B. A. Omirov

On cubic equations over p-adic fields

We provide a solvability criterion for a cubic equation in domains

SOLVABILITY OF CUBIC EQUATIONS IN p-ADIC INTEGERS ( p> 3)

mathbb{Z}_{p}^{*}, mathbb{Z}_p, mathbb{Q}_p

Leibniz algebras associated with representations of filiform Lie algebras

. We show that, in principal, the Cardano method is not always applicable for such equations. Moreover, the numbers of solutions of the cubic equation in domains

Some radicals, Frattini and Cartan subalgebras of Leibniz n-algebras

mathbb{Z}_{p}^{*}, mathbb{Z}_p, mathbb{Q}_p

Infinitesimal deformations of null-filiform Leibniz superalgebras

are provided. Since 𝔽p is a subgroup of ℚp, we generalize Serres and Suns results concerning with cubic equations over the finite field 𝔽p. Finally, all cubic equations, for which the Cardano method could be applied, are described and the p-adic Cardano formula is provided for those cubic equations.

On real chains of evolution algebras

We give a criterion for the existence of solutions to an equation of the form x3 + ax = b, where a, b ∈ ℚp, in p-adic integers for p > 3. Moreover, in the case when the equation x3 + ax = b is solvable, we give necessary and sufficient recurrent conditions on a p-adic number x ∈ ℤ*p under which x is a solution to the equation.

Dibaric and evolution algebras in biology

Abstract In this paper we investigate Leibniz algebras whose quotient Lie algebra is a naturally graded filiform Lie algebra n n , 1 . We introduce a Fock module for the algebra n n , 1 and provide classification of Leibniz algebras L whose corresponding Lie algebra L / I is the algebra n n , 1 with condition that the ideal I is a Fock n n , 1 -module, where I is the ideal generated by squares of elements from L . We also consider Leibniz algebras with corresponding Lie algebra n n , 1 and such that the action I × n n , 1 → I gives rise to a minimal faithful representation of n n , 1 . The classification up to isomorphism of such Leibniz algebras is given for the case of n = 4 .

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

Abstract In the present work, we introduce notions such as -solvability, - and -nilpotency and the corresponding radicals. We prove that these radicals are invariant under derivations of Leibniz -algebras. The Frattini and Cartan subalgebras of Leibniz -algebras are studied. In particular, we construct examples that show a classical result on conjugacy of Cartan subalgebras of Lie algebras, which also holds in Leibniz algebras and Lie -algebras, is not true for Leibniz -algebras.

Infinitesimal deformations of naturally graded filiform Leibniz algebras

Abstract In this paper we describe the infinitesimal deformations of null-filiform Leibniz superalgebras over a field of zero characteristic. It is known that up to isomorphism in each dimension there exist two such superalgebras N F n , m . One of them is a Leibniz algebra (that is m = 0 ) and the second one is a pure Leibniz superalgebra (that is m ≠ 0 ) of maximum nilindex. We show that the closure of the union of orbits of single-generated Leibniz algebras forms an irreducible component of the variety of Leibniz algebras. We prove that any single-generated Leibniz algebra is a linear integrable deformation of the algebra N F n . Similar results for the case of Leibniz superalgebras are obtained.

Leibniz algebras associated with representations of the Diamond Lie algebra

In this paper, we define a chain of -dimensional evolution algebras corresponding to a permutation of numbers. We show that a chain of evolution algebras (CEA) corresponding to a permutation is non-trivial if and only if the permutation has a fixed point. We show that a CEA is a chain of nilpotent algebras (independently on time) if it is trivial. We construct a wide class of chains of three-dimensional EAs and a class of symmetric -dimensional CEAs. A construction of arbitrary dimensional CEAs is given. Moreover, for a chain of three-dimensional EAs, we study the behaviour of the baric property, the behaviour of the set of absolute nilpotent elements and dynamics of the set of idempotent elements depending on the time.