Rutwig Campoamor-Stursberg
The invariants of all complex solvable rigid Lie algebras up to dimension eight are computed. Moreover we show, for rank one solvable algebras, some criteria to deduce to non-existence of non-trivial invariants or the existence of fundamental sets of invariants formed by rational functions of the Casimir invariants of the associated nilradical.
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We determine fundamental systems of invariants for complex solvable rigid Lie algebras having nonsplit nilradicals of characteristic sequence $(3,1,..,1)$, these algebras being the natural followers of solvable algebras having Heisenberg nilradicals. A special case of this allows us to obtain a criterion to determine the number of functionally independent invariants of rank one subalgebras of (real or complex) solvable Lie algebras. Finally, we give examples of the inverse procedure, obtaining fundamental systems of an algebra starting from rank one subalgebras.
We finish the determination of the invariants of the coadjoint representation of six dimensional real Lie algebras, by determining a fundamental set of invariants for the 99 isomorphism classes of solvable Lie algebras with five dimensional nilradical. We also give some results on the invariants of solvable Lie algebras in arbitrary dimension whose nilradical has codimension one.
We study Novikov algebras and Novikov structures on finite-dimensional Lie algebras. We show that a Lie algebra admitting a Novikov structure must be solvable. Conversely we present an example of a nilpotent 2-step solvable Lie algebra without any Novikov structure. We construct Novikov structures on certain Lie algebras via classical r-matrices and via extensions. In the latter case we lift Novikov structures on an abelian Lie algebra A and a Lie algebra B to certain extensions of B by A. We apply this to prove the existence of affine and Novikov structures on several classes of 2-step solvab…
All finite-dimensional indecomposable solvable Lie algebras $L(n,f)$, having the triangular algebra T(n) as their nilradical, are constructed. The number of nonnilpotent elements $f$ in $L(n,f)$ satisfies $1\leq f\leq n-1$ and the dimension of the Lie algebra is $\dim L(n,f)=f+{1/2}n(n-1)$.
In this paper, we find a family $SL^{n,m}$, in any arbitrary dimensions, of cohomologically rigid solvable Lie superalgebras with nilradical the model filiform Lie superalgebra $L^{n,m}$. Moreover, we exhibit a family of cohomologically rigid solvable Lie superalgebras with nilradical the model nilpotent Lie superalgebra of generic characteristic sequence. Both cases correspond to solvable Lie superalgebras of maximal dimension for a given nilradical. Contrariwise, we will show that the family of Lie superalgebras $SL^{n,m}$ can be deformed if defined over a field of odd characteristic.
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.
Cohomological rigidity of solvable Lie algebras of maximal ran
本文研究了可解李代数最大范围的上同调刚性，通过分析根系结构和上同调群的非平凡性，提出了一种统一的计算方法，扩展了经典条件并揭示了非刚性李代数的广泛家族。
The author has previously shown that solvable Lie A-algebras and complemented solvable Lie algebras decompose as a vector space direct sum of abelian subalgebras, and their ideals relate nicely to this decomposition. However, neither of these classes is contained in the other. The object of this paper is to find a larger class of algebras, containing each of these classes, in which these same results hold.
The description of complex solvable Leibniz algebras whose nilradical is a naturally graded filiform algebra is already known. Unfortunately, a mistake was made in that description. Namely, in the case where the dimension of the solvable Leibniz algebra with nilradical $F_n^1$ is equal to $n+2$, it was asserted that there is no such algebra. However, it was possible for us to find a unique $(n+2)$-dimensional solvable Leibniz algebra with nilradical $F_n^1$. In addition, we establish the triviality of the second group of cohomology for this algebra with coefficients in itself, which implies it…
Lie affgebra structures on three-dimensional solvable Lie algebras
本文完整分类了三维可解李代数上的复李 affgebra 结构。
The present paper is devoted to the description of rigid solvable Leibniz algebras. In particular, we prove that solvable Leibniz algebras under some conditions on the nilradical are rigid and we describe four-dimensional solvable Leibniz algebras with three-dimensional rigid nilradical. We show that the Grunewald-O'Halloran's conjecture "any $n$-dimensional nilpotent Lie algebra is a degeneration of some algebra of the same dimension" holds for Lie algebras of dimensions less than six and for Leibniz algebras of dimensions less than four. The algebra of level one, which is omitted in the 1991…
We illustrate some simple ideas that can be used for obtaining a classification of small-dimensional solvable Lie algebras.Using these we obtain the classification of 3 and 4 dimensional solvable Lie algebras (over fields of any characteristic). Precise conditions for isomorphism are given.
Lie solvable restricted enveloping algebras were characterized by Riley and Shalev except when the ground field is of characteristic 2. We resolve the characteristic 2 case here which completes the classification. As an application of our result, we obtain a characterization of ordinary Lie algebras over any field whose enveloping algebra is Lie solvable.
On solvable Lie superalgebras of maximal rank
本文研究了李超代数的超导子性质，在给定幂零根的条件下估计了补空间维数上界，并证明了极大秩可解李超代数同构于极大秩幂零根的极大可解扩张。
We show that the isomorphism class of a two-step solvable Lie poset subalgebra of a semisimple Lie algebra is determined by its dimension. We further establish that all such algebras are absolutely rigid.
We give a classification of real solvable Lie algebras whose non-trivial coadjoint orbits of corresponding simply connected Lie groups are all of codimension 2. These Lie algebras belong to a well-known class, called the class of MD-algebras.
Recently in \cite{s-n}, we have investigated Lie algebras and abelian Lie algebras derived from Lie hyperalgebras using the fundamental relations $\mathcal{L}$ and $\mathcal{A}$, respectively. In the present paper, continuing this method we obtain solvable Lie algebras from Lie hyperalgebras by $\mathcal{S}_n$-relations. We show that $\bigcap_{n\geq 1}\mathcal{S}^*_n$ is the smallest equivalence relation on a Lie hyperalgebra such that the quotient structure is a solvable Lie algebra. We also provide some necessary and sufficient conditions for transitivity of the relation $\mathcal{S}_n$ usin…
All finite-dimensional indecomposable solvable Lie algebras g, having the filiform Lie algebra Q_(2m+1) as the nilradical, are studied and classified. It turns out that the dimension of g is at most dimQ_(2m+1)+2.
We study numerical invariants of identities of finite-dimensional solvable Lie superalgebras. We define new series of finite-dimensional solvable Lie superalgebras $L$ with non-nilpotent derived subalgebra $L'$ and discuss their codimension growth. For the first algebra of this series we prove the existence and integrality of $exp(L)$.
In this work we consider extensions of solvable Lie algebras with naturally graded filiform nilradicals. Note that there exist two naturally graded filiform Lie algebras $n_{n, 1}$ and $Q_{2n}.$ We find all one-dimensional central extensions of the algebra $n_{n, 1}$ and show that any extension of $Q_{2n}$ is split. After that we find one-dimensional extensions of solvable Lie algebras with nilradical $n_{n, 1}$. We prove that there exists a unique non-split central extension of solvable Lie algebras with nilradical $n_{n, 1}$ of maximal codimension. Moreover, all one-dimensional extensions of…
The goal of this paper is to show that many key results found in the study of Einstein Lorentzian nilpotent Lie algebras can still hold in the more general settings of unimodular Lie algebras and (completely) solvable Lie algebras.
The Solvabilizers and Solvable Graphs in Lie Superalgebras
本文研究了李超代数的可解化剂与可解图，探讨了其作为同构不变量的应用及可解性度量的定义。
We show that in the class of solvable Lie algebras there exist algebras which admit local derivations which are not ordinary derivation and also algebras for which every local derivation is a derivation. We found necessary and sufficient conditions under which any local derivation of solvable Lie algebras with abelian nilradical and one-dimensional complementary space is a derivation. Moreover, we prove that every local derivation on a finite-dimensional solvable Lie algebra with model nilradical and maximal dimension of complementary space is a derivation.
Extraction Algorithm of Hom-Lie Algebras Based on Solvable and Nilpotent Groups
本文借鉴群论中成熟的可解群、幂零群性质研究，构建了对应的可解与幂零Hom-Lie代数理论，并辅以示例说明，为Hom-Lie代数研究提供了新方向。
In the present paper we prove the existence outer derivations finite-dimensional solvable Lie algebras with nilradical of maximal rank and complementary subspace to nilardical of dimension less than rank of the nilradical.
In this paper, we classify solvable quadratic Lie algebras up to dimension 6. In dimensions smaller than 6, we use the Witt decomposition given in \cite{Bou59} and a result in \cite{PU07} to obtain two non-Abelian indecomposable solvable quadratic Lie algebras. In the case of dimension 6, by applying the method of double extension given in \cite{Kac85} and \cite{MR85} and the classification result of singular quadratic Lie algebras in \cite{DPU}, we have three families of solvable quadratic Lie algebras which are indecomposable and not isomorphic.
We give a characterization of symplectic quadratic Lie algebras that their Lie algebra of inner derivations has an invertible derivation. A family of symplectic quadratic Lie algebras is introduced to illustrate this situation. Finally, we calculate explicitly the Betti numbers of a family of solvable Lie algebras in two ways: using the cohomology of quadratic Lie algebras and applying a Pouseele's result on extensions of the one-dimensional Lie algebra by Heisenberg Lie algebras
Along this paper we show that under certain conditions the method for describing of solvable Lie and Leibniz algebras with maximal codimension of nilradical is also extensible to Lie and Leibniz superalgebras, respectively. In particular, we totally determine the solvable Lie and Leibniz superalgebras with maximal codimension of model filiform and model nilpotent nilradicals. Finally, it is established that the superderivations of the obtained superalgebras are inner.
In this paper, we classify solvable Lie algebras of dimensions $\leq 8$ endowed with a nondegenerate invariant symmetric bilinear form over an algebraically closed field. This classification (up to isometrically isomorphisms) is mainly based on the double extension method.
The purpose of this paper is to consider when two maximal subalgebras of a finite-dimensional solvable Lie algebra $L$ are conjugate, and to investigate their intersection.
In this paper solvable Leibniz algebras with naturally graded non-Lie $p$-filiform $(n-p\geq4)$ nilradical and with one-dimensional complemented space of nilradical are described. Moreover, solvable Leibniz algebras with abelian nilradical and extremal (minimal, maximal) dimensions of complemented space nilradical are studied. The rigidity of solvable Leibniz algebras with abelian nilradical and maximal dimension of its complemented space is proved.
We study solvable Lie algebras in prime characteristic $p$ that admit non-singular derivations. We show that Jacobson's Theorem remains true if the quotients of the derived series have dimension less than~$p$. We also study the structure of Lie algebras with non-singular derivations in which the derived subalgebra is abelian and has codimension~one. The paper presents some new examples of solvable, but not nilpotent, Lie algebras of derived length~3 with non-singular derivations.
On maximal solvable extensions of nilpotent Lie algebras
本文完整刻画了一类幂零李代数的极大可解扩张，证明其在d-局部可对角化条件下唯一并实现为半直积，解决Šnobl猜想，并推广至李超代数。
Codimensions of identities of solvable Lie superalgebras
该研究针对特征零域上的李超代数恒等式问题，构造了换位子代数非幂零的有限维可解李超代数实例，证明了其PI指数存在且为整数。
Transposed Poisson structures on complex Galilean type Lie algebras and superalgebras are described. It was proven that all principal Galilean Lie algebras do not have non-trivial $\frac{1}{2}$-derivations and as it follows they do not admit non-trivial transposed Poisson structures. Also, we proved that each complex finite-dimensional solvable Lie algebra admits a non-trivial transposed Poisson structure and a non-trivial ${\rm Hom}$-Lie structure.
On solvable Lie algebras of small breadth
本文研究了有限维可解李代数中广度小于等于2的结构，并对复数域上纯且非 nilpotent 的李代数进行了完整分类。
In this paper we introduce the notion of pure non-characteristically nilpotent Lie algebra and under a condition we prove that a complex maximal extension of a finite-dimensional pure non-characteristically nilpotent Lie algebra is isomorphic to a semidirect sum of the nilradical and its maximal torus. We also prove that such solvable Lie algebras are complete and we specify a subclass of the maximal solvable extensions of pure non-characteristically nilpotent Lie algebras that have trivial cohomology group. Some comparisons with the results obtained earlier are given.
The paper is devoted to the study of pro-solvable Lie algebras whose maximal pro-nilpotent ideal is either $\mathfrak{m}_0$ or $\mathfrak{m}_2$. Namely, we describe such Lie algebras and establish their completeness. Triviality of the second cohomology group for one of the obtained algebra is established.
Semisimple Lie algebras have been completely classified by Cartan and Killing. The Levi theorem states that every finite dimensional Lie algebra is isomorphic to a semidirect sum of its largest solvable ideal and a semisimple Lie algebra. These focus the classification of solvable Lie algebras as one of the main challenges of Lie algebra research. One approach towards this task is to take a class of nilpotent Lie algebras and construct all extensions of these algebras to solvable ones. In this paper, we propose another approach, i.e., to decompose a solvable nonnilpotent Lie algebra to two nil…
This paper presents a classification of 7-dimensional real and complex indecomposable solvable Lie algebras having some 5-dimensional nilradicals. Afterwards, we combine our results with those of Rubin and Winternitz (1993), Ndogmo and Winternitz (1994), Snobl and Winternitz (2005, 2009), Snobl and Karásek (2010) to obtain a complete classification of 7-dimensional real and complex indecomposable solvable Lie algebras with 5-dimensional nilradicals. In association with Gong (1998), Parry (2007), Hindeleh and Thompson (2008), we achieve a classification of 7-dimensional real and complex indecom…
Transposed Poisson structures on solvable Lie algebras with filiform nilradical
该研究利用具有线状幂零根基的可解李代数的 1/2-导子，建立交换结合乘法，从而构造出关联给定李代数的非平凡转置 Poisson 代数。
We compare the maximal dimension of abelian subalgebras and the maximal dimension of abelian ideals for finite-dimensional Lie algebras. We show that these dimensions coincide for solvable Lie algebras over an algebraically closed field of characteristic zero. We compute this invariant for all complex nilpotent Lie algebras of dimension n less than 8. Furthermore we study the case where there exists an abelian subalgebra of codimension 2. Here we explicitly construct an abelian ideal of codimension 2 in case of nilpotent Lie algebras.
Radicals of Lie-solvable Novikov algebras
研究李可解诺维科夫代数，证明其贝尔根与右幂零元集重合、安德拉诺维奇根与最大左拟正则理想重合，还探讨应用格尔范德 - 多尔夫曼构造后带导子交换代数性质的稳定性。
This paper concerns the problem of classifying finite-dimensional real solvable Lie algebras whose derived algebras are of codimension 1 or 2. On the one hand, we present an effective method to classify all $(n+1)$-dimensional real solvable Lie algebras having 1-codimensional derived algebras provided that a full classification of $n$-dimensional nilpotent Lie algebras is given. On the other hand, the problem of classifying all $(n+2)$-dimensional real solvable Lie algebras having 2-codimensional derived algebras is proved to be wild. In this case, we provide a method to classify a subclass of…
This paper is a contribution to the isomorphism problem for universal enveloping algebras of finite-dimensional Lie algebras. We focus on solvable Lie algebras of small dimensions over fields of arbitrary characteristic. We prove, over an arbitrary field, that the isomorphism type of a metabelian Lie algebra whose derived subalgebra has codimension one is determined by its universal enveloping algebra. As an application of the results in this paper, we solve the isomorphism problem for solvable Lie algebras of dimension four over fields of characteristic zero and also point out the problems th…
In this paper we study a Lie-theoretic analogue of a generalisation of the prefrattini subgroups introduced by W. Gaschütz. The approach follows that of P. Hauck and H. Kurtzweil for groups, by first considering complements in subalgebra intervals. Conjugacy of these subalgebras is established for a large class of solvable lie algebras.
In this paper, we first study properties of the lower central chains for Novikov algebras. Then we show that for every Lie nilpotent Novikov algebra~$\mathcal{N}$, the ideal of~$\mathcal{N}$ generated by the set~$\{ab - ba\mid a, b\in \mathcal{N}\}$ is nilpotent. We secondly provide necessary and sufficient conditions on the graph $E$ and the field $K$ for which the Leavitt path algebra $L_K(E)$ is Lie solvable. Consequently, we obtain a complete description of Lie nilpotent Leavitt path algebras, and show that the Lie solvability of~$L_K(E)$ and the Lie nilpotency of $[L_K(E),L_K(E)]$ are the…
Infinite dimensional analogues of nilpotent and solvable Lie algebras
该文研究幂零与可解李代数的无穷维类似物，推广经典定理与秩的概念，构造特定剩余可解扩张，证明极大秩投射幂零代数的极大剩余可解扩张的导子均为内导子，还给出保持投射幂零性的标准构造。
In this paper solvable Leibniz algebras whose nilradical is quasi-filiform Lie algebra of maximum length, are classified. The rigidity of such Leibniz algebras with two-dimensional complemented space to nilradical is proved.
Solvable Quadratic Lie algebras up to dimension 7
本文借助双扩张方法，对特征不为2的任意域上维数不超过7的可解不可分解二次李代数，在代数闭域和实闭域上完成了等距同构意义下的完整分类。
Zeta functions of solvable Lie algebras over finite fields -- with calculations in detail
本文研究了有限域上维数小于等于4的可解李代数的子代数和理想计数的ζ函数，并探讨了其在李代数理论中的意义。
We exhibit a Cayley-Hamilton trace identity for $2\times2$ matrices with entries in a ring $R$ satisfying $[[x,y],[x,z]]=0$ and 1/2 \in R$.
By definition, a quadratic Lie superalgebra is a Lie superalgebra endowed with a non-degenerate supersymmetric bilinear form which satisfies the even and invariant properties. In this paper we calculate all of the second cohomology group of elementary quadratic Lie superalgebras which have been classified in \cite{DU14} by applying the super-Poisson bracket on the super exterior algebra. Besides, we give the classification of 8-dimensional solvable quadratic Lie superalgebras having 6-dimensional indecomposable even part. The method is based on the double extension and classification results o…
Fixed-point-free automorphisms of solvable Lie algebras
研究可解李代数中无不动点自同构的存在性，证明此类李代数必为强单模的，且为复近近阿贝尔李代数存在无不动点自同构的充要条件，复filiform李代数中无不动点自同构等价于非特征 nilpotent。
By using the concept of weight graph associated to certain nilpotent Lie algebras $\frak{g}$, we find necessary and sufficient conditions for a semidirect product $\frak{g}\oplus T_{i}$, where $T_{i}<T$ is a subalgebra of a maximal torus of derivations $T$ of $\frak{g}$ which induces a decomposition of $\frak{g}$ into one dimensional weight spaces, to be 2-step solvable. In particular we show that the semidirect product of such a Lie algebra with its torus of derivations cannot be itself 2-step solvable.
Let $L$ be a finite dimensional Lie algebra over a field $F$. It is well known that the solvable radical $S(L)$ of the algebra $L$ is a characteristic ideal of $L$ if $\char F=0$ and there are counterexamples to this statement in case $\char F=p>0$. We prove that the sum $S(L)$ of all solvable ideals of a Lie algebra $L$ (not necessarily finite dimensional) is a characteristic ideal of $L$ in the following cases: 1) $\char F=0;$ 2) $S(L)$ is solvable and its derived length is less than $\log_{2}p.$ Some estimations (in characteristic 0) for the derived length of ideals $I+D(I)+... +D^{k}(I)$ a…
The present article is a part of the study of solvable Leibniz algebras with a given nilradical. In this paper solvable Leibniz algebras, whose nilradicals is naturally graded $p$-filiform non-Lie Leibniz algebra $(n-p\geq4)$ and the complemented space to nilradical has maximal dimension, are described up to isomorphism. Moreover, among obtained algebras we indicate the rigid and complete algebras
In this paper a characterisation is given of solvable complemented Lie algebras. They decompose as a direct sum of abelian subalgebras and their ideals relate nicely to this decomposition. The class of such algebras is shown to be a formation whose residual is the ideal closure of the prefrattini subalgebras.
Transposed Poisson structures on solvable and perfect Lie algebras
本文分类描述了振荡李代数等多种李代数上的所有转置Poisson代数结构，并构造性地回答了关于转置Poisson代数的一个遗留问题，找到了一个具有非平凡$\ rac{1}{2}$-导数但无非平凡转置Poisson结构的有限维李代数。
We give the description of three-dimensional Lie triple systems and their corresponding Lie algebras with invomorphisme, The description of three-dimensional Bol algebras linked with the distinguished Lie triple systems above is given. The construction of the object describing the isotopy of Bol algebras. The description of Bol 3-Webs, connected with the selected Bol algebras.
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