We study the p-adic equation x q = a over the field of p-adic numbers. We construct an algorithm which gives a solvability criteria in the case of q = p m and present a computer program to compute the criteria for any fixed value of m ≤ p − 1. Moreover, using this solvability criteria for q = 2; 3; 4; 5; 6, we classify p-adic 6-dimensional filiform Leibniz algebras.
@article{bwmeta1.element.doi-10_2478_s11533-013-0225-9,
author = {Manuel Ladra and Bakhrom Omirov and Utkir Rozikov},
title = {Classification of p-adic 6-dimensional filiform Leibniz algebras by solutions of x q = a},
journal = {Open Mathematics},
volume = {11},
year = {2013},
pages = {1083-1093},
zbl = {1272.17007},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0225-9}
}
Manuel Ladra; Bakhrom Omirov; Utkir Rozikov. Classification of p-adic 6-dimensional filiform Leibniz algebras by solutions of x q = a. Open Mathematics, Tome 11 (2013) pp. 1083-1093. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0225-9/
[1] Akbaraliev B.B., Classification of six-dimensional complex filiform Leibniz algebras, Uzbek. Mat. Zh., 2004, 2, 17–22 (in Russian)
[2] Albeverio S., Cianci R., Khrennikov A.Yu., p-adic valued quantization, p-adic Numbers Ultrametric Anal. Appl., 2009, 1(2), 91–104 http://dx.doi.org/10.1134/S2070046609020010[Crossref] | Zbl 1187.81137
[3] Aref’eva I.Ya., Dragovich B., Frampton P.H., Volovich I.V., The wave function of the universe and p-adic gravity, Internat. J. Modern Phys. A, 1991, 6(24), 4341–4358 http://dx.doi.org/10.1142/S0217751X91002094[Crossref] | Zbl 0733.53039
[4] Ayupov Sh.A., Kurbanbaev T.K., The classification of 4-dimensional p-adic filiform Leibniz algebras, TWMS J. Pure Appl. Math., 2010, 1(2), 155–162 | Zbl 1246.17004
[5] Casas J.M., Omirov B.A., Rozikov U.A., Solvability criteria for the equation x q = a in the field of p-adic numbers, preprint available at http://128.84.158.119/abs/1102.2156v1 | Zbl 1296.11157
[6] Dragovich B., Khrennikov A.Yu., Kozyrev S.V., Volovich I.V., On p-adic mathematical physics, p-adic Numbers Ultrametric Anal. Appl., 2009, 1(1), 1–17 http://dx.doi.org/10.1134/S2070046609010014[Crossref] | Zbl 1187.81004
[7] Fekak A., Srhir A., On the p-adic algebra and its applications, Int. Math. Forum, 2009, 4(25–28), 1267–1280 | Zbl 1273.13045
[8] Felipe R., López-Reyes N., Ongay F., R-matrices for Leibniz algebras, Lett. Math. Phys., 2003, 63(2), 157–164 http://dx.doi.org/10.1023/A:1023067727095[Crossref] | Zbl 1053.17003
[9] Freund P.G.O., Witten E., Adelic string amplitudes, Phys. Lett. B, 1987, 199(2), 191–194 http://dx.doi.org/10.1016/0370-2693(87)91357-8[Crossref]
[10] Gómez J.R., Omirov B.A., On classification of complex filiform Leibniz algebras, preprint available at http://128.84.158.119/abs/math/0612735v1
[11] Hagiwara Y., Nambu-Jacobi structures and Jacobi algebroids, J. Phys. A, 2004, 37(26), 6713–6725 http://dx.doi.org/10.1088/0305-4470/37/26/008[Crossref]
[12] Haskell D., A transfer theorem in constructive p-adic algebra, Ann. Pure Appl. Logic, 1992, 58(1), 29–55 http://dx.doi.org/10.1016/0168-0072(92)90033-V[Crossref]
[13] Ibáñez R., de León M., Marrero J.C., Padrón E., Leibniz algebroid associated with a Nambu-Poisson structure, J. Phys. A, 1999, 32(46), 8129–8144 http://dx.doi.org/10.1088/0305-4470/32/46/310[Crossref] | Zbl 0962.53047
[14] Khrennikov A.Yu., p-adic quantum mechanics with p-adic valued functions, J. Math. Phys., 1991, 32(4), 932–937 http://dx.doi.org/10.1063/1.529353[Crossref] | Zbl 0746.46067
[15] Khrennikov A.Yu., p-adic Valued Distributions in Mathematical Physics, Math. Appl., 309, Kluwer, Dordrecht, 1994 | Zbl 0833.46061
[16] Khrennikov A.Yu., Rakić Z., Volovich I.V. (Eds.), p-adic Mathematical Physics, AIP Conference Proceedings, 826, American Institue of Physics, New York, 2006
[17] Khudoyberdiyev A.Kh., Kurbanbaev T.K., Omirov B.A., Classification of three-dimensional solvable p-adic Leibniz algebras, p-adic Numbers Ultrametric Anal. Appl., 2010, 2(3), 207–221 http://dx.doi.org/10.1134/S2070046610030039[Crossref] | Zbl 1305.17002
[18] Koblitz N., p-adic Numbers, p-adic Analysis, and Zeta-Functions, Grad. Texts in Math., 58, Springer, New York-Heidelberg, 1977 http://dx.doi.org/10.1007/978-1-4684-0047-2[Crossref]
[19] Kozyrev S.V., Khrennikov A.Yu., Localization in space for a free particle in ultrametric quantum mechanics, Dokl. Math., 2006, 74(3), 906–909 http://dx.doi.org/10.1134/S1064562406060305[Crossref] | Zbl 1209.81115
[20] Kuku A.O., Some finiteness theorems in the K-theory of orders in p-adic algebras, J. London Math. Soc., 1976, 13(1), 122–128 http://dx.doi.org/10.1112/jlms/s2-13.1.122[Crossref] | Zbl 0323.18011
[21] Loday J.-L., Une version non commutative des algèbres de Lie: les algèbres de Leibniz, Enseign. Math., 1993, 39(3–4), 269–293
[22] Marinari E., Parisi G., On the p-adic five-point function, Phys. Lett. B, 1988, 203(1–2), 52–54
[23] Mukhamedov F., Rozikov U., On rational p-adic dynamical systems, Methods Funct. Anal. Topology, 2004, 10(2), 21–31 | Zbl 1053.37022
[24] Mukhamedov F.M., Rozikov U.A., On Gibbs measures of p-adic Potts model on the Cayley tree, Indag. Math. (N.S.), 2004, 15(1), 85–99 http://dx.doi.org/10.1016/S0019-3577(04)90007-9[Crossref] | Zbl 1161.82311
[25] Omirov B.A., Rakhimov I.S., On Lie-like complex filiform Leibniz algebras, Bull. Aust. Math. Soc., 2009, 79(3), 391–404 http://dx.doi.org/10.1017/S000497270900001X[Crossref][WoS] | Zbl 1194.17001
[26] Schikhof W.H., Ultrametric Calculus, Cambridge Stud. Adv. Math., 4, Cambridge University Press, Cambridge, 1984
[27] Vladimirov V.S., Volovich I.V., Zelenov E.I., p-adic Analysis and Mathematical Physics, Ser. Soviet East European Math., 1, World Scientific, River Edge, 1994 http://dx.doi.org/10.1142/1581[Crossref]