Filiform李代数Q_n的Hom-结构

首先证明了有限维Z-阶化李代数上的一个线性算子是Hom-结构的充分必要条件,即它的每个齐次分支也是Hom-结构.然后计算了特征零代数闭域上一类有限维Z-阶化Filiform李代数Qn的齐次Hom-结构,从而决定了Qn的所有Hom-结构.     

Simplicity of quadratic Lie conformal algebras

In this paper, simplicity of quadratic Lie conformal algebras is investigated. From the view point of the corresponding Gel’fand–Dorfman bialgebras, some su?cient conditions and necessary conditions to ensure simplicity of quadratic Lie conformal algebras are presented. By these observations, we present several new classes of infinite simple Lie conformal algebras. These results will be useful for classification purposes.     

Finite-dimensional solvable Lie algebras generated by normal operators are commutative

It is proved that any finite-dimensional solvable Lie algebra generated by normal operators is commutative, and some other related results are given.     

Two generator subalgebras of Lie algebras

In [Thompson, J., 1968, Non-solvable finite groups all of whose local subgroups are solvable. Bulletin of the American Mathematical Society, 74, 383–437.], Thompson showed that a finite group G is solvable if and only if every two-generated subgroup is solvable (Corollary 2, p. 388). Recently, Grunevald et al. [Grunewald et al., 2000, Two-variable identities in groups and Lie algebras. Rossiiskaya Akademiya Nauk POMI, 272, 161–176; 2003. Journal of Mathematical Sciences (New York), 116, 2972–2981.] have shown that the analogue holds for finite-dimensional Lie algebras over infinite fields of characteristic greater than 5. It is a natural question to ask to what extent the two-generated subalgebras determine the structure of the algebra. It is to this question that this article is addressed. Here, we consider the classes of strongly-solvable and of supersolvable Lie algebras, and the property of triangulability.     

Filiform李超代数L_(1,2)上的Yang-Baxter方程

求解一般Filiform李超代数L_(n,m)的Yang-Baxter方程尚无一般方法.通过计算,刻画了特征零的代数闭域上四维Filiform李超代数L_(1,2)上的所有Yang-Baxter方程的解.     

Novikov-Poisson algebras and associative commutative derivation algebras

We describe Novikov-Poisson algebras in which a Novikov algebra is not simple while its corresponding associative commutative derivation algebra is differentially simple. In particular, it is proved that a Novikov algebra is simple over a field of characteristic not 2 iff its associative commutative derivation algebra is differentially simple. The relationship is established between Novikov-Poisson algebras and Jordan superalgebras. Supported by RFBR (grant No. 05-01-00230), by SB RAS (Integration project No. 1.9), and by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (project NSh-344.2008.1). __________ Translated from Algebra i Logika, Vol. 47, No. 2, pp. 186–202, March–April, 2008.     

A Sufficient and Necessary Condition for a Solvable Lie Algebra to Be Complete

In this paper,we give a sufficient and neccesary condition under which a solvable Lie algebra is complete.     

四维Filiform李超代数的Yang-Baxter方程

利用复数域C上四维Filiform李超代数的分类,通过计算刻画了复数域C上四维Filiform李超代数上的所有Yang-Baxter方程的解.     

Characterizing n-Isoclinism Classes of Lie Algebras

In this article, we introduce the notion of the equivalence relation, n-isoclinism, between Lie algebras, and obtain some criterions under which Lie algebras are n-isoclinic. In particular, we show that n-isoclinic Lie algebras can be isoclinically embedded into one Lie algebra. Also, we present the notion of an n-stem Lie algebra and prove its existence within an arbitrary n-isoclinism class. In addition, similar to a result of Hekster [6 Hekster , N. S. ( 1986 ). On the structure of n-isoclinam classes of groups . J. Pure Appl. Algebra 40 : 63 – 85 .[Crossref], [Web of Science ®] , [Google Scholar]] in the group case, we characterize the n-stem Lie algebras in the n-isoclinism classes which contains at least one finitely generated Lie algebra L with dim (L ⁿ⁺¹) finite.     

A Class of Solvable Lie Algebras with Triangular Decompositions

So far there has been elementary proof for Frobenius's theorem only in special cases: if the complement is solvable, see e.g. [3], if the complement is of even order, see e.g. [6]. In the first section we consider the case, when the order of the complement is odd. We define a graph the vertices of which are the set K^# of elements of our Frobenius group with 0 fixed points. Two vertices are connected with an edge if and only if the corresponding elements commute. We prove with elementary methods that K is a normal subgroup in G if and only if there exists an element x in K^# such that all elements of K^# belonging to the connected component C of K^# containing x are at most distance 2 from c and N^G(C) is not a -group, where is the set of prime divisors of the Frobenius complement of G. In the second section we generalize the case when the order of the complement is even, proving that the Frobenius kernel is a normal subgroup, if a fixed element a of the complement, the order of which is a minimal prime divisor of the order of the complement, generates a solvable subgroup together with any ofits conjugates. In the third section we prove a generalization of the Glauberman-Thompson normal p-complement theorem, and using this wegive another sufficient condition for the Frobenius kernel to be a normal subgroup for |G| odd, namely we prove this under the conditionthat all the Sylow normalizers in G intersect some of the complements     

Linearly compact algebraic Lie algebras and coalgebraic Lie coalgebras

It is proved that if the dual Lie algebra of a Lie coalgebra is algebraic, then it is algebraic of bounded degree. This result is an analog of the D.Radford's result for associative coalgebras.

     

Characterizations for Supersolvable Lie Algebras

In (2009), Towers [10 Towers , D. A. ( 2009 ). C-ideals of Lie algebras . Comm. Algebra 37 : 4366 – 4373 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] presented the notion of c-ideality of a subalgebra of a Lie algebra, and gave some characterizations of solvable and supersolvable Lie algebras. In this article, we further investigate the influence of c-ideality of some subalgebras on the structure of Lie algebras. We also obtain some equivalent conditions for supersolvability of a finite dimensional Lie algebra.     

Combinatorial structures and Lie algebras of upper triangular matrices

This work shows how to associate the Lie algebra h_n, of upper triangular matrices, with a specific combinatorial structure of dimension 2, for n∈N. The properties of this structure are analyzed and characterized. Additionally, the results obtained here are applied to obtain faithful representations of solvable Lie algebras.     

Lie algebras with few centralizers

We will characterize all finite dimensional Lie algebras with at most |F|²+|F|+2 centralizers, where F is the underlying field of Lie algebras under consideration.     

Solvable Indecomposable Extensions of Two Nilpotent Lie Algebras

A pair of sequences of nilpotent Lie algebras denoted by N_{n, 7} and N_{n, 16} are introduced. Here, n denotes the dimension of the algebras that are defined for n ≥ 6; the first terms in the sequences are denoted by 6.7 and 6.16, respectively, in the standard list of six-dimensional Lie algebras. For each of them, all possible solvable extensions are constructed so that N_{n, 7} and N_{n, 16} serve as the nilradical of the corresponding solvable algebras. The construction continues Winternitz’ and colleagues’ program of investigating solvable Lie algebras using special properties rather than trying to extend one dimension at a time.     

Lie algebras and Lie super algebra for the integrable couplings of NLS–MKdV hierarchy

Lie algebras and Lie super algebra are constructed and integrable couplings of NLS–MKdV hierarchy are obtained. Furthermore, its Hamiltonian and Super-Hamiltonian are presented by using of quadric-form identity and super-trace identity. The method can be used to produce the Hamiltonian structures of the other integrable and super-integrable systems.