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.     

Lie triple derivations on nest algebras

Let δ be a Lie triple derivation from a nest algebra ?? into an ??‐bimodule ??. We show that if ?? is a weak* closed operator algebra containing ?? then there are an element S ∈ ?? and a linear functional f on ?? such that δ (A) = SA – AS + f (A)I for all A ∈ ??, and if ?? is the ideal of all compact operators then there is a compact operator K such that δ (A) = KA – AK for all A ∈ ??. As applications, Lie derivations and Jordan derivations on nest algebras are characterized. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Characterizing centralizers and generalized derivations on triangular algebras by acting on zero product

Let U = Tri(A,M,B) be a triangular ring, where A and B are unital rings, and M is a faithful (A, B)-bimodule. It is shown that an additive map Φ on U is centralized at zero point (i.e., Φ(A)B = AΦ(B) = 0 whenever AB = 0) if and only if it is a centralizer. Let δ : U → U be an additive map. It is also shown that the following four conditions are equivalent: (1) δ is specially generalized derivable at zero point, i.e., δ(AB) = δ(A)B + Aδ(B) Aδ(I)B whenever AB = 0; (2) δ is generalized derivable at zero point, i.e., there exist additive maps τ1 and τ2 on U derivable at zero point such that δ(AB) = δ(A)B + Aτ1 (B) = τ2 (A)B + Aδ(B) whenever AB = 0; (3) δ is a special generalized derivation; (4) δ is a generalized derivation. These results are then applied to nest algebras of Banach space.     

Characterization of Lie higher derivations on triangular algebras

Let $\mathcal{A}$ and $\mathcal{B}$ be unital rings, and $\mathcal{M}$ be an $\left( {\mathcal{A},\mathcal{B}} \right)$ -bimodule, which is faithful as a left $\mathcal{A}$ -module and also as a right $\mathcal{B}$ -module. Let $\mathcal{U} = Tri\left( {\mathcal{A},\mathcal{M},\mathcal{B}} \right)$ be the triangular algebra. In this paper, we give some different characterizations of Lie higher derivations on $\mathcal{U}$ .     

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.     

一秩算子和三角代数

Abstract, In this paper, a necessary condition for a maximal triangular algebra to be closed is given, A necessary and sufficient condition for a maxima] triangular algebra to he strongly reducible is obtained,     

Additive maps preserving rank-one operators on nest algebras

In this article, we give a thorough discussion of additive maps between nest algebras acting on Banach spaces which preserve rank-one operators in both directions.     

Isomorphisms between sub-strongly reducible maximal triangular algebras

In this paper, we introduce the concept of sub-strongly maximal triangular algebras which form a large class of maximal triangular algebras, and prove that every algebraic isomorphism of sub-strongly maximal triangular algebras is spatially implemented, which generalizes the result by Ringrose in two respects.

     

c-Graded filiform Lie algebras

In this paper we generalize naturally graded filiform Lie algebras as well as filiform Lie algebras admitting a connected gradation of maximal length, by introducing the concept of c-graded complex filiform Lie algebras. We deal with the particular case of 3-graded filiform Lie algebras and we obtain their classification in arbitrary dimension. We finally show a link among derived algebras, graded filiform and rigid solvable Lie algebras.     

Involutions of reductive Lie algebras in positive characteristic

Let G be a reductive group over a field k of characteristic ≠2, let , let θ be an involutive automorphism of G and let be the associated symmetric space decomposition. For the case of a ground field of characteristic zero, the action of the isotropy group G^θ on is well understood, since the well-known paper of Kostant and Rallis [B. Kostant, S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971) 753–809]. Such a theory in positive characteristic has proved more difficult to develop. Here we use an approach based on some tools from geometric invariant theory to establish corresponding results in (good) positive characteristic.Among other results, we prove that the variety of nilpotent elements of has a dense open orbit, and that the same is true for every fibre of the quotient map . However, we show that the corresponding statement for G, conjectured by Richardson, is not true. We provide a new, (mostly) calculation-free proof of the number of irreducible components of , extending a result of Sekiguchi for . Finally, we apply a theorem of Skryabin to describe the infinitesimal invariants .     

Additivity of multiplicative maps on triangular rings

In this paper we shall give a unified technique in the discussion of the additivity of n-multiplicative automorphisms, n-multiplicative derivations, n-elementary surjective maps, and Jordan multiplicative surjective maps on triangular rings.