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 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).