Let R be a commutative ring with unity and T be a triangular algebra over R.Let a sequence G={G_n}_(n∈N)of nonlinear mappings G_n:T→T associated with nonlinear Lie triple higher derivations∆={δ_n}_(n∈N)by local ac...Let R be a commutative ring with unity and T be a triangular algebra over R.Let a sequence G={G_n}_(n∈N)of nonlinear mappings G_n:T→T associated with nonlinear Lie triple higher derivations∆={δ_n}_(n∈N)by local actions be a generalized Lie triple higher derivation by local actions satisfying Gn([[x,y],z])=Σ_(i+j+k=n)[[Gi(x),δj(y)],δk(z)]for all x,y,z∈T with xyz=0.Under some mild conditions on T,we prove in this paper that every nonlinear generalized Lie triple higher derivation by local actions on triangular algebras is proper.As an application we shall give a characterization of nonlinear generalized Lie triple higher derivations by local actions on upper triangular matrix algebras and nest algebras,respectively.At the same time,it also improves some interesting conclusions,such as[J.Algebra Appl.22(3),2023,Paper No.2350059],[Axioms,11,2022,1–16].展开更多
In this paper, we prove that any nonlinear Jordan higher derivation on triangular algebras is an additive higher derivation. As a byproduct, we obtain that any nonlinear Jordan derivation on nest algebras over infinit...In this paper, we prove that any nonlinear Jordan higher derivation on triangular algebras is an additive higher derivation. As a byproduct, we obtain that any nonlinear Jordan derivation on nest algebras over infinite dimensional Hilbert suaces is inner.展开更多
By using properties of triangular algebra, we prove that if derivations D and G on a triangular algebra T satisfy certain generalized identities, then both D and G are zero mappings. As a corollary we get that if D an...By using properties of triangular algebra, we prove that if derivations D and G on a triangular algebra T satisfy certain generalized identities, then both D and G are zero mappings. As a corollary we get that if D and G are cocentralizing on T, then both D and G are zero mappings.展开更多
In this paper, we introduce the concept of weakly reducible maxi mal triangular algebras S*!which form a large class of maximal t riangular algebras. Let B be a weakly closed algebra containing S, we prove that the co...In this paper, we introduce the concept of weakly reducible maxi mal triangular algebras S*!which form a large class of maximal t riangular algebras. Let B be a weakly closed algebra containing S, we prove that the cohomology spaces Hn(S , B) ( n≥1) are trivial.展开更多
Let T,U be two Artin algebras and_(U)M_(T)be a U-T-bimodule.In this paper,we get a necessary and sufficient condition such that the formal triangular matrix algebra Λ=(T 0 M U)is(m,n)-Igusa-Todorov when_(U)M,M_(T)are...Let T,U be two Artin algebras and_(U)M_(T)be a U-T-bimodule.In this paper,we get a necessary and sufficient condition such that the formal triangular matrix algebra Λ=(T 0 M U)is(m,n)-Igusa-Todorov when_(U)M,M_(T)are projective.We also study the Igusa-Todorov dimension of Λ.More specifically,it is proved that max{IT.dim T,IT.dim U}≤IT.dim Λ≤min{max{gl.dim T,IT.dim U},max{gl.dim U,IT.dim T}}.展开更多
We construct a triangular algebra whose diagonals form a noncommutative algebra and its lattice of invariant projections contains only two nontrivial projections. Moreover we prove that our triangular algebra is maximal.
In this paper, the structure of Jordan higher derivable maps on triangular algebras by commutative zero products is given. As an application, the form of Jordan higher derivable maps of nest algebras by commutative ze...In this paper, the structure of Jordan higher derivable maps on triangular algebras by commutative zero products is given. As an application, the form of Jordan higher derivable maps of nest algebras by commutative zero products is obtained.展开更多
Let A and B be unital rings, and M be an （A, B）-bimodule, which is faithful as a left A-module and also as a right B-module. Let U = Tri（A,M, B） be the triangular algebra. In this paper, we give some different cha...Let A and B be unital rings, and M be an （A, B）-bimodule, which is faithful as a left A-module and also as a right B-module. Let U = Tri（A,M, B） be the triangular algebra. In this paper, we give some different characterizations of Lie higher derivations on U.展开更多
In this paper, a necessary condition for a maximal triangular algebra to be closed is given. A necessary and sufficient condition for a maximal triangular algebra to be strongly reducible is obtained.
Let U = Tri（fit, 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 = A,φ（B）...Let U = Tri（fit, 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 = A,φ（B） = 0 whenever AB = 0） if and only if it is a centralizer. Let 5 ： U →U be an additive map. It is also shown that the following four conditions are equivalent： （1） 5 is specially generalized derivable at zero point, i.e., 5（AB） = δ（A）B ＋ AS（B） - Aδ（I）B whenever AB = 0; （2） 5 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展开更多
Let A and B be unital Banach algebra and M be Banach A, B-module. Then T＇ = （AB^M） becomes a triangular Banach algebra when equipped with the Banach space norm ｜｜（ ab^m）｜｜ = ｜｜a｜｜A ＋｜｜m｜｜M ＋ ｜｜b｜...Let A and B be unital Banach algebra and M be Banach A, B-module. Then T＇ = （AB^M） becomes a triangular Banach algebra when equipped with the Banach space norm ｜｜（ ab^m）｜｜ = ｜｜a｜｜A ＋｜｜m｜｜M ＋ ｜｜b｜｜m A Banach algebra T is said to be Lie n-weakly amenable if all Lie derivations from T into its nth dual space T^（n） are standard. In this paper we investigate Lie n-weak amenability of a triangular Banach algebra T in relation to that of the algebras A, B and their action on the module M.展开更多
In this paper, we study a class of subalgebras of the Lie algebra of vector fields on n-dimensional torus, which are called the Triangular derivation Lie algebra. We give the structure and the central extension of Tri...In this paper, we study a class of subalgebras of the Lie algebra of vector fields on n-dimensional torus, which are called the Triangular derivation Lie algebra. We give the structure and the central extension of Triangular derivation Lie algebra.展开更多
Let T(R) be a two-order upper matrix algebra over the semilocal ring R which is determined by R=F×F where F is a field such that CharF=0. In this paper, we prove that any Jordan automorphism of T(R) can be decomp...Let T(R) be a two-order upper matrix algebra over the semilocal ring R which is determined by R=F×F where F is a field such that CharF=0. In this paper, we prove that any Jordan automorphism of T(R) can be decomposed into a product of involutive, inner and diagonal automorphisms.展开更多
Let F be a field, n ≥ 3, N（n,F） the strictly upper triangular matrix Lie algebra consisting of the n × n strictly upper triangular matrices and with the bracket operation {x, y} = xy-yx. A linear map φ on N（...Let F be a field, n ≥ 3, N（n,F） the strictly upper triangular matrix Lie algebra consisting of the n × n strictly upper triangular matrices and with the bracket operation {x, y} = xy-yx. A linear map φ on N（n,F） is said to be a product zero derivation if {φ（x）,y] ＋ [x, φ（y）] = 0 whenever {x, y} = 0,x,y ∈ N（n,F）. In this paper, we prove that a linear map on N（n, F） is a product zero derivation if and only if φ is a sum of an inner derivation, a diagonal derivation, an extremal product zero derivation, a central product zero derivation and a scalar multiplication map on N（n, F）.展开更多
Let （C,α） and （H, β） be Hom-bialgebras and ω ： C × H → H × C a linear map. We introduce a Horn-ω-smash coproduct （Cω H, γ） and give necessary and sufficient conditions for （Cω H, γ） to be...Let （C,α） and （H, β） be Hom-bialgebras and ω ： C × H → H × C a linear map. We introduce a Horn-ω-smash coproduct （Cω H, γ） and give necessary and sufficient conditions for （Cω H, γ） to be a Hom-bialgebra. We study the quasi-triangular structures over （Cω H, γ） and show the necessary and sufficient conditions for （Cω H, γ R） to be a quasi-triangular Hom-Hopf algebra. As applications of our results, we introduce the concept of D（H）＊ and construct quasi-triangular structures over D（H）＊.展开更多
基金Supported by Open Research Fund of Hubei Key Laboratory of Mathematical Sciences(Central China Normal University)the Natural Science Foundation of Anhui Province(Grant No.2008085QA01)the University Natural Science Research Project of Anhui Province(Grant No.KJ2019A0107)。
文摘Let R be a commutative ring with unity and T be a triangular algebra over R.Let a sequence G={G_n}_(n∈N)of nonlinear mappings G_n:T→T associated with nonlinear Lie triple higher derivations∆={δ_n}_(n∈N)by local actions be a generalized Lie triple higher derivation by local actions satisfying Gn([[x,y],z])=Σ_(i+j+k=n)[[Gi(x),δj(y)],δk(z)]for all x,y,z∈T with xyz=0.Under some mild conditions on T,we prove in this paper that every nonlinear generalized Lie triple higher derivation by local actions on triangular algebras is proper.As an application we shall give a characterization of nonlinear generalized Lie triple higher derivations by local actions on upper triangular matrix algebras and nest algebras,respectively.At the same time,it also improves some interesting conclusions,such as[J.Algebra Appl.22(3),2023,Paper No.2350059],[Axioms,11,2022,1–16].
文摘In this paper, we prove that any nonlinear Jordan higher derivation on triangular algebras is an additive higher derivation. As a byproduct, we obtain that any nonlinear Jordan derivation on nest algebras over infinite dimensional Hilbert suaces is inner.
基金The NSF(11101175,11371165) of China985 Project211 Project
文摘By using properties of triangular algebra, we prove that if derivations D and G on a triangular algebra T satisfy certain generalized identities, then both D and G are zero mappings. As a corollary we get that if D and G are cocentralizing on T, then both D and G are zero mappings.
文摘In this paper, we introduce the concept of weakly reducible maxi mal triangular algebras S*!which form a large class of maximal t riangular algebras. Let B be a weakly closed algebra containing S, we prove that the cohomology spaces Hn(S , B) ( n≥1) are trivial.
基金Supported by the National Natural Science Foundation of China(Grant No.12301041)the Science Foundation for Distinguished Young Scholars of Anhui Province(Grant No.2108085J01)。
文摘Let T,U be two Artin algebras and_(U)M_(T)be a U-T-bimodule.In this paper,we get a necessary and sufficient condition such that the formal triangular matrix algebra Λ=(T 0 M U)is(m,n)-Igusa-Todorov when_(U)M,M_(T)are projective.We also study the Igusa-Todorov dimension of Λ.More specifically,it is proved that max{IT.dim T,IT.dim U}≤IT.dim Λ≤min{max{gl.dim T,IT.dim U},max{gl.dim U,IT.dim T}}.
基金Shaanxi Natural Science Foundation of China (Grant No. 2006A17)
文摘We construct a triangular algebra whose diagonals form a noncommutative algebra and its lattice of invariant projections contains only two nontrivial projections. Moreover we prove that our triangular algebra is maximal.
基金Supported by National Natural Science Foundation of China(Grant Nos.11471199 and 11371233)Research Fund for the Doctoral Program of Higher Education of China(Grant No.20110202110002)the Innovation Funds of Graduate Programs of Shaanxi Normal University(Grant No.2015CXB007)
文摘In this paper, the structure of Jordan higher derivable maps on triangular algebras by commutative zero products is given. As an application, the form of Jordan higher derivable maps of nest algebras by commutative zero products is obtained.
基金Supported by National Natural Science Foundation of China(Grant No.11101250)Youth Foundation of Shanxi Province(Grant No.2012021004)Young Talents Plan for Shanxi University
文摘Let A and B be unital rings, and M be an （A, B）-bimodule, which is faithful as a left A-module and also as a right B-module. Let U = Tri（A,M, B） be the triangular algebra. In this paper, we give some different characterizations of Lie higher derivations on U.
文摘In this paper, a necessary condition for a maximal triangular algebra to be closed is given. A necessary and sufficient condition for a maximal triangular algebra to be strongly reducible is obtained.
基金supported by National Natural Science Foundation of China (Grant No. 11101250)supported by National Natural Science Foundation of China (Grant No. 11171249)Youth Foundation of Shanxi Province (Grant No. 2012021004)
文摘Let U = Tri（fit, 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 = A,φ（B） = 0 whenever AB = 0） if and only if it is a centralizer. Let 5 ： U →U be an additive map. It is also shown that the following four conditions are equivalent： （1） 5 is specially generalized derivable at zero point, i.e., 5（AB） = δ（A）B ＋ AS（B） - Aδ（I）B whenever AB = 0; （2） 5 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
基金Supported by the National Natural Science Foundation of China(Grant Nos.1117124411601010)
文摘Let A and B be unital Banach algebra and M be Banach A, B-module. Then T＇ = （AB^M） becomes a triangular Banach algebra when equipped with the Banach space norm ｜｜（ ab^m）｜｜ = ｜｜a｜｜A ＋｜｜m｜｜M ＋ ｜｜b｜｜m A Banach algebra T is said to be Lie n-weakly amenable if all Lie derivations from T into its nth dual space T^（n） are standard. In this paper we investigate Lie n-weak amenability of a triangular Banach algebra T in relation to that of the algebras A, B and their action on the module M.
基金Supported by the National Natural Science Foundation of China (Grant No. 11171294)the Natural Science Foundation of Heilongjiang Province (Grant No. A201013)the Fund of Heilongjiang Education Committee(Grant No. 11541268)
文摘In this paper, we study a class of subalgebras of the Lie algebra of vector fields on n-dimensional torus, which are called the Triangular derivation Lie algebra. We give the structure and the central extension of Triangular derivation Lie algebra.
文摘Let T(R) be a two-order upper matrix algebra over the semilocal ring R which is determined by R=F×F where F is a field such that CharF=0. In this paper, we prove that any Jordan automorphism of T(R) can be decomposed into a product of involutive, inner and diagonal automorphisms.
基金Supported by the National Natural Science Foundation of China(Grant No.11101084)the Natural Science Foundation of Fujian Province(Grant No.2013J01005)
文摘Let F be a field, n ≥ 3, N（n,F） the strictly upper triangular matrix Lie algebra consisting of the n × n strictly upper triangular matrices and with the bracket operation {x, y} = xy-yx. A linear map φ on N（n,F） is said to be a product zero derivation if {φ（x）,y] ＋ [x, φ（y）] = 0 whenever {x, y} = 0,x,y ∈ N（n,F）. In this paper, we prove that a linear map on N（n, F） is a product zero derivation if and only if φ is a sum of an inner derivation, a diagonal derivation, an extremal product zero derivation, a central product zero derivation and a scalar multiplication map on N（n, F）.
基金Supported by the National Natural Science Foundation of China(60873267)the Ningbo Natural Science Foundation of China(2011A610172)K.C.Wang Magna Fund in Ningbo University
文摘Let （C,α） and （H, β） be Hom-bialgebras and ω ： C × H → H × C a linear map. We introduce a Horn-ω-smash coproduct （Cω H, γ） and give necessary and sufficient conditions for （Cω H, γ） to be a Hom-bialgebra. We study the quasi-triangular structures over （Cω H, γ） and show the necessary and sufficient conditions for （Cω H, γ R） to be a quasi-triangular Hom-Hopf algebra. As applications of our results, we introduce the concept of D（H）＊ and construct quasi-triangular structures over D（H）＊.
