Leibniz Algebras Research Articles

On the Structure of Automorphism Groups of Some Low-Dimensional Leibniz Algebras

On the Structure of Automorphism Groups of Some Low-Dimensional Leibniz Algebras

On the Structure of the Algebra of Derivations for Some Low-Dimensional Leibniz Algebras

On the Structure of the Algebra of Derivations for Some Low-Dimensional Leibniz Algebras

Twisted 𝒪-operator families on Leibniz algebras and NS-Leibniz family algebras

Twisted 𝒪-operator families on Leibniz algebras and NS-Leibniz family algebras

On the Gabriel quiver of extensions of Leibniz algebras

We compute the Gabriel quiver of simple objects in the category of bimodules over a simple Leibniz algebra and over the trivial 1-dimensional Leibniz algebra. Vertices of the quiver are the classes of simple objects, arrows are given by the dimensions of Ext 1 -groups.

Leibniz algebras in which all centralizers of nonzero elements are zero algebras

This paper is concerned with generalizing the results for Lie CT-algebras to Leibniz algebras. In some cases our results give a generalization even for the case of a Lie algebra. Results on A-algebras are used to show every Leibniz CT-algebra over an algebraically closed field of characteristic different from 2,3 is solvable or is isomorphic to s l 2 ( F ) . A characterization is then given for solvable Leibniz CT-algebras. It is also shown that the class of solvable Leibniz CT-algebras is factor closed.

Subinvariance and ascendancy in Leibniz Algebras

Leibniz algebras are certain generalizations of Lie algebras. Motivated by the concept of subinvariance and ascendancy in group theory, Schenkman studied properties of subinvariant subalgebras of a Lie algebra. Kawamoto studied ascendency in Lie algebras. In this paper we define subinvariance and asendency in Leibniz algebras and study their properties. It is shown that the signature results on subinvariance and ascendency in Lie algebras have analogs for Leibniz algebras.

Quasi-triangular, factorizable Leibniz bialgebras and relative Rota–Baxter operators

Abstract We introduce the notion of quasi-triangular Leibniz bialgebras, which can be constructed from solutions of the classical Leibniz Yang–Baxter equation (CLYBE) whose skew-symmetric parts are invariant. In addition to triangular Leibniz bialgebras, quasi-triangular Leibniz bialgebras contain factorizable Leibniz bialgebras as another subclass, which lead to a factorization of the underlying Leibniz algebras. Relative Rota–Baxter operators with weights on Leibniz algebras are used to characterize solutions of the CLYBE whose skew-symmetric parts are invariant. On skew-symmetric quadratic Leibniz algebras, such operators correspond to Rota–Baxter type operators. Consequently, we introduce the notion of skew-symmetric quadratic Rota–Baxter Leibniz algebras, such that they give rise to triangular Leibniz bialgebras in the case of weight 0, while they are in one-to-one correspondence with factorizable Leibniz bialgebras in the case of nonzero weights.

NECESSARY CONDITION FOR $IA$-AUTOMORPHISMS IN LEIBNIZ ALGEBRAS

Let $F$ be the free Leibniz algebra generated by the set $% X=\{x_{1},...,x_{n}\}$ over the field $K$ of characteristic $0$. consider $R$ as an ideal of $F$. This study initially derives an explicit matrix representation for the $IA$-automorphisms of the Leibniz algebra $F/R^{\prime }$. Subsequently, we establish a necessary condition for an $IA$% -endomorphism of $F/R^{\prime }$ to be an $IA$-automorphism. This method is explicitly based on Dieudonn\'{e} determinant.

Generalized NS-algebras

We generalize the notion of an NS-algebra, which was previously only considered for associative, Lie and Leibniz algebras, to arbitrary categories of binary algebras with one operation. We do this by defining these algebras using a bimodule property, as we did in our earlier work for defining the notions of a dendriform and tridendriform algebra for such categories of algebras. We show that several types of operators lead to NS-algebras: Nijenhuis operators, twisted Rota-Baxter operators and relative Rota-Baxter operators of arbitrary weight. We thus provide a general framework in which several known results and constructions for associative, Lie and Leibniz-NS-algebras are unified, along with some new examples and constructions that we also present.

Almost-reductive and almost-algebraic Leibniz algebra

This paper examines whether the concept of an almost-algebraic Lie algebra developed by Auslander and Brezin in [J. Algebra, 8(1968), 295-313] can be introduced for Leibniz algebras. Two possible analogues are considered: almost-reductive and almost-algebraic Leibniz algebras. For Lie algebras these two concepts are the same, but that is not the case for Leibniz algebras, the class of almost-algebraic Leibniz algebras strictly containing that of the almost-reductive ones. Various properties of these two classes of algebras are obtained, together with some relationships between $\phi$-free, elementary, $E$-algebras and $A$-algebras.

Automorphism groups of some non-nilpotent Leibniz algebras

Let $L$ be an algebra over a field $F$ with the binary operations $+$ and $[,]$. Then $L$ is called a left Leibniz algebra if it satisfies the left Leibniz identity: $[a,[b,c]]=[[a,b],c]+[b,[a,c]]$ for all $a,b,c\in L$. A linear transformation $f$ of $L$ is called an endomorphism of $L$, if $f([a,b])=[f(a),f(b)]$ for all elements $a,b\in L$. A bijective endomorphism of $L$ is called an automorphism of $L$. It is easy to show that the set of all automorphisms of Leibniz algebra is a group with respect to the operation of multiplication of automorphisms. The description of the structure of the automorphism groups of Leibniz algebras is one of the natural and important problems of the general Leibniz algebra theory. The main goal of this article is to describe the structure of the automorphism group of a certain type of non-nilpotent three-dimensional Leibniz algebras.

On Poisson (2-3)-algebras which are finite-dimensional over the center

One of the classic results of group theory is the so-called Schur theorem. It states that if the central factor-group $G/\zeta(G)$ of a group $G$ is finite, then its derived subgroup $[G,G]$ is also finite. This result has numerous generalizations and modifications in group theory. At the same time, similar investigations were conducted in other algebraic structures, namely in modules, linear groups, topological groups, $n$-groups, associative algebras, Lie algebras, Lie $n$-algebras, Lie rings, Leibniz algebras. In 2021, L.A. Kurdachenko, O.O. Pypka and I.Ya. Subbotin proved an analogue of Schur theorem for Poisson algebras: if the center of the Poisson algebra $P$ has finite codimension, then $P$ includes an ideal $K$ of finite dimension such that $P/K$ is abelian. In this paper, we continue similar studies for another algebraic structure. An analogue of Schur theorem for Poisson (2-3)-algebras is proved.

Non-Nilpotent Leibniz Algebras with One-Dimensional Derived Subalgebra

In this paper we study non-nilpotent non-Lie Leibniz F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {F}$$\\end{document}-algebras with one-dimensional derived subalgebra, where F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {F}$$\\end{document} is a field with char(F)≠2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {char}}(\\mathbb {F}) \ e 2$$\\end{document}. We prove that such an algebra is isomorphic to the direct sum of the two-dimensional non-nilpotent non-Lie Leibniz algebra and an abelian algebra. We denote it by Ln\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_n$$\\end{document}, where n=dimFLn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n=\\dim _\\mathbb {F}L_n$$\\end{document}. This generalizes the result found in Demir et al. (Algebras and Representation Theory 19:405-417, 2016), which is only valid when F=C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {F}=\\mathbb {C}$$\\end{document}. Moreover, we find the Lie algebra of derivations, its Lie group of automorphisms and the Leibniz algebra of biderivations of Ln\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_n$$\\end{document}. Eventually, we solve the coquecigrue problem for Ln\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_n$$\\end{document} by integrating it into a Lie rack.

Omni-Representations of Leibniz Algebras

Omni-Representations of Leibniz Algebras

On Inner Derivations of Leibniz Algebras

Leibniz algebras are generalizations of Lie algebras. Similar to Lie algebras, inner derivations play a crucial role in characterizing complete Leibniz algebras. In this work, we demonstrate that the algebra of inner derivations of a Leibniz algebra can be decomposed into the sum of the algebra of left multiplications and a certain ideal. Furthermore, we show that the quotient of the algebra of derivations of the Leibniz algebra by this ideal yields a complete Lie algebra. Our results independently establish that any derivation of a semisimple Leibniz algebra can be expressed as a combination of three derivations. Additionally, we compare the properties of the algebra of inner derivations of Leibniz algebras with the algebra of central derivations.

Jacobi–Lie Models and Supergravity Equations

Abstract Poisson–Lie T-duality/plurality was recently generalized to Jacobi–Lie T-plurality formulated in terms of double field theory and based on Leibniz algebras given by the structure coefficients fabc, fcab, and Za, Za. We investigate three- and four-dimensional sigma models corresponding to six-dimensional Leibniz algebras with fbba ≠ 0, Za = 0. We show that these algebras are plural one to another and, moreover, to an algebra with fbba = 0, Za = 0. These pluralities are used for construction of Jacobi–Lie models. It was conjectured that plural models should satisfy generalized supergravity equations. We have found examples of models satisfying “true” generalized supergravity equations where no trivialization to usual supergravity equations is possible. On the other hand, we show that there are also models corresponding to algebras with fbba ≠ 0, Za = 0 where the Killing vector appearing in generalized supergravity equations either vanishes or can be removed by suitable gauge transformation. Such models then satisfy usual supergravity equations, i.e. vanishing beta-function equations.

On the second relative homology of Leibniz algebras

On the second relative homology of Leibniz algebras

Isotopisms of nilpotent Leibniz algebras and Lie racks

In this paper we study the isotopism classes of two-step nilpotent algebras. We show that every nilpotent Leibniz algebra g with dim [ g , g ] = 1 is isotopic to the Heisenberg Lie algebra or to the Heisenberg algebra l 2 n + 1 J 1 , where J 1 is the n × n Jordan block of eigenvalue 1. We also prove that two such algebras are isotopic if and only if the Lie racks integrating them are isotopic. This gives the classification of Lie racks whose tangent space at the unit element is a nilpotent Leibniz algebra with one-dimensional commutator ideal. Eventually, we introduce new isotopism invariants for Leibniz algebras and Lie racks.

Symplectic structures, product structures and complex structures on Leibniz algebras

In this paper, a symplectic structure on a Leibniz algebra is defined to be a symmetric nondegenerate bilinear form satisfying certain compatibility condition, and a phase space of a Leibniz algebra is defined to be a symplectic Leibniz algebra satisfying certain conditions. We show that a Leibniz algebra has a phase space if and only if there is a compatible Leibniz-dendriform algebra, and phase spaces of Leibniz algebras are one-to-one corresponds to Manin triples of Leibniz-dendriform algebras. Product (paracomplex) structures and complex structures on Leibniz algebras are studied in terms of decompositions of Leibniz algebras. A para-Kähler structure on a Leibniz algebra is defined to be a symplectic structure and a paracomplex structure satisfying a compatibility condition. We show that a symplectic Leibniz algebra admits a para-Kähler structure if and only if the Leibniz algebra is the direct sum of two Lagrangian subalgebras as vector spaces. A complex product structure on a Leibniz algebra consists of a complex structure and a product structure satisfying a compatibility condition. A pseudo-Kähler structure on a Leibniz algebra is defined to be a symplectic structure and a complex structure satisfying a compatibility condition. Various properties and relations of complex product structures and pseudo-Kähler structures are studied. In particular, Leibniz-dendriform algebras give rise to complex product structures and pseudo-Kähler structures naturally.

On classification of $7$-dimensional odd-nilpotent Leibniz algebras

In this paper we extend the method of canonical form for congruence of bilinear forms to give the classification of some subclasses of $7-$dimensional nilpotent Leibniz algebras. Odd-nilpotent Leibniz algebras are defined as that its even dimensional ideals in lower central series are all zero and the classification of $7-$dimensional complex odd-nilpotent Leibniz algebras with one dimensional Leib ideal is obtained by applying the aforementioned method.