Additivity of Lie multiplicative maps on triangular algebras

Let 𝒜 and ? be unital algebras over a commutative ring ?, and ? be a (𝒜,??)-bimodule, which is faithful as a left 𝒜-module and also as a right ?-module. Let 𝒰?=?Tri(𝒜,??,??) be the triangular algebra and 𝒱 any algebra over ?. Assume that Φ?:?𝒰?→?𝒱 is a Lie multiplicative isomorphism, that is, Φ satisfies Φ(ST???TS)?=?Φ(S)Φ(T)???Φ(T)Φ(S) for all S, T?∈?𝒰. Then Φ(S?+?T)?=?Φ(S)?+?Φ(T)?+?Z _S,T for all S, T?∈?𝒰, where Z _S,T is an element in the centre 𝒵(𝒱) of 𝒱 depending on S and T.     

Idempotence preserving maps between matrix spaces

Suppose 𝔽 is an arbitrary field of characteristic not 2 and 𝔽?≠?𝔽₃. Let M _n (𝔽) be the space of all n?×?n full matrices over 𝔽 and P _n (𝔽) the subset of M _n (𝔽) consisting of all n?×?n idempotent matrices and GL _n (𝔽) the subset of M _n (𝔽) consisting of all n?×?n invertible matrices. Let Φ_𝔽(n,?m) denote the set of all maps from M _n (𝔽) to M _m (𝔽) satisfying A???λB?∈?P _n (𝔽)???φ(A)???λφ(B)?∈?P _m (𝔽) for every A,?B?∈?M _n (𝔽) and λ?∈?𝔽, where m and n are integers with 3?≤?n?≤?m. It is shown that if φ?∈?Φ_𝔽(n,?m), then there exists T?∈?GL _m (𝔽) such that φ(A)?=?T?[A???I _p ?⊕?A ^t ???I _q ?⊕?0]T?^?1 for every A?∈?M _n (𝔽), where I ₀?=?0. This improves the results of some related references.     

Zero product determined Jordan algebras,I

We show that the Jordan algebra 𝒮 of symmetric matrices with respect to either transpose or symplectic involution is zero product determined. This means that if a bilinear map {.,?.} from 𝒮?×?𝒮 into a vector space X satisfies {x, y}?=?0 whenever x?○?y?=?0, then there exists a linear map T : 𝒮?→?X such that {x,?y}?=?T(x?○?y) for all x, y?∈?𝒮 (here, x?○?y?=?xy?+?yx).     

Strong commutativity preserving generalized derivations on Lie ideals

We apply elementary matrix computations and the theory of differential identities to prove the following: let R be a prime ring with extended centroid C and L a noncommutative Lie ideal of R. Suppose that f?:?L?→?R is a map and g is a generalized derivation of R such that [f(x),?g(y)]?=?[x,?y] for all x,?y?∈?L. Then there exist a nonzero α?∈?C and a map μ?:?L?→?C such that g(x)?=?αx for all x?∈?R and f(x)?=?α^?1 x?+?μ(x) for all x?∈?L, except when R???M ₂(F), the 2?×?2 matrix ring over a field F.     

Characterizations of Jordan derivations and Jordan homomorphisms

Let 𝒜 be a unital Banach algebra and ? be a unital 𝒜-bimodule. We show that if δ is a linear mapping from 𝒜 into ? satisfying δ(ST)?=?δ(S)T?+Sδ(T) for any S, T?∈?𝒜 with ST?=?W, where W is a left or right separating point of ?, then δ is a Jordan derivation. Also, it is shown that every linear mapping h from 𝒜 into a unital Banach algebra ? which satisfies h(S)h(T)?=?h(ST) for any S,?T?∈?𝒜 with ST?=?W is a Jordan homomorphism if h(W) is a separating point of ?.     

Characterization of derivations on reflexive algebras

Let Alg? be a reflexive algebra on a Hilbert space H. We say that a linear map δ: Alg??→?Alg? is derivable at Ω?∈?Alg? if δ(A)B?+?Aδ(B)?=?δ(Ω) for every A, B?∈?Alg? with AB?=?Ω. In this article, we give a necessary and sufficient condition for a map δ on Alg? to be derivable at Ω. In particular, we show that every linear map δ derivable at Ω?≠?0 from an irreducible CDC algebra (in particular, a nest algebra) into itself is a derivation. Moreover, if Alg? is a CSL algebra, and if for some nontrivial projection P?∈??, PΩP and (I???P)Ω(I???P) are left or right separating points in PAlg?P and (I???P)Alg?(I???P) respectively, then a linear map δ on Alg? is derivable at Ω if and only if δ is a derivation.     

Commuting Derivations and Automorphisms of Certain Nilpotent Lie Algebras Over Commutative Rings

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 𝒩(R) be the nilpotent subalgebra of L _R spanned by the root vectors associated with positive roots. A map ? of 𝒩(R) is called commuting if [?(x), x] = 0 for all x ∈ 𝒩(R). In this article, we prove that under some conditions for R, if Φ is not of type A ₂, then a derivation (resp., an automorphism) of 𝒩(R) is commuting if and only if it is a central derivation (resp., automorphism), and if Φ is of type A ₂, then a derivation (resp., an automorphism) of 𝒩(R) is commuting if and only if it is a sum (resp., a product) of a graded diagonal derivation (resp., automorphism) and a central derivation (resp., automorphism).     

Linear spaces and preservers of symmetric matrices of bounded rank-two

Let 𝔽 be a field of characteristic two. Let S _n (𝔽) denote the vector space of all n?×?n symmetric matrices over 𝔽. We characterize i. subspaces of S _n (𝔽) all whose elements have rank at most two where n???3,

ii. linear maps from S _m (𝔽) to S _n (𝔽) that sends matrices of rank at most two into matrices of rank at most two where m, n???3 and |𝔽|?≠?2.

     

Vertex Poisson Algebras Associated with Courant Algebroids

We study relationships between vertex Poisson algebras and Courant algebroids. For any ?-graded vertex Poisson algebra A = ? _n∈? A _(n), we show that A ₍₁₎ is a Courant A ₍₀₎-algebroid. On the other hand, for any Courant 𝒜-algebroid ?, we construct an ?-graded vertex Poisson algebra A = ? _n∈? A _(n) such that A ₍₀₎ is 𝒜 and the Courant 𝒜-algebroid A ₍₁₎ is isomorphic to ? as a Courant 𝒜-algebroid.     

On the Simplicity of Lie Algebras Associated to Leavitt Algebras

For any field 𝕂 and integer n ≥ 2, we consider the Leavitt algebra L _𝕂(n); for any integer d ≥ 1, we form the matrix ring S = M _d (L _𝕂(n)). S is an associative algebra, but we view S as a Lie algebra using the bracket [a, b] = ab ? ba for a, b ∈ S. We denote this Lie algebra as S ^?, and consider its Lie subalgebra [S ^?, S ^?]. In our main result, we show that [S ^?, S ^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1 and char(𝕂) does not divide d. In particular, when d = 1, we get that [L _𝕂(n)^?, L _𝕂(n)^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1.     

The Semigroup Generated by the Similarity Class of a Singular Matrix

Let A be a singular matrix of M _n (𝕂), where 𝕂 is an arbitrary field. Using canonical forms, we give a new proof that the sub-semigroup of ( _n (𝕂), ×) generated by the similarity class of A is the set of matrices of M _n (𝕂) with a rank lesser than or equal to that of A.     

Formal Expansion of Harish-Chandra Homomorphism

Abstract

Let 𝔤 be a complex semisimple Lie algebra with adjoint group G and let 𝔥 be a Cartan subalgebra of 𝔤. Let Â(𝔤) and Â(𝔥) denote the algebra of differential operators with formal power series coefficients on 𝔤 and 𝔥 respectively. We construct a subalgebra A _𝔤 of Â(𝔤) containing all the pull-backs of the differential operators in G attached to any element x in 𝔤. We also consider the projection P: A _𝔤 → Â _𝔥. Then, we calculate explicity the pull-back of the differential operator in G attached to an element h in 𝔥 modulo Ker P.     

On the spectral radius of weighted trees with given number of pendant vertices and a positive weight set

Let 𝒯(n,?r;?W _n?1) be the set of all n-vertex weighted trees with r vertices of degree 2 and fixed positive weight set W _n?1, 𝒫(n,?γ;?W _n?1) the set of all n-vertex weighted trees with q pendants and fixed positive weight set W _n?1, where W _n?1?=?{w ₁,?w ₂,?…?,?w _n?1} with w ₁???w ₂???···???w _n?1?>?0. In this article, we first identify the unique weighted tree in 𝒯(n,?r;?W _n?1) with the largest adjacency spectral radius. Then we characterize the unique weighted trees with the largest adjacency spectral radius in 𝒫(n,?γ;?W _n?1).     

Nonlinear generalized Lie triple derivation on triangular algebras

Let ? be a commutative ring with identity and let 𝔄 = Tri(𝒜,?,?) be a triangular algebra consisting of unital algebras 𝒜,? over ? and an (𝒜,?)-bimodule ? which is faithful as a left 𝒜-module as well as a right ?-module. In this paper, we prove that under certain assumptions every nonlinear generalized Lie triple derivation G_L:𝔄→𝔄 is of the form G_L = δ+τ, where δ:𝔄→𝔄 is an additive generalized derivation on 𝔄 and τ is a mapping from 𝔄 into its center which annihilates all Lie triple products [[x,y],z].