In this paper,we first give the general forms of skew commuting maps and skew anti-commuting maps by the Peirce decomposition on a unital ring with a nontrivial idempotent,respectively,and then,as applications,we obta...In this paper,we first give the general forms of skew commuting maps and skew anti-commuting maps by the Peirce decomposition on a unital ring with a nontrivial idempotent,respectively,and then,as applications,we obtain the concrete characterizations of all nonadditive skew(anti-)commuting maps on some operator algebras.展开更多
Let R be a unital ＊-ring with the unit I. Assume that R contains a symmetric idempotent P which satisfies ARP= 0 implies A = 0 and AR（I - P） = 0 implies A = 0. In this paper, it is shown that a surjective map Ф：...Let R be a unital ＊-ring with the unit I. Assume that R contains a symmetric idempotent P which satisfies ARP= 0 implies A = 0 and AR（I - P） = 0 implies A = 0. In this paper, it is shown that a surjective map Ф： R →R is strong skew commutativity preserving （that is, satisfies Ф（A）Ф（B） - Ф（B）Ф（A）＊ ： AB- BA＊ for all A, B ∈R） if and only if there exist a map f ： R → ZSz（R） and an element Z ∈ ZS（R） with Z^2 =I such that Ф（A） =ZA ＋ f（A） for all A ∈ R, where ZS（R） is the symmetric center of R. As applications, the strong skew commutativity preserving maps on unital prime ＊-rings and von Neumann algebras with no central summands of type I1 are characterized.展开更多
We know that in Ringel-Hall algebra of Dynkin type,the set of all skew commutator relations between the iso-classes of indecomposable modules forms a minimal Grobner-Shirshov basis,and the corresponding irreducible el...We know that in Ringel-Hall algebra of Dynkin type,the set of all skew commutator relations between the iso-classes of indecomposable modules forms a minimal Grobner-Shirshov basis,and the corresponding irreducible elements forms a PBW type basis of the Ringel-Hall algebra.We aim to generalize this result to the derived Hall algebra DH(An)of type An.First,we compute all skew commutator relations between the iso-classes of indecomposable objects in the bounded derived category D^b(An)using the Auslander-Reiten quiver of D^b(An),and then we prove that all possible compositions between these skew commutator relations are trivial.As an application,we give a PBW type basis of DH(An).展开更多
基金supported by the National Natural Science Foundation of China (Nos.12171290,12301152)the Natural Science Foundation of Shanxi Province (No.202203021222018)。
文摘In this paper,we first give the general forms of skew commuting maps and skew anti-commuting maps by the Peirce decomposition on a unital ring with a nontrivial idempotent,respectively,and then,as applications,we obtain the concrete characterizations of all nonadditive skew(anti-)commuting maps on some operator algebras.
基金Supported by Natural Science Foundation of Shandong Province,China(Grant No.ZR2015Item PA010)National Natural Science Foundation of China(Grant Nos.11526123 and 11401273)
文摘Let R be a unital ＊-ring with the unit I. Assume that R contains a symmetric idempotent P which satisfies ARP= 0 implies A = 0 and AR（I - P） = 0 implies A = 0. In this paper, it is shown that a surjective map Ф： R →R is strong skew commutativity preserving （that is, satisfies Ф（A）Ф（B） - Ф（B）Ф（A）＊ ： AB- BA＊ for all A, B ∈R） if and only if there exist a map f ： R → ZSz（R） and an element Z ∈ ZS（R） with Z^2 =I such that Ф（A） =ZA ＋ f（A） for all A ∈ R, where ZS（R） is the symmetric center of R. As applications, the strong skew commutativity preserving maps on unital prime ＊-rings and von Neumann algebras with no central summands of type I1 are characterized.
基金Supported by the Natural Science Foundation of China(Grant No.11861061)。
文摘We know that in Ringel-Hall algebra of Dynkin type,the set of all skew commutator relations between the iso-classes of indecomposable modules forms a minimal Grobner-Shirshov basis,and the corresponding irreducible elements forms a PBW type basis of the Ringel-Hall algebra.We aim to generalize this result to the derived Hall algebra DH(An)of type An.First,we compute all skew commutator relations between the iso-classes of indecomposable objects in the bounded derived category D^b(An)using the Auslander-Reiten quiver of D^b(An),and then we prove that all possible compositions between these skew commutator relations are trivial.As an application,we give a PBW type basis of DH(An).
