Continuous maps on triangular matrices that preserve commutativity

We investigate continuous injective maps defined on the space of upper triangular matrices over a field F that preserve commutativity of matrices in both directions. We show that every such map is a composition of either an inner automorphism and a locally polynomial map or the two latter and one more automorphism of upper triangular matrices.     

Adjacency preserving bijective maps on triangular matrices over any division ring

Let 𝕋 _n (D) be the set of n × n upper triangular matrices over a division ring D. We characterize the adjacency preserving bijective maps in both directions on 𝕋 _n (D) (n ≥ 3). As applications, we describe the ring semi-automorphisms and the Jordan automorphisms on upper triangular matrices over a simple Artinian ring.     

Commutativity preserving linear maps and Lie automorphisms of strictly triangular matrix space

In this paper we classify linear maps preserving commutativity in both directions on the space N(F) of strictly upper triangular (n+1)×(n+1) matrices over a field F. We show that for n3 a linear map on N(F) preserves commutativity in both directions if and only if =^′+f where ^′ is a product of standard maps on N(F) and f is a linear map of N(F) into its center.     

保体上上三角幂等矩阵的诱导映射

假设T_m(D)是体D上所有上三角m×m矩阵的集合.首先分别给出诱导映射和保幂等性的定义.然后为了刻画T_m(D)的保幂等的诱导映射,提出类序列的概念,同时描述类序列的性质.最后,使用矩阵技术和初等方法,借助于分类讨论得到了T_m(D)的保幂等的诱导映射的一般形式并且给出了某些例子,用以解释某些结果之间的关系.     

The maximal Abelian dimension of linear algebras formed by strictly upper triangular matrices

We compute the largest dimension of the Abelian Lie subalgebras contained in the Lie algebra of n×n strictly upper triangular matrices, where n ∈ ℕ \ {1}. We do this by proving a conjecture, which we previously advanced, about this dimension. We introduce an algorithm and use it first to study the two simplest particular cases and then to study the general case. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 3, pp. 419–429, September, 2007.     

Maps preserving square-zero matrices on the algebra consisting of all upper triangular matrices

Let F be a field, T _n (F) (respectively, N _n (F)) the matrix algebra consisting of all n × n upper triangular matrices (respectively, strictly upper triangular matrices) over F. A ∈ T _n (F) is said to be square zero if A ² = 0. In this article, we firstly characterize non-singular linear maps on N _n (F) preserving square-zero matrices in both directions, then by using it we determine non-singular linear maps on T _n (F) preserving square-zero matrices in both directions.     

The group of commutativity preserving maps on strictly upper triangular matrices

Let N = N _n (R) be the algebra of all n × n strictly upper triangular matrices over a unital commutative ring R. A map φ on N is called preserving commutativity in both directions if xy = yx ? φ(x)φ(y) = φ(y)φ(x). In this paper, we prove that each invertible linear map on N preserving commutativity in both directions is exactly a quasi-automorphism of N, and a quasi-automorphism of N can be decomposed into the product of several standard maps, which extains the main result of Y. Cao, Z. Chen and C. Huang (2002) from fields to rings.     

关于域上上三角矩阵的诱导映射

令F是一个域,T_n(F)是F上所有n×n上三角矩阵的集合.本文分别给出了矩阵保相似性及保交换性的定义,并使用矩阵技术和初等方法,得到了T_n(F)的保相似性及保交换性的诱导映射的一般形式,并且给出了例子,来解释一些结果之间的关系.     

Nonlinear Lie derivations on upper triangular matrices

Let M _n (𝔸) and T _n (𝔸) be the algebra of all n?×?n matrices and the algebra of all n?×?n upper triangular matrices over a commutative unital algebra 𝔸, respectively. In this note we prove that every nonlinear Lie derivation from T _n (𝔸) into M _n (𝔸) is of the form A?→?AT???TA?+?A _??+?ξ(A)I _n , where T?∈?M _n (𝔸), ??:?𝔸?→?𝔸 is an additive derivation, ξ?:?T _n (𝔸)?→?𝔸 is a nonlinear map with ξ(AB???BA)?=?0 for all A,?B?∈?T _n (𝔸) and A _? is the image of A under???applied entrywise.     

On commuting varieties of upper triangular matrices

It is known that the variety of the pairs of n×n commuting upper triangular matrices is not a complete intersection for infinitely many values of n; we show that there exists m such that this happens if and only if n>m. We also show that m<18 and that m could be found by determining the dimension of the variety of the pairs of commuting strictly upper triangular matrices. Then, we define an embedding of any commuting variety into a grassmannian of subspaces of codimension 2.     

Preserving diagonalisability on upper triangular matrices

We characterise all bijective linear mappings on the algebra of upper triangular _n × n matrices that preserve diagonalisability.     

Three linear preservers on infinite triangular matrices

We consider three linear preserver problems on the algebra of infinite triangular matrices over fields. We characterize the maps preserving invertible and noninvertible matrices, the surjective maps preserving inverses and the surjective maps preserving rank permutability.     

Additivity of Jordan maps on triangular algebras

The aim of this article is to prove a result on the additivity of Jordan maps on triangular algebras. As a consequence the additivity of Jordan maps on upper triangular matrix algebras over a faithful commutative ring of 2-torsion free is determined.     

Derivations of the Lie algebra of infinite strictly upper triangular matrices over a commutative ring

Let 𝒩(∞,R) be the Lie algebra of infinite strictly upper triangular matrices over a commutative ring R. We show that every derivation of 𝒩(∞,R) is a sum of diagonal and inner derivations.