Factorization of loop algebras over so(4) and integrable nonlinear differential equations

We consider factoring subalgebras for loop algebras over so(4). Given a factoring subalgebra, we find (in terms of coefficients of commutator relations) an explicit form of (1) the corresponding system of the chiral-field-equation type, (2) the corresponding two-spin model of the Landau-Lifshitz equation, and (3) the corresponding Hamiltonian system of ordinary differential equations with homogeneous quadratic Hamiltonian and linear so(4)-Poisson brackets. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 3, pp. 79–94, 2005.     

Conjugation-uniqueness of exact Borel subalgebras

It is proved that the exact Borel subalgebras of a basic quasi-hereditary algebra are conjugate to each other. Moreover, the inner automorphism group of a basic quasi-hereditary algebra acts transitively on the set of its exact Borel subalgebras. Project supported by the National Natural Science Foundation of China (Grant No. 19771070). and partly supported by the NSF of Hainan Province (Grant No. 19702) and by the Natural Science Foundation of Education Department of Hainan province.     

Involutions of Iwahori–Hecke Algebras and Representations of Fixed Subalgebras

We establish branching rules between some Iwahori–Hecke algebra of type B and their subalgebras which are defined as fixed subalgebras by involutions including Goldman involution. The Iwahori–Hecke algebra of type D is one of such fixed subalgebras. We also obtain branching rules between those fixed subalgebras and their intersection subalgebra. We determine basic sets of irreducible representations of those fixed subalgebras and their intersection by making use of generalized Clifford theory.     

On the normalizers of subalgebras in an infinite Lie algebra

An algorithm for calculating the normalizer of subalgebra in an infinite Lie symmetry algebra is proposed. The classification problem for a subalgebra spanned by generators that depend on arbitrary functions is formulated. This problem lies in finding the specifications of arbitrary functions and calculating the normalizers of the subalgebras so obtained. As an example, we consider the Lie symmetry algebra L admitted by the thermal diffusion equations. The first-order optimal system of subalgebras Θ₁L is constructed and the normalizers of finite subalgebras from this system are found. The classification of subalgebras depending on arbitrary functions is made.     

Invariant operators of four-dimensional nonconjugate subalgebras of the Lie algebra of the Poincaré group <Emphasis Type="Italic">P</Emphasis>(1,4)

The classification of four-dimensional nonconjugate subalgebras of the Lie algebra of the Poincare group P(1, 4) into classes of isomorphic subalgebras is performed. Using this classification, we construct invariant operators (generalized Casimir operators) [30] for all four-dimensional nonconjugate subalgebras of the Lie algebra of the group P(1, 4) and present them in the explicit form.     

Computational Experiments with Nilpotent Lie Algebras

Results of computer experiments on the study of properties of generic Lie subalgebras with two generators in the Lie algebra of nilpotent matrices whose order does not exceed 10 are presented. The calculations carried out have made it possible to formulate several statements (so-called virtual theorems) on properties of the Lie subalgebras in question. The dimensions of the lower and upper central series and of the series of commutator subalgebras and the characteristic nilpotency property of the Lie subalgebras constructed here and of generic Lie subalgebras of codimension 1 in these Lie subalgebras are studied.

     

The Ideal Index of Subalgebras of a Finite Dimensional Lie Algebra

D. A. Towers introduced the notion of ideal index of a maximal subalgebra of a Lie algebra, and used it to analyze the influence of maximal subalgebras on the structure of a finite dimensional Lie algebras.

In this article, we generalize the ideal index from maximal subalgebras to all subalgebras, and obtain some new characterizations of solvable and supersolvable Lie algebras by the ideal indices of some certain subalgebras.     

Cartan Subalgebras of Leibniz n-Algebras

The present article is devoted to the investigation of properties of Cartan subalgebras and regular elements in Leibniz n-algebras. The relationship between Cartan subalgebras and regular elements of given Leibniz n-algebra and Cartan subalgebras and regular elements of the corresponding factor n-Lie algebra is established.     

Partially-2-Homogeneous Monounary Algebras

This paper is a continuation of [5], where k-homogeneous and k-set-homogeneous algebras were defined. The definitions are analogous to those introduced by Fraïssé [2] and Droste, Giraudet, Macpherson, Sauer [1] for relational structures. In [5] we found all 2-homogeneous and all 2-set-homogeneous monounary algebras when the homogenity is considered with respect to subalgebras, to connected subalgebras and with respect to connected partial subalgebras, respectively. The results of [3], where all homogeneous monounary algebras were characterized, were applied in [4] for 1-homogeneity.The aim of the present paper is to describe all monounary algebras which are 2-homogeneous and 2-set-homogeneous with respect to partial subalgebras, respectively; we will say that they are partially-2-homogeneous and partially-2-set-homogeneous.     

Poisson-Nijenhuis structures and the Vinogradov bracket

We express the compatibility conditions that a Poisson bivector and a Nijenhuis tensor must fulfil in order to be a Poisson-Nijenhuis structure by means of a graded Lie bracket. This bracket is a generalization of Schouten and Frölicher-Nijenhuis graded Lie brackets defined on multivector fields and on vector valued differential forms respectively.Partially supported by Fundació Caixa Castelló.Partially supported by the Spanish DGICYT grant #P B91-0324.     

Grassmann manifold of aJH-algebra

We study the set of rankp idempotents in a topologically simple Hilbert Jordan algebra (JH-algebra for short). To produce the differential geometric structure on, we establish Jordan algebraic results concerning the structure of some two-generator subalgebras. We identify geodesics, the Riemannian distance and the sectional curvature of by using the Jordan algebraic structure.     

On 2-Homogeneity of Monounary Algebras

Fraïssé introduced the notion of a k-set-homogeneous relational structure. In the present paper the following classes of monounary algebras are described: --the class of all algebras which are 2-set-homogeneous with respect to subalgebras, connected subalgebras, connected partial subalgebras, respectively, and --the class of all algebras which are 2-homogeneous with respect to subalgebras, connected subalgebras, connected partial subalgebras, respectively.     

Invariant complex structures on four-dimensional solvable real lie groups

In this paper, invariant complex structures on four-dimensional, solvable, simply-connected real Lie groups are classified where the dimension of the commutator is less than three. The resulting complex surfaces corresponding to these structures are also determined. The classification problem is reduced to determining certain complex “structure” subalgebras of the complexifications of the four-dimensional, solvable real Lie algebras. Most of the eleven types of non-abelian solvable real Lie algebras do have complex structure subalgebras; three do not. Only three types of algebras have solvable complex structure subalgebras, and only one possesses both abelian and solvable complex structure subalgebras. Each of the possible homogeneous surfaces is represented in the list of resulting manifolds.     

布尔代数的直觉T-S模糊子代数及理想

在布尔代数中引入了的直觉T-S模糊子代数和直觉T-S模糊理想的概念,给出了布尔代数的直觉T-S模糊子代数的两个等价定义,进一步讨论了它们的性质.证明了布尔代数的两个直觉T-S模糊子代数(理想)的模交与直积也是直觉T-S模糊子代数(理想).     

On Limits of Mishchenko--Fomenko Subalgebras in Poisson Algebras of Semisimple Lie Algebras

In this paper, the commutative (with respect to the Poisson bracket) subalgebras in the Poisson algebras of the semisimple Lie algebras are considered on condition that these subalgebras are limits of Mishchenko--Fomenko subalgebras. We study the case of the degeneration within a fixed Cartan subalgebra. The structure of the limit subalgebras is described (i.e., it is proved that these subalgebras are free, and their generators are found). The classification of the limit subalgebras of the above type is also established.     

On classical artinian localizations

ABSTRACT

We present some results about Lie algebras, which can be written as the sum of two subalgebras in two cases: where both subalgebras are simple or both are nilpotent. In the first case we suggest new examples of simple Lie algebras admitting decomposition into the sum of simple subalgebras and give explicit realizations where the existence of such decompositions was established earlier. We single out cases where such decomposition is not possible. We also construct examples of solvable Lie algebras, which are the sums of two nilpotent subalgebras, and the derived length of the sum is greater than the sum of the nilpotent indexes of the summands.     

Strong conjugation actions induced by flags of exact Borel subalgebras

Let A be a quasi-hereditary algebra with a strong exact Borel subalgebra. It is proved that for any standard semisimple subalgebra T there exists an exact Borel subalgebra B of A such that T is a maximal semisimple subalgebra of B. It is shown that the maximal length of flags of exact Borel subalgebras of A is the difference of the radium and the rank of Grothendic group of A plus 2. The number of conjugation-classes of exact Borel subalgebras is 1 if and only if A is basic; the number is 2 if and only if A is semisimple. For all other cases, this number is 0 or no less than 3. Furthermore, it is shown that all the exact Borel subalgebras are idempotent-conjugate to each other, that is, for any exact Borel subalgebras B and C of A, there exists an idempotent e of A, and an invertible element u of A, such that eBe = u^-1eCeu.     

格蕴涵代数的区间值模糊子代数

将区间值模糊集的概念应用于格蕴涵代数,引入区间值模糊格蕴涵子代数的概念并研究它们的性质.讨论了区间值模糊格蕴涵子代数与（模糊）格蕴涵子代数之间的关系;定义了区间值模糊集的象和原象,获得了区间值模糊格蕴涵子代数的象和原象成为区间值模糊格蕴涵子代数的条件.     

Leibniz n-超代数的Frattini-子代数

给出了Leibniz n-超代数的Frattini-子代数的一些重要性质,确定了Leibniz n-超代数的Frattini-子代数的分解定理,并且利用所得到的Frattini-子代数的重要性质,Leibniz n-超代数是幂零的一个必要条件被给出.