2005.10128911 | Zbl 1173.17300
[8] Head N., Formal language theory and DNA: an analysis of the generative capacity of specific recombinant behaviors, Bull. Math. Biol., 1987, 49, 737–759 http://dx.doi.org/10.1007/BF02481771 | Zbl 0655.92008
[9] Holland J., Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, Michigan, 1975
[10] Mishchenko S., Varieties of centre-by-metabelian Lie algebras over a field of characteristic zero, Math. Notes, 1986, 40(6), 901–905 http://dx.doi.org/10.1007/BF01158347
[11] Landweber L., Kary L., The evolution of cellular computing: nature’s solution to a computational problem, Biosystems, 1999, 52(1–3), 3–13 http://dx.doi.org/10.1016/S0303-2647(99)00027-1
[12] Pãun Gh., Rozenberg G, Salomaa A., Computing by splicing, Theoretical Computer Science, 1996, 161, 321–336 http://dx.doi.org/10.1016/S0304-3975(96)00082-5 | Zbl 0874.68117
[13] Pãun Gh., Salomaa A., DNA computing based on the splicing operation, Mathematica Japonica, 1996, 43(3), 607–632 | Zbl 0852.68028
[14] Reed M., Algebraic structure of genetic inheritance. Bull. Amer. Math. Soc., 1997, 34(2), 107–130 http://dx.doi.org/10.1090/S0273-0979-97-00712-X | Zbl 0876.17040
[15] Sverchkov S., Structure and representation of Jordan algebras arising from intermolecular recombination, Contemp. Math., 2009, 483, 261–285 http://dx.doi.org/10.1090/conm/483/09450 | Zbl 1196.17024
[16] Sverchkov S., Structure and representations of n-ary algebras of DNA recombination, Cent. Eur. J. Math., 2011, 9(6), 1193–1216 http://dx.doi.org/10.2478/s11533-011-0087-y | Zbl 1252.17016
[17] Zhevlakov K., Slinko A., Shestakov I., Shirshov A., Rings That Are Nearly Associative, AcademicPress, New York, 1982 | Zbl 0487.17001