Pair Of Lie Algebras Research Articles

Post-Lie algebra structures for perfect Lie algebras

We study the existence of post-Lie algebra structures on pairs of Lie algebras ( g , n ) , where one of the algebras is perfect non-semisimple, and the other one is abelian, nilpotent non-abelian, solvable non-nilpotent, simple, semisimple non-simple, reductive non-semisimple or complete non-perfect. We prove several nonexistence results, but also provide examples in some cases for the existence of a post-Lie algebra structure. Among other results we show that there is no post-Lie algebra structure on ( g , n ) , where g is perfect non-semisimple, and n is s l 3 ( C ) . We also show that there is no post-Lie algebra structure on ( g , n ) , where g is perfect and n is reductive with a 1-dimensional center.

Polynilpotent capability of a pair of Lie algebras

The notion of capability for pairs of groups was defined by Ellis in 1996. Also, the concept of capability of Lie algebras was introduced by some authors, and is analogous to the group case. In this paper, we generalize the concept of capability of a Lie algebra L to the polynilpotent capability of a pair (N,L) of Lie algebras and introduce a criterion for polynilpotent capability of (N,L).

Rigidity results for Lie algebras admitting a post-Lie algebra structure

We study rigidity questions for pairs of Lie algebras [Formula: see text] admitting a post-Lie algebra structure. We show that if [Formula: see text] is semisimple and [Formula: see text] is arbitrary, then we have rigidity in the sense that [Formula: see text] and [Formula: see text] must be isomorphic. The proof uses a result on the decomposition of a Lie algebra [Formula: see text] as the direct vector space sum of two semisimple subalgebras. We show that [Formula: see text] must be semisimple and hence isomorphic to the direct Lie algebra sum [Formula: see text]. This solves some open existence questions for post-Lie algebra structures on pairs of Lie algebras [Formula: see text]. We prove additional existence results for pairs [Formula: see text], where [Formula: see text] is complete, and for pairs, where [Formula: see text] is reductive with [Formula: see text]-dimensional center and [Formula: see text] is solvable or nilpotent.

Diagonal reduction algebra for $$\mathfrak{osp}(1|2)$$

The problem of providing complete presentations of reduction algebras associated to a pair of Lie algebras $$( \mathfrak{G} , \mathfrak{g} )$$ has previously been considered by Khoroshkin and Ogievetsky in the case of the diagonal reduction algebra for $$\mathfrak{gl}(n)$$ . In this paper, we consider the diagonal reduction algebra of the pair of Lie superalgebras $$( \mathfrak{osp}(1|2) \times \mathfrak{osp}(1|2) , \mathfrak{osp}(1|2) )$$ as a double coset space having an associative $$ \mathrel{\scriptstyle\lozenge} $$ -product and give a complete presentation in terms of generators and relations. We also provide a PBW basis for this reduction algebra along with Casimir-like elements and a subgroup of automorphisms.

On the $c$-nilpotent multiplier of a pair of Lie algebras

On the $c$-nilpotent multiplier of a pair of Lie algebras

Some properties of the c-nilpotent multiplier of a pair of Lie algebras

The concept of the Schur multiplier of a Lie algebra was introduced by Batten et al. (1996). This concept was extended to that of a c-nilpotent multiplier of a Lie algebra, and then further extended to a theory of a c-nilpotent multiplier of a pair of Lie algebras, by the author and others. In this paper we present some new inequalities for the dimension of the c-nilpotent multiplier of a pair of Lie algebras. We also give a necessary and sufficient condition for the c-nilpotent multiplier of a pair of Lie algebras to embed into the c-nilpotent multiplier of their quotient. Finally we provide a sufficient condition for the c-nilpotent multiplier of a pair of Lie algebras to be finite dimensional.

On characterizing pairs of nilpotent Lie algebras by their second relative homologies

Let (L,N) be a pair of Lie algebras, in which N is an ideal of L. We prove if dim⁡(N)=n and dim⁡(L/N)=m, then the dimension of the second relative homology of (L,N) is equal to 12n(n+2m−1)−t, for some non-negative integer t. Also, we characterize all pairs of nilpotent Lie algebras for which t=0,1,2,3.

Some upper bounds for the dimension of the c-nilpotent multiplier of a pair of Lie algebras

Some upper bounds for the dimension of the c-nilpotent multiplier of a pair of Lie algebras

Atiyah Classes of Strongly Homotopy Lie Pairs

The subject of this paper is strongly homotopy (SH) Lie algebras, also known as L∞-algebras. We extract an intrinsic character, the Atiyah class, which measures the nontriviality of an SH Lie algebra A when it is extended to L. In fact, with such an SH Lie pair (L, A) and any A-module E, there is associated a canonical cohomology class, the Atiyah class [αE], which generalizes the earlier known Atiyah classes out of Lie algebra pairs. We show that the Atiyah class [αL/A] induces a graded Lie algebra structure on [Formula: see text], and the Atiyah class [αE] of any A-module E induces a Lie algebra module structure on [Formula: see text]. Moreover, Atiyah classes are invariant under gauge equivalent A-compatible infinitesimal deformations of L.

Isoclinic relative extensions of pairs of Lie algebras

In the present paper we introduce the concept of isoclinism on the relative central extensions of pairs of Lie algebras, which forms an equivalence relation and we show that each equivalence class contains a stem relative central extension. Also, we determine the structure of all relative central extensions in an isoclinism family of a given relative central extension. Moreover, we show that under some conditions, two concepts of isoclinism and isomorphism between the relative central extensions of pairs of finite dimensional Lie algebras are coincide.

On stem covers and the universal central extensions of Lie crossed modules

In this article, we investigate stem covers and the universal central extensions of Lie crossed modules. In particular, a cover for the direct sum of two Lie crossed modules in term of given covers of them will be constructed. Using the results obtained, we give some properties of covers and the universal relative central extensions of pairs of Lie algebras.

Rota–Baxter operators and post-Lie algebra structures on semisimple Lie algebras

Rota–Baxter operators R of weight 1 on are in bijective correspondence to post-Lie algebra structures on pairs , where is complete. We use such Rota–Baxter operators to study the existence and classification of post-Lie algebra structures on pairs of Lie algebras , where is semisimple. We show that for semisimple and , with or simple, the existence of a post-Lie algebra structure on such a pair implies that and are isomorphic, and hence both simple. If is semisimple, but is not, it becomes much harder to classify post-Lie algebra structures on , or even to determine the Lie algebras which can arise. Here only the case was studied. In this paper, we determine all Lie algebras such that there exists a post-Lie algebra structure on with .

Some Results on the c-Nilpotent Multiplier of a Pair of Lie Algebras

The Schur multiplier of a Lie algebra L, was introduced by Batten et al. (Commun Algebra 24:4319–4330, 1996). Salemkar and colleagues generalized the concept of the Schur multiplier to the c-nilpotent multiplier. Recently, the author introduced some exact sequences and gived some upper bounds for the dimension of the c-nilpotent multiplier of a pair of Lie algebras. In the present paper, we will extend these results. Moreover, we give some isomorphisms for the c-nilpotent multiplier of a pair of Lie algebras.

Some properties of c-covers of a pair of Lie algebras

In [H. Safa and H. Arabyani, On c-nilpotent multiplier and c-covers of a pair of Lie algebras, Commun. Algebra 45(10) (2017), 4429–4434], we characterized the structure of the c-nilpotent multiplier of a pair of Lie algebras in terms of its c-covering pairs and discussed some results on the existence of c-covers of a pair of Lie algebras. In the present paper, it is shown under some conditions that a relative c-central extension of a pair of Lie algebras is a homomorphic image of a c-covering pair. Moreover, we prove that a c-cover of a pair of finite dimensional Lie algebras, under some assumptions, has a unique domain up to isomorphism and also that every perfect pair of Lie algebras admits at least one c-cover. Finally, we discuss a result concerning the isoclinism of c-covering pairs.

Three-dimensional non-reductive homogeneous spaces of solvable groups Lie, admitting affine connections

When a homogeneous space admits an invariant affine connection? If there exists at least one invariant connection then the space is isotropy-faithful, but the isotropy-faithfulness is not sufficient for the space in order to have invariant connections. If a homogeneous space is reductive, then the space admits an invariant connection. The purpose of the work is the classification of three-dimensional non-reductive homogeneous spaces, admitting invariant affine connections. We concerned only case, when Lie group is solvable. The local classification of homogeneous spaces is equivalent to the description of effective pairs of Lie algebras. The peculiarity of techniques presented in the work is the application of purely algebraic approach, the compound of different methods of differential geometry, theory of Lie groups, Lie algebras and homogeneous spaces.

Regularization of Mickelsson generators for nonexceptional quantum groups

Let g′ ⊂ g be a pair of Lie algebras of either symplectic or orthogonal infinitesimal endomorphisms of the complex vector spaces C N−2 ⊂ C N and U q (g′) ⊂ U q (g) be a pair of quantum groups with a triangular decomposition U q (g) = U q (g-)U q (g+)U q (h). Let Z q (g, g′) be the corresponding step algebra. We assume that its generators are rational trigonometric functions h ∗ → U q (g±). We describe their regularization such that the resulting generators do not vanish for any choice of the weight.

On dimension and homological methods of the (higher) Schur multiplier of a pair of Lie algebras

ABSTRACTIn this article, some inequalities of the dimension of Schur multiplier of pairs of Lie algebras will be obtained and new upper bound will be compared with the existing ones in the literature. Furthermore, by using homological methods, we will derive some properties of the higher Schur multiplier of a pair of Lie algebras and give some isomorphisms that generalize some known results of Stallings in group theory setting.

Capability and Schur multiplier of a pair of Lie algebras

The aim of this work is to find some criteria for detecting the capability of a pair of Lie algebras. We characterize the exact structure of all pairs of capable Lie algebras in the class of abelian and Heisenberg ones. Among the other results, we also give some exact sequences on the Schur multiplier and exterior product of Lie algebras.

On c-nilpotent multiplier and c-covers of a pair of Lie algebras

ABSTRACTIn this paper, we characterize the structure of the c-nilpotent multiplier of a pair of Lie algebras concerning the c-covering pairs and discuss some results on the existence of c-covers of a pair of Lie algebras. Moreover, we show that every nilpotent pair of Lie algebras of class at most k with non-trivial c-nilpotent multiplier does not admit any c-covering pair, if c>k. Also, we obtain the structure of c-covers of a c-capable pair of Lie algebras.

A non-restricted counterexample to the first Kac–Weisfeiler conjecture

In 1971 Kac and Weisfeiler made two important conjectures regarding the representation theory of restricted Lie algebras over fields of positive characteristic. The first of these predicts the maximal dimension of the simple modules, and can be stated without the hypothesis that the Lie algebra is restricted. In this short article we construct the first example of a non-restricted Lie algebra for which the prediction of the first Kac--Weisfeiler conjecture fails. Our method is to present pairs of Lie algebras which have isomorphic enveloping algebras but distinct indexes.