Zhengxin Chen

• Doctor of Philosophy
• Fujian Normal University

The n-th Schrödinger algebra 𝔰𝔠𝔥n := 𝔰𝔩2 ⋉ 𝔥n is the semidirect product of the Lie algebra 𝔰𝔩2 with the n-th Heisenberg Lie algebra 𝔥n. In this paper, we will show that 𝔰𝔠𝔥n has neither nontrivial 1 2-derivations nor nontrivial transposed Poisson algebra structures, and doesn’t have nonzero 1 2-biderivations. In addition, all δ-derivations and δ-bi...

The mirror Heisenberg–Virasoro algebra 𝔇 is the semi-direct product of the Virasoro algebra and the twisted Heisenberg algebra. In this paper, all biderivations of the mirror Heisenberg–Virasoro algebra are determined by using the gradation of biderivations. As an application, we show that any linear commuting map f on the mirror Heisenberg–Virasor...

The n-th Schrödinger algebra schn:=sl2⋉hn is the semi-direct of the Lie algebra sl2 with the n-th Heisenberg Lie algebra hn. 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 schn are obtained.

Let n ≥ 3, Mn(F) be the set of all n × n matrices over a finite field F, and Rn(F) the subset of Mn(F) consisting of all rank one matrices. In this paper, we first determine the automorphism group and the fixing number of the orthogonality graph of Rn(F), and then characterize the automorphism group and the fixing number of the orthogonality graph...

A linear map φ on a Lie algebra g is called a triple derivation if φ([[x,y],z])=[[φ(x),y],z]+[[x,φ(y)],z]+[[x,y],φ(z)] for all x,y,z∈g. Let g be a Kac-Moody algebra over an algebraically closed field of characteristic 0, and p an arbitrary parabolic subalgebra of g. In this paper, we prove that any triple derivation of p is a derivation, which gene...

A map φ on a Lie algebra L is called commuting if [φ(x),x]=0 for all x∈L. Let g be a Kac-Moody algebra over an algebraically closed field of characteristic 0. In this paper, we determine the skew-symmetric biderivations and the linear commuting maps of g. As applications, the commuting derivations and the commuting automorphisms of g are gained.

A Hom-structure on a Lie algebra (𝔤, [⋅,⋅]) is a linear map σ : 𝔤 → 𝔤 which satisfies the Hom–Jacobi identity [[x,y],σ(z)] + [[z,x],σ(y)] + [[y,z],σ(x)] = 0 for all x,y,z ∈ 𝔤. A Hom-structure is called regular if σ is also a Lie algebra isomorphism. Let 𝒩 be the Lie algebra consisting of all strictly upper triangular (n + 1) × (n + 1) matrices over...

A linear map ψ on a Lie algebra g over a field F with char ( F ) ≠ 2 is called to be commuting (resp., skew-commuting) if [ ψ ( x ) , y ] = [ x , ψ ( y ) ] (resp., [ ψ ( x ) , y ] = − [ x , ψ ( y ) ] ) for all x , y ∈ g , and to be strong commutativity-preserving if [ ψ ( x ) , ψ ( y ) ] = [ x , y ] for all x , y ∈ g . Let L be a finite-dimensional...

A Hom-structure on a Lie algebra (g,[·]) is a linear map φ:g→g which satisfies the Hom-Jacobi identity [[x,y],φ(z)]+[[z,x],φ(y)]+[[y,z],φ(x)]=0 for all x,y,z∈g. A Hom-structure is called regular if φ is also a Lie algebra isomorphism. Let B be a Borel subalgebra of a finite-dimensional simple Lie algebra L over an algebraically closed field of char...

Let F be an algebraically closed field of characteristic 0, let L be a finite-dimensional simple Lie algebra over F, and let B be the standard Borel subalgebra of L. In this paper we prove that every local derivation of B is a derivation.

Let N(𝔽) be the Lie algebra consisting of all strictly upper triangular (n + 1) × (n + 1) matrices over a field 𝔽. An invertible linear map φ on N(𝔽) is called to be strong commutativity preserving (simply denoted by SCP) if [φ(x),φ(y)] = [x,y] for any x,y ∈ N(𝔽). We show that for n ≥ 4, an invertible linear map φ preserves strong commutativity if...

Let n ≥ 3. The complex Lie algebra, which is attached to a unit form q(x1, x2,..., xn) = \({\sum\nolimits_{i = 1}^n {x_i^2 + \sum\nolimits_{1 \leqslant i \leqslant j \leqslant n} {\left( { - 1} \right)} } ^{j - i}}{x_i}{x_j}\) and defined by generators and generalized Serre relations, is proved to be a finite-dimensional simple Lie algebra of type...

Let W be a simple generalized Witt algebras over a field of characteristic zero. In this paper, it is proved that each anti-symmetric biderivation of W is inner. As an application of biderivations, it is shown that a linear map ψ on W is commuting if and only if ψ is a scalar multiplication map on W. The commuting automorphisms and derivations of W...

Let L be a finite-dimensional complex simple Lie algebra, L be the -span of a Chevalley basis of L, and L R = R ⊗L be a Chevalley algebra of type L over a commutative ring R. Let (R) be the nilpotent subalgebra of L R spanned by the root vectors associated with positive roots. A map Φ of (R) is called commuting if [Φ(x), x] = 0 for all x ∈ (R). In...

Let P be a parabolic subalgebra of a general linear Lie algebra gl(n, ) over a field , where n ≥ 3, contains at least n different elements, and char() ≠ 2. In this article, we prove that generalized derivations, quasiderivations, and product zero derivations of P coincide, and any generalized derivation of P is a sum of an inner derivation, a centr...

Let M n be the algebra of all n×n complex matrices and gl(n,ℂ) be the general linear Lie algebra, where n≥2. An invertible linear map ϕ:gl(n,ℂ)→gl(n,ℂ) preserves solvability in both directions if both ϕ and ϕ -1 map every solvable Lie subalgebra of gl(n,ℂ) to some solvable Lie subalgebra. In this paper, we classify the invertible linear maps preser...

Let g be a finite-dimensional simple Lie algebra of rank l over an algebraically closed field of characteristic zero. An invertible linear map phi on g is called preserving commutativity in both directions if, for any x, y is an element of g, [x, y] = 0 double left right arrow [phi(x), phi(y)] = 0. The group of all such maps on g is denoted by Pzp(...

The purpose of this paper is to construct quotient algebras L(A) 1ℂ/I(A) of complex degenerate composition Lie algebras L(A) 1ℂ by some ideals, where L(A 1ℂ is defined via Hall algebras of tubular algebras A, and to prove that the quotient algebras L(A) 1ℂ/I(A) are isomorphic to the corresponding affine Kac-Moody algebras. Moreover, it is shown tha...

A conceptual framework for describing expert systems is introduced. This framework takes a layered viewpoint. Based on this framework, current research directions in knowledge engineering can be organized appropriately. In this paper, we summarize this framework, and report our experiment of teaching a graduate level expert system course using this...