Classification of p-adic 6-dimensional filiform Leibniz algebras by solutions of x q = a

Abstract

We study the -adic equation = over the field of -adic numbers. We construct an algorithm which gives a solvability criteria in the case of = and present a computer program to compute the criteria for any fixed value of ≤ − 1. Moreover, using this solvability criteria for = 2 3 4 5 6, we classify -adic 6-dimensional filiform Leibniz algebras. MSC:11S05, 17A32

On Isomorphism Classes of Low Dimensional Non-Lie Filiform Leibniz Algebras over Z_p

“…Besides the classification of algebras over complex field, there are also classifications over a special field which is p-adic (has zero characteristic). The classifications of filiform Leibniz algebras over p-adic have been done for the dimension up to eight (Ayupov and Kurbanbaev, 2010;Ladra et al, 2013;Khudoyberdiyev et al, 2010). Other than that Rakhimov et al, (2018) classified three dimensional Leibniz algebras over arbitrary field where = ℝ, ℤ 3 , ℤ 5 and ℤ 7 .…”

Leibniz algebras of Heisenberg type

“…Since the 1993 Loday's work many researchers have been attracted by this category of algebras, being remarkable the great activity in this field developed in the last years. This activity has been mainly focussed in the frameworks of low dimensional algebras, nilpotence and physics applications (see [2,5,6,14,15,16,17,21,22,23,25,28,33,40,41]). for all x, y, z ∈ L.…”

On the Solutions of the Equation <i>x</i><sup>3</sup> + <i>Ax</i> = <i>B</i> in <span style="font-family: Euclid Math Two">Z</span><sub>3</sub><sup style="margin-left:-6px;">*</sup> with Coefficients from <span style="font-family: Euclid Math Two"&gt

“…In similar complex case, the problem of classification in p -adic case is reduced to the solution of the Equations in the field. The classifications of Leibniz algebras over the field of p -adic numbers have been obtained in [11][12][13].…”