A_(n)型扩张扭导出Hall代数的Grobner-Shirshov基

Grobner-Shirshov Basis of Extended Twisted Derived Hall Algebra of Type A_(n)

在Dynkin型Ringel-Hall代数中,不可分解表示同构类之间的拟交换关系的集合构成一个极小Grobner-Shirshov基,并且相应的不可约元素构成此Ringel-Hall代数的一组PBW型基.本文的目的是把此结果推广到A_(n)型扩张扭导出Hall代数上去.为此,首先计算A_(n)的不可分解表示同构类之间的斜交换关系,并且证明这些关系之间的所有合成是平凡的,从而是A_(n)型扩张扭导出Hall代数的一个极小Grobner-Shirshov基.其次,用扩张扭导出Hall代数与格代数之间的同构来给出A_(n)型格代数的一个极小Grobner-Shirshov基.最后,作为一个应用,通过取不可约元素分别给出A_(n)型扩张扭导出Hall代数和格代数的PBW基.

In Ringel-Hall algebra of Dynkin type,the set of all skew commutator relations between the iso-classes of indecomposable representations forms a minimal Grobner-Shirshov basis and the corresponding irreducible elements form a PBW-type basis of the Ringel-Hall algebra.The aim of this paper is to generalize this result to the extended twisted derived Hall algebra of type A_(n).First,a minimal Grobner-Shirshov basis for the extended twisted derived Hall algebra is given by computing all skew commutator relations between the iso-classes of indecomposable representations of A_(n)and then it is proved that all possible compositions between these skew commutator relations are trivial.Next,by using the isomorphism between the extended twisted derived Hall algebra and the lattice algebra,a minimal Grobner-Shirshov basis for the lattice algebra of type A_(n)is given.Finally,as an application,by taking the corresponding irreducible monomials,the PBW-type base of the extended derived Hall algebra and lattice algebra are constructed,respectively.