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Bakhrom Omirov

Publications and source records attributed to Bakhrom Omirov.

12 recordsLinked to original sources
arXiv2026

Poisson $n$-Lie algebras: constructions and the structure of solvable algebras

In this paper, we develop a construction of Poisson $n$-Lie algebras that generalizes the Jacobian $n$-Lie construction. Using the Grassmann--Plücker relations, we derive necessary and sufficient conditions under which the resulting bracket defines a Poisson $n$-Lie algebra. We also prove that suitable quotients of tensor products of Poisson algebras carry natural Poisson $n$-Lie structures. Conversely, we give a tensor-type procedure that associates a Poisson algebra to a given Poisson $n$-Lie algebra. The quotient and converse constructions thus provide two systematic methods for relating Poisson algebras to Poisson $n$-Lie algebras. We further establish analogues of Engel's theorem and Lie's theorem and characterize solvability and nilpotency of Poisson $n$-Lie algebras in terms of their underlying associative and $n$-Lie structures. We introduce hypo-nilpotent ideals and investigate maximal such ideals in finite-dimensional solvable Poisson $n$-Lie algebras. Finally, we prove that the generalized eigenspaces of multiplication operators are ideals.

math.RA↗
arXiv2026

On Lie Algebras with Only Inner Derivations

This paper is devoted to the study of non-semisimple Lie algebras of the form $\mathcal{L} = \mathcal{S} \ltimes \mathcal{N}$ whose derivations are all inner. By generalizing the methods of Sato and Angelopoulos, we introduce new families of Lie algebras and establish the vanishing of their first adjoint cohomology. As an application, we construct a family of complete non-perfect Lie algebras, thereby providing examples that yield a positive answer to Carles' question on the existence of such algebras. In addition, we reduce the dimension of known examples of perfect Lie algebras with non-trivial center and only inner derivations to $31$. Furthermore, we employ the Hochschild--Serre factorization theorem to analyze the second adjoint cohomology groups, providing insights non-vanishing of the second adjoint cohomology groups for the algebras obtained through the paper.

math.RA↗
arXiv2026

Irreducible cuspidal modules of simple $n$-Lie algebras

This work devoted to the description of irreducible cuspidal modules over simple $n$-Lie algebras. Since the description of irreducible modules over $n$-Lie algebra $O^n$ are already well understood, we focus here on the irreducible cuspidal modules over $n$-Lie algebras of Wronskians and Jacobians. First, for a given $n$-Lie algebra $\mathcal{L}$, we analyze the possible Lie and Leibniz structures on $\wedge^{n-1} \mathcal{L}$ and $\otimes^{n-1} \mathcal{L}$ by thoroughly examining existing structures. Next, we classify the irreducible cuspidal modules over the $n$-Lie algebra of Wronskians defined on Laurent polynomials with degree-preserving derivations. Furthermore, we prove that these modules remain irreducible over the $n$-Lie algebra of Jacobians.

math.RA↗
arXiv2025

On maximal solvable extensions of nilpotent Lie algebras

In this paper, we provide a complete description of complex maximal solvable extensions for a certain class of nilpotent Lie algebras. In particular, we show that, up to isomorphism, a solvable extension of a $d$-locally diagonalizable nilpotent Lie algebra is unique and is realized as the semidirect product of its nilradical with a maximal torus. This result resolves a conjecture of Šnobl concerning the uniqueness of maximal solvable extensions under the condition $d$-locally diagonalizability on the nilradical. Moreover, we extend this description to the setting of Lie superalgebras and present an alternative method for constructing such maximal solvable extensions. Finally, we discuss further aspects and open questions related to maximal solvable extensions of nilpotent Lie algebras

math.RA↗
arXiv2019

A rigid Leibniz algebra with non-trivial HL^2

In this article, we generalize Richardson's example of a rigid Lie algebra with non-trivial $H^2$ to the Leibniz setting. Namely, we consider the hemisemidirect product ${\mathfrak h}$ of a semidirect product Lie algebra $M_k\rtimes{\mathfrak g}$ of a simple Lie algebra ${\mathfrak g}$ with some non-trivial irreducible ${\mathfrak g}$-module $M_k$ with a non-trivial irreducible ${\mathfrak g}$-module $I_l$. Then for ${\mathfrak g}={\mathfrak s}{\mathfrak l}_2({\mathbb C})$, we take $M_k$ (resp. $I_l$) to be the standard irreducible ${\mathfrak s}{\mathfrak l}_2({\mathbb C})$-module of dimension $k+1$ (resp. $l+1$). Assume $\frac{k}{2}>5$ is an odd integer and $l>2$ is odd, then we show that the Leibniz algebra ${\mathfrak h}$ is geometrically rigid and has non-trivial $HL^2$ with adjoint coefficients. We close the article with an appendix where we record further results on the question whether $H^2({\mathfrak g},{\mathfrak g})=0$ implies $HL^2({\mathfrak g},{\mathfrak g})=0$.

math.KT↗
arXiv2019

Some cohomologically rigid solvable Leibniz algebras

In this paper we describe solvable Leibniz algebras whose quotient algebra by one-dimensional ideal is a Lie algebra with rank equal to the length of the characteristic sequence of its nilpotent radical. We prove that such Leibniz algebra is unique and centerless. Also it is proved that the first and the second cohomology groups of the algebra with coefficients in itself is trivial.

math.RA↗
arXiv2018

Derivations and automorphisms of nilpotent evolution algebras with maximal nilindex

In this paper is devoted to nilpotent finite-dimensional evolution algebras E with $dimE^2 = dimE-1$. We described Lie algebras associated with evolution algebras whose nilindex is maximal. Moreover, in terms of this Lie algebra we fully construct nilpotent evolution algebra with maximal index of nilpotency. Furthermore, this result allowed us fully characterize all local and 2-local derivations of the considered evolution algebras. All automorphisms and local automorphisms of the nilpotent evolution algebras with maximal nilindex are found.

math.RA↗
arXiv2017

On Levi-Malcev theorem for Leibniz algebras

The present paper is devoted to provide conditions for the Levi--Malcev theorem to hold or not to hold (i.e. for two Levi subalgebras to be or not conjugate by an inner automorphism) in the context of finite-dimensional Leibniz algebras over a field of characteristic zero. Particularly, in the case of the field $\mathbb{C}$ of complex numbers, we consider all possible cases in which Levi subalgebras are conjugate and not conjugate.

math.RT↗
arXiv2017

Local and 2-local derivations and automorphisms on simple Leibniz algebras

The present paper is devoted to local and 2-local derivations and automorphism of complex finite-dimensional simple Leibniz algebras. We prove that all local derivations and 2-local derivations on a finite-dimensional complex simple Leibniz algebra are automatically derivations. We show that nilpotent Leibniz algebras as a rule admit local derivations and 2-local derivations which are not derivations. Further we consider automorphisms of simple Leibniz algebras. We prove that every 2-local automorphism on a complex finite-dimensional simple Leibniz algebra is an automorphism and show that nilpotent Leibniz algebras admit 2-local automorphisms which are not automorphisms. A similar problem concerning local automorphism on simple Leibniz algebras is reduced to the case of simple Lie algebras.

math.RA↗
arXiv2012

On cubic equations over $P-$adic field

We provide a solvability criteria for a depressed cubic equation in domains $\bz_p^{*},\bz_p,\bq_p$. We show that, in principal, the Cardano method is not always applicable for such equations. Moreover, the numbers of solutions of the depressed cubic equation in domains $\bz_p^{*},\bz_p,\bq_p$ are provided. Since $\bbf_p\subset\bq_p,$ we generalize J.-P. Serre's \cite{JPSJ} and Z.H.Sun's \cite{ZHS1,ZHS3} results concerning with depressed cubic equations over the finite field $\bbf_p$. Finally, all depressed cubic equations, for which the Cardano method could be applied, are described and the $p-$adic Cardano formula is provided for those cubic equations.

math.NT↗