Commutativity-preserving Maps Research Articles

Zero product determined n-th Schrödinger algebra

The n-th Schrödinger algebra schn:=sl2⋉hn is the semi-direct product of the Lie algebra sl2 with the n-th Heisenberg Lie algebra hn. Let K be an algebraically closed field of characteristic zero. A Lie algebra L is zero product determined over K, if for every K-linear space V and every bilinear map ϕ:L×L→V with the property that ϕ(x,y)=0 whenever [x,y]=0, then there exists a linear map f:[L,L]→V such that ϕ(x,y)=f([x,y]) for all x,y∈L. This article shows that the n-th Schrödinger algebra schn is zero product determined. Applying this result, product zero derivations and two-sided commutativity-preserving maps on schn are determined. Furthermore, quasi-derivations, linear anti-commuting maps, quasi-automorphisms and strong commutativity-preserving maps of schn are all obtained.

Biderivations and strong commutativity-preserving maps on parabolic subalgebras of simple Lie algebras

ABSTRACT A linear map ψ on a Lie algebra over a field F with char is called to be commuting (resp., skew-commuting) if (resp., ) for all , and to be strong commutativity-preserving if for all . Let L be a finite-dimensional simple Lie algebra over an algebraically closed field F of characteristic 0, P a parabolic subalgebra of L. In this paper, firstly, we improve existing results about skew-symmetric biderivations on P by determining related linear commuting maps. Secondly, we classify the linear skew-commuting maps and the related symmetric biderivations on P, and so the biderivations of P are characterized. Finally, we classify the invertible linear strong commutativity-preserving maps of P.

Nonlinear Lie higher derivations on triangular algebras

Let ℛ be a commutative ring with identity, A, B be unital algebras over ℛ and M be a unital (A, B)-bimodule, which is faithful as a left A-module and also as a right B-module. Let be the triangular algebra consisting of A, B and M. Motivated by the powerful works of Brešar [M. Brešar, Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings, Trans. Amer. Math. Soc. 335 (1993), pp. 525–546] and Yu et al. [W.-Y. Yu and J.-H. Zhang, Nonlinear Lie derivations of triangular algebras, Linear Algebra Appl. 432 (2010), pp. 2953–2960], we will study nonlinear Lie higher derivations on 𝒯 in this article. Let D = {L n } n∈ℕ be a Lie higher derivation on 𝒯 without additive condition. Under mild assumptions, it is shown that D = {L n } n∈ℕ is of standard form, i.e. each component L n (n ≥ 1) can be expressed through an additive higher derivation and a nonlinear functional vanishing on all commutators of 𝒯. As applications, nonlinear Lie higher derivations on some classical triangular algebras are characterized.

A class of zero product determined Lie algebras

Let L be a Lie algebra over a field F. We say that L is zero product determined if, for every F-linear space V and every bilinear map φ : L × L → V , the following condition holds. If φ ( x , y ) = 0 whenever [ x , y ] = 0 , then there exists a linear map f from [ L , L ] to V such that φ ( x , y ) = f ( [ x , y ] ) for all x , y ∈ L . This article shows that every parabolic subalgebra p of a (finite-dimensional) simple Lie algebra defined over an algebraically closed field is always zero product determined. Applying this result, we present a method different from that of Wang et al. (2010) [9] to determine zero product derivations of p , and we obtain a definitive solution for the problem of describing two-sided commutativity-preserving maps on p .

Nonlinear commutativity-preserving maps on Hermitian matrices

Let $H_n$ , $n\ge3$ , be the space of all $n\times n$ Hermitian matrices. Assume that a map $\phi:H_n\to H_n$ preserves commutativity in both directions (no linearity or bijectivity of $\phi$ is assumed). Then $\phi$ is a unitary similarity transformation composed with a locally polynomial map possibly composed by the transposition. The same result holds for injective continuous maps on $H_n$ preserving commutativity in one direction only. We give counter-examples showing that these two theorems cannot be improved or extended to the infinite-dimensional case.

Commutativity-preserving maps on Lie ideals of prime algebras

One of the most studied linear preserver problems is the problem of characterizing maps preserving commutativity. In this paper we study maps preserving commutativity on a Lie ideal L of a prime algebra A. We show by an example that in general we cannot expect the same standard conclusion as in the case when L= A. Therefore we confine ourselves to some special classes of algebras where the usual result can be proved.

Spectrum and commutativity preserving mappings on H2

The continuous mappings on 2×2 real symmetric or complex hermitian matrices that are spectrum and commutativity preserving are characterized.

Spectrum and commutativity preserving mappings on triangular matrices

We give the description of continuous mappings on the algebra of all upper-triangular complex matrices that preserve spectrum and commutativity.

A note on additive commutativity-preserving mappings

A note on additive commutativity-preserving mappings

Local Derivations, Automorphisms and Commutativity Preserving Maps on $\mathscr L^+(\mathscr D)$

Local Derivations, Automorphisms and Commutativity Preserving Maps on $\mathscr L^+(\mathscr D)$

Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings

Biadditive mappings B: R × R → R where R is a prime ring with certain additional properties, satisfying B(x, x)x = xB(x, x) for all x ∈ R, are characterized. As an application we determine the structures of commutativity-preserving mappings, Lie isomorphisms, and Lie derivations of certain prime rings

Commuting Traces of Biadditive Mappings, Commutativity-Preserving Mappings and Lie Mappings

Commuting Traces of Biadditive Mappings, Commutativity-Preserving Mappings and Lie Mappings

Linear maps preserving commutativity

Commutativity-preserving maps on the real space of all real symmetric or complex self-adjoint matrices are characterized. Related results are given for adjoint-preserving maps defined on all n × n matrices. These results are extended to infinite dimensions in the case of invertible maps.