post-Lie Algebra Structures Research Articles

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 .

Biderivations and Commutative Post-Lie Algebra Structures on the Lie Algebra $\mathcal{W}(a,b)$

For $a,b \in \mathbb{C}$, the Lie algebra $\mathcal{W}(a,b)$ is the semidirect product of the Witt algebra and a module of the intermediate series. In this paper, all biderivations of $\mathcal{W}(a,b)$ are determined. Surprisingly, these Lie algebras have symmetric (and skew-symmetric) non-inner biderivations. As an application, commutative post-Lie algebra structures on $\mathcal{W}(a,b)$ are obtained.

Biderivations of the Galilean Conformal Algebra and their applications

In this paper, the biderivations of the Galiean conformal algebra are determined. As an application, the forms of the commutative post-Lie algebra structures on the Galiean conformal algebra are obtained.

Post-Lie algebra structures for nilpotent Lie algebras

We study post-Lie algebra structures on [Formula: see text] for nilpotent Lie algebras. First, we show that if [Formula: see text] is nilpotent such that [Formula: see text], then also [Formula: see text] must be nilpotent, of bounded class. For post-Lie algebra structures [Formula: see text] on pairs of [Formula: see text]-step nilpotent Lie algebras [Formula: see text] we give necessary and sufficient conditions such that [Formula: see text] defines a CPA-structure on [Formula: see text], or on [Formula: see text]. As a corollary, we obtain that every LR-structure on a Heisenberg Lie algebra of dimension [Formula: see text] is complete. Finally, we classify all post-Lie algebra structures on [Formula: see text] for [Formula: see text], where [Formula: see text] is the three-dimensional Heisenberg Lie algebra.

Biderivations of the higher rank Witt algebra without anti-symmetric condition

Abstract The Witt algebra 𝔚 d of rank d(≥ 1) is the derivation algebra of Laurent polynomial algebras in d commuting variables. In this paper, all biderivations of 𝔚 d without anti-symmetric condition are determined. As an applications, commutative post-Lie algebra structures on 𝔚 d are obtained. Our conclusions recover and generalize results in the related papers on low rank or anti-symmetric cases.

Biderivations of the planar Galilean conformal algebra and their applications

ABSTRACTIn this paper, the biderivations without skew-symmetric condition of the planar Galilean conformal algebra are presented. As applications, the characterizations of the forms of linear commuting maps and the commutative post-Lie algebra structures on the planar Galilean conformal algebra are given.

Derivations of the Schrödinger algebra and their applications

The Schrodinger algebra is a non-semisimple Lie algebra and plays an important role in mathematical physics and its applications. In this paper, all derivations of the Schrodinger algebra are determined. As applications, all biderivations, linear commuting maps and commutative post-Lie algebra structures on the Schrodinger algebra are obtained.

Biderivations of the twisted Heisenberg–Virasoro algebra and their applications

ABSTRACTIn this paper, the biderivations without the skew-symmetric condition of the twisted Heisenberg–Virasoro algebra are presented. We find some non-inner and non-skew-symmetric biderivations. As applications, the characterizations of the forms of linear commuting maps and the commutative post-Lie algebra structures on the twisted Heisenberg–Virasoro algebra are given. It is also proved that every biderivation of the graded twisted Heisenberg–Virasoro left-symmetric algebra is trivial.

Biderivations, linear commuting maps and commutative post-Lie algebra structures on W-algebras

ABSTRACTIn this paper, the biderivations without the skew-symmetric condition of W-algebras including the Witt algebra, the algebra W(2,2) and their central extensions are characterized. Some classes of non-inner biderivations are presented. As applications, the forms of linear commuting maps and the commutative post-Lie algebra structures on aforementioned W-algebras are given.

Commutative post-Lie algebra structures on Lie algebras

We show that any CPA-structure (commutative post-Lie algebra structure) on a perfect Lie algebra is trivial. Furthermore we give a general decomposition of inner CPA-structures, and classify all CPA-structures on complete Lie algebras. As a special case we obtain the CPA-structures of parabolic subalgebras of semisimple Lie algebras.

Post-Lie algebra structures on pairs of Lie algebras

We study post-Lie algebra structures on pairs of Lie algebras (g,n), which describe simply transitive nil-affine actions of Lie groups. We prove existence results for such structures depending on the interplay of the algebraic structures of g and n. We consider the classes of simple, semisimple, reductive, perfect, solvable, nilpotent, abelian and unimodular Lie algebras. Furthermore we consider commutative post-Lie algebra structures on perfect Lie algebras. Using Lie algebra cohomology we can classify such structures in several cases. We also study commutative structures on low-dimensional Lie algebras and on nilpotent Lie algebras.

Rota–Baxter operators on Witt and Virasoro algebras

The homogeneous Rota-Baxter operators on the Witt and Virasoro algebras are classified. As applications, the induced solutions of the classical Yang-Baxter equation and the induced pre-Lie and PostLie algebra structures are obtained.

Derivation double Lie algebras

We study classical [Formula: see text]-matrices [Formula: see text] for Lie algebras [Formula: see text] such that [Formula: see text] is also a derivation of [Formula: see text]. This yields derivation double Lie algebras [Formula: see text]. The motivation comes from recent work on post-Lie algebra structures on pairs of Lie algebras arising in the study of nil-affine actions of Lie groups. We prove that there are no nontrivial simple derivation double Lie algebras, and study related Lie algebra identities for arbitrary Lie algebras.

Post-Lie algebra structures on solvable Lie algebra [formula omitted

The post-Lie algebra is an enriched structure of the Lie algebra introduced by Vallette. In this paper we give a complete classification of post-Lie algebra structures on solvable Lie algebra t(2,C), the Lie algebra of 2×2 upper triangular matrices. Furthermore, we discuss their isomorphism classes and obtain one necessary and sufficient condition.

Post-Lie Algebra Structures and Generalized Derivations of Semisimple Lie Algebras

We study post-Lie algebra structures on pairs of Lie algebras (g,n), and prove existence results for the case that one of the Lie algebras is semisimple. For semisimple g and solvable n we show that there exist no post-Lie algebra structures on (g,n). For semisimple n and certain solvable g we construct canonical post-Lie algebra structures. On the other hand we prove that there are no post-Lie algebra structures for semisimple n and solvable, unimodular g. We also determine the generalized $(\al,\be,\ga)$-derivations of n in the semisimple case. As an application we classify post-Lie algebra structures induced by generalized derivations.

Post-Lie Algebra Structures on the Lie Algebragl(2,C)

The post-Lie algebra is an enriched structure of the Lie algebra. We give a complete classification of post-Lie algebra structures on the Lie algebragl(2,C)up to isomorphism.

Affine actions on Lie groups and post-Lie algebra structures

We introduce post-Lie algebra structures on pairs of Lie algebras (g,n) defined on a fixed vector space V. Special cases are LR-structures and pre-Lie algebra structures on Lie algebras. We show that post-Lie algebra structures naturally arise in the study of NIL-affine actions on nilpotent Lie groups. We obtain several results on the existence of post-Lie algebra structures, in terms of the algebraic structure of the two Lie algebras g and n. One result is, for example, that if there exists a post-Lie algebra structure on (g,n), where g is nilpotent, then n must be solvable. Furthermore special cases and examples are given. This includes a classification of all complex, two-dimensional post-Lie algebras.

PostLie algebra structures on the Lie Algebra SL(2,Ca)

The PostLie algebra is an enriched structure of the Lie algebra that has recently arisen from operadic study. It is closely related to pre-Lie algebra, Rota-Baxter algebra, dendriform trialgebra, modified classical Yang-Baxter equations and integrable systems. This paper gives a complete classification of PostLie algebra structures on the Lie algebra sl(2,C) up to isomorphism. The classification problem is first reduced to solving an equation of 3 × 3 matrices. Then the latter problem is solved by making use of the classification of complex symmetric matrices up to the congruent action of orthogonal groups.