Commutative post-Lie Algebra Structures Research Articles

Biderivations, commuting linear maps, post-Lie algebra structure on solvable Lie algebras of maximal rank

In this paper, all biderivations of solvable Lie algebras of maximal rank g = Q ⊕ N are characterized. Namely, we consider a solvable Lie algebra of the form g = Q ⊕ N , where Q is the maximal torus subalgebra of g , N is the nilradical of g and dim Q = dim N / N 2 .In the case dim Q = N we characterize the form of skew-symmetric and symmetric biderivations of g . As applications, the forms of linear commuting (skew-commuting) maps and the commutative post-Lie algebra structures on g are given.

Derivations and biderivations of the non-square linear Lie algebra gl(m×n, 𝔽)

Let m, n be integers with m > n . Denote M m , n by the set of all m × n matrices over the field F of characteristic zero. Let I be an n × m matrix with ( i , i ) -position 1 for any 1 ≤ i ≤ n , and 0 in other position. Define a bracket [ A , B ] = AIB − BIA , where A , B ∈ M m , n . Then M m , n with this bracket is a Lie algebra, called non-square linear Lie algebra, denoted by gl ( m × n , F ) . In this paper, all derivations and biderivations of gl ( m × n , F ) are determined. As applications of biderivations, the linear commuting maps and commutative post-Lie algebra structures on gl ( m × n , F ) are given.

Biderivations of a Lie algebra of Block type and applications

Let be a Lie algebra of Block type with basis and relations . In the present paper, the biderivations of are explicitly described. In particular, it is shown that any biderivation of is inner. As applications, the linear commuting maps and commutative post-Lie algebra structures on this algebra are given.

Derivations and biderivations of the n-th Schrödinger algebra

The n-th Schrödinger algebra is the semi-direct of the Lie algebra with the n-th Heisenberg Lie algebra . In this paper, all derivations and biderivations of the n-th Schrödinger algebra are determined. As applications, all linear commuting maps and commutative post-Lie algebra structures on are obtained.

Derivations and Biderivations of the Schrödinger algebra in (n + 1)-dimensional space-time

In this paper, all derivations of the Schrödinger algebra in -dimensional space-time of the Schrödinger Lie group are determined. As applications, all biderivations are obtained. Finally, as applications of biderivations, the linear commuting maps and commutative post-Lie algebra structures on this algebra are given.

N-Derivations of the Extended Schrödinger–Virasoro Lie Algebra

Let [Formula: see text] be the extended Schrödinger–Virasoro Lie algebra and [Formula: see text] an integer. A map [Formula: see text] is called an [Formula: see text]-derivation if it is a derivation in one variable while other variables fixed. We investigate [Formula: see text]-derivations of the extended Schrödinger–Virasoro Lie algebra [Formula: see text]. The main result when [Formula: see text] is then applied to characterize the linear commuting maps and the commutative post-Lie algebra structures on [Formula: see text].

Biderivations and linear commuting maps on the restricted contact Lie algebras K(n; )

Let K be the restricted contact Lie algebras K(n, ) over an algebraically closed field F of characteristic p > 3. We prove that each skew-symmetric biderivation of K is inner and show that commuting maps on K are scalar multiplication maps. Moreover, it is showed that the commuting automorphisms and dertivations of K are proved to be the identity mappings and zero mappings, respectively. Meanwhile, the commutative post-Lie algebra structures on K are given.

A Zassenhaus conjecture and CPA-structures on simple modular Lie algebras

Commutative post-Lie algebra structures on Lie algebras, in short CPA structures, have been studied over fields of characteristic zero, in particular for real and complex numbers motivated by geometry. A perfect Lie algebra in characteristic zero only admits the trivial CPA-structure. In this article we study these structures over fields of characteristic p>0. We show that every perfect modular Lie algebra in characteristic p>2 having a solvable outer derivation algebra admits only the trivial CPA-structure. This involves a conjecture by Hans Zassenhaus, saying that the outer derivation algebra Out(g) of a simple modular Lie algebra g is solvable. We try to summarize the known results on the Zassenhaus conjecture and prove some new results using the classification of simple modular Lie algebras by Premet and Strade for algebraically closed fields of characteristic p>3. As a corollary we obtain that every central simple modular Lie algebra of characteristic p>3 admits only the trivial CPA-structure.

Commutative post-Lie algebra structures on nilpotent Lie algebras and Poisson algebras

We give an explicit description of commutative post-Lie algebra structures on some classes of nilpotent Lie algebras. For non-metabelian filiform nilpotent Lie algebras and Lie algebras of strictly upper-triangular matrices we show that all CPA-structures are associative and induce an associated Poisson-admissible algebra.

Biderivations of the deformative Schrödinger-Virasoro algebras

For , s = 0 or , the deformative Schrödinger-Virasoro algebra is the semi-direct product Lie algebra of the Witt algebra and its tensor density module. It contains W-algebra as its subalgebras, and Schrödinger-Virasoro algebra is his special case. In this paper, all biderivations of are determined. It is observed that these Lie algebras have rich variety of symmetric (and skewsymmetric) non-inner biderivations. As an application, commutative post-Lie algebra structures on are obtained.

Commutative post-Lie algebra structures on Kac–Moody algebras

We determine commutative post-Lie algebra structures on some infinite-dimensional Lie algebras. We show that all commutative post-Lie algebra structures on loop algebras are trivial. This extends the results for finite-dimensional perfect Lie algebras. Furthermore, we show that all commutative post-Lie algebra structures on affine Kac–Moody Lie algebras are “almost trivial”.

Biderivations and Commutative Post-Lie Algebra Structure on Schrödinger–Virasoro Lie Algebras

In this paper, the biderivations of Schrodinger–Virasoro Lie algebras are characterized. It is obtained a class of non-inner and non-skewsymmetric biderivations. As an application, the commutative post-Lie algebra structures on the Schrodinger–Virasoro Lie algebra are presented.

Commutative post-Lie algebra structures and linear equations for nilpotent Lie algebras

We show that for a given nilpotent Lie algebra g with Z(g)⊆[g,g] all commutative post-Lie algebra structures, or CPA-structures, on g are complete. This means that all left and all right multiplication operators in the algebra are nilpotent. Then we study CPA-structures on free-nilpotent Lie algebras Fg,c and discover a strong relationship to solving systems of linear equations of type [x,u]+[y,v]=0 for generator pairs x,y∈Fg,c. We use results of Remeslennikov and Stöhr concerning these equations to prove that, for certain g and c, the free-nilpotent Lie algebra Fg,c has only central CPA-structures.

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.

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.