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
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
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.
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 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
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
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
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
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 A be a unital algebra and M be a unital .A-bimodule. A linear map δ ： A →M is said to be Jordan derivable at a nontrivial idempotent P ∈A if δ（A） o B ＋ A o δ（B） =δ（A o B） for any A,B ∈ .4 with A o B...Let A be a unital algebra and M be a unital .A-bimodule. A linear map δ ： A →M is said to be Jordan derivable at a nontrivial idempotent P ∈A if δ（A） o B ＋ A o δ（B） =δ（A o B） for any A,B ∈ .4 with A o B = P, here A o B = AB ＋ BA is the usual Jordan product. In this article, we show that if ,A = AlgAN is a Hilbert space nest Mgebra and M = B（H）, or A =M= B（X）, then, a linear mapδ： A→M is Jordan derivable at a nontrivial projection P ∈ N or an arbitrary but fixed nontrivial idempotent P∈ B（X） if and only if it is a derivation. New equivalent characterization of derivations on these operator algebras was obtained.展开更多
The AR-quiver and derived equivalence are two important subjects in the repre- sentation theory of finite dimensional algebras, and for them there are two important research tools-AR-sequences and ：D-split sequences....The AR-quiver and derived equivalence are two important subjects in the repre- sentation theory of finite dimensional algebras, and for them there are two important research tools-AR-sequences and ：D-split sequences. So in order to study the representations of triangular matrix algebra T2（T） - （T O,T T）whereTis a finite dimensional algebra over afield, it is important to determine its AR-sequences and ：D-split sequences. The aim of this paper is to construct the right（left） almost split morphisms, irreducible morphisms, almost split sequences and V-split sequences of T2（T） through the corresponding morphisms and sequences of T. Some interesting results are obtained.展开更多
Let R be a commutative ring with identity, Tn （R） the R-algebra of all upper triangular n by n matrices over R. In this paper, it is proved that every local Jordan derivation of Tn （R） is an inner derivation and t...Let R be a commutative ring with identity, Tn （R） the R-algebra of all upper triangular n by n matrices over R. In this paper, it is proved that every local Jordan derivation of Tn （R） is an inner derivation and that every local Jordan automorphism of Tn（R） is a Jordan automorphism. As applications, we show that local derivations and local automorphisms of Tn （R） are inner.展开更多
The aim of this paper is mainly to build a new representation-theoretic realization of finite root systems through the so-called Frobenius-type triangular matrix algebras by the method of reflection functors over any ...The aim of this paper is mainly to build a new representation-theoretic realization of finite root systems through the so-called Frobenius-type triangular matrix algebras by the method of reflection functors over any field. Finally, we give an analog of APR-tilting module for this class of algebras. The major conclusions contains the known results as special cases, e.g., that for path algebras over an algebraically closed field and for path algebras with relations from symmetrizable cartan matrices. Meanwhile, it means the corresponding results for some other important classes of algebras, that is, the path algebras of quivers over Frobenius algebras and the generalized path algebras endowed by Frobenius algebras at vertices.展开更多
For any positive integer N,we clearly describe all finite-dimensional algebras A such that the upper triangular matrix algebras TN(A)are piecewise hereditary.Consequently,we describe all finite-dimensional algebras A ...For any positive integer N,we clearly describe all finite-dimensional algebras A such that the upper triangular matrix algebras TN(A)are piecewise hereditary.Consequently,we describe all finite-dimensional algebras A such that their derived categories of N-complexes are triangulated equivalent to derived categories of hereditary abelian categories,and we describe the tensor algebras A⊗K[X]/(X^(N))for which their singularity categories are triangulated orbit categories of the derived categories of hereditary abelian categories.展开更多
Gluing techniques with respect to a recollement have long been studied. Recently, ladders of recollements of abelian categories were introduced as important tools. In this paper, we present explicit constructions of g...Gluing techniques with respect to a recollement have long been studied. Recently, ladders of recollements of abelian categories were introduced as important tools. In this paper, we present explicit constructions of gluing support τ-tilting modules via symmetric ladders of height two. Moreover, we apply the result to triangular matrix algebras to give a detailed version of the known Jasso's reduction and study maximal green sequences.展开更多
Let R be an Artin algebra and e be an idempotent of R. Assume that Tor_(i)^(eRe)(Re, G) = 0 for any G ∈ GprojeRe and i sufficiently large. Necessary and sufficient conditions are given for the Schur functor S_(e) to ...Let R be an Artin algebra and e be an idempotent of R. Assume that Tor_(i)^(eRe)(Re, G) = 0 for any G ∈ GprojeRe and i sufficiently large. Necessary and sufficient conditions are given for the Schur functor S_(e) to induce a triangle-equivalence ■. Combining this with a result of Psaroudakis et al.(2014),we provide necessary and sufficient conditions for the singular equivalence ■ to restrict to a triangle-equivalence ■. Applying these to the triangular matrix algebra ■,corresponding results between candidate categories of T and A(resp. B) are obtained. As a consequence,we infer Gorensteinness and CM(Cohen-Macaulay)-freeness of T from those of A(resp. B). Some concrete examples are given to indicate that one can realize the Gorenstein defect category of a triangular matrix algebra as the singularity category of one of its corner algebras.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.
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 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.
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.
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. 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.
基金Supported by National Natural Foundation of China(11001194)Provincial International Cooperation Project of Shanxi(2014081027-2)
文摘Let A be a unital algebra and M be a unital .A-bimodule. A linear map δ ： A →M is said to be Jordan derivable at a nontrivial idempotent P ∈A if δ（A） o B ＋ A o δ（B） =δ（A o B） for any A,B ∈ .4 with A o B = P, here A o B = AB ＋ BA is the usual Jordan product. In this article, we show that if ,A = AlgAN is a Hilbert space nest Mgebra and M = B（H）, or A =M= B（X）, then, a linear mapδ： A→M is Jordan derivable at a nontrivial projection P ∈ N or an arbitrary but fixed nontrivial idempotent P∈ B（X） if and only if it is a derivation. New equivalent characterization of derivations on these operator algebras was obtained.
基金Supported by the National Natural Science Foundation of China(Grant No.10971172)the Natural Science Foundation of Beijing(Grant Nos.10920021122002)
文摘The AR-quiver and derived equivalence are two important subjects in the repre- sentation theory of finite dimensional algebras, and for them there are two important research tools-AR-sequences and ：D-split sequences. So in order to study the representations of triangular matrix algebra T2（T） - （T O,T T）whereTis a finite dimensional algebra over afield, it is important to determine its AR-sequences and ：D-split sequences. The aim of this paper is to construct the right（left） almost split morphisms, irreducible morphisms, almost split sequences and V-split sequences of T2（T） through the corresponding morphisms and sequences of T. Some interesting results are obtained.
基金Supported by the Doctor Foundation of Henan Polytechnic University (Grant No. B2010-93)
文摘Let R be a commutative ring with identity, Tn （R） the R-algebra of all upper triangular n by n matrices over R. In this paper, it is proved that every local Jordan derivation of Tn （R） is an inner derivation and that every local Jordan automorphism of Tn（R） is a Jordan automorphism. As applications, we show that local derivations and local automorphisms of Tn （R） are inner.
基金Supported by National Natural Science Foundation of China(Grant Nos.11271318 and 11571173)the Zhejiang Provincial Natural Science Foundation of China(Grant No.LZ13A010001)
文摘The aim of this paper is mainly to build a new representation-theoretic realization of finite root systems through the so-called Frobenius-type triangular matrix algebras by the method of reflection functors over any field. Finally, we give an analog of APR-tilting module for this class of algebras. The major conclusions contains the known results as special cases, e.g., that for path algebras over an algebraically closed field and for path algebras with relations from symmetrizable cartan matrices. Meanwhile, it means the corresponding results for some other important classes of algebras, that is, the path algebras of quivers over Frobenius algebras and the generalized path algebras endowed by Frobenius algebras at vertices.
文摘For any positive integer N,we clearly describe all finite-dimensional algebras A such that the upper triangular matrix algebras TN(A)are piecewise hereditary.Consequently,we describe all finite-dimensional algebras A such that their derived categories of N-complexes are triangulated equivalent to derived categories of hereditary abelian categories,and we describe the tensor algebras A⊗K[X]/(X^(N))for which their singularity categories are triangulated orbit categories of the derived categories of hereditary abelian categories.
基金supported by National Natural Science Foundation of China (Grant No. 12201211)the China Scholarship Council (Grant No. 202109710002)。
文摘Gluing techniques with respect to a recollement have long been studied. Recently, ladders of recollements of abelian categories were introduced as important tools. In this paper, we present explicit constructions of gluing support τ-tilting modules via symmetric ladders of height two. Moreover, we apply the result to triangular matrix algebras to give a detailed version of the known Jasso's reduction and study maximal green sequences.
基金supported by National Natural Science Foundation of China (Grant Nos. 11626179, 12101474, 12171206 and 11701455)Natural Science Foundation of Jiangsu Province (Grant No. BK20211358)+1 种基金Natural Science Basic Research Plan in Shaanxi Province of China (Grant Nos. 2017JQ1012 and 2020JM-178)Fundamental Research Funds for the Central Universities (Grant Nos. JB160703 and 2452020182)。
文摘Let R be an Artin algebra and e be an idempotent of R. Assume that Tor_(i)^(eRe)(Re, G) = 0 for any G ∈ GprojeRe and i sufficiently large. Necessary and sufficient conditions are given for the Schur functor S_(e) to induce a triangle-equivalence ■. Combining this with a result of Psaroudakis et al.(2014),we provide necessary and sufficient conditions for the singular equivalence ■ to restrict to a triangle-equivalence ■. Applying these to the triangular matrix algebra ■,corresponding results between candidate categories of T and A(resp. B) are obtained. As a consequence,we infer Gorensteinness and CM(Cohen-Macaulay)-freeness of T from those of A(resp. B). Some concrete examples are given to indicate that one can realize the Gorenstein defect category of a triangular matrix algebra as the singularity category of one of its corner algebras.
