The Non-weight Modules Over The Schr(?)dinger-Virasoro Algebras And The Crossed Modules Of Five-dimensional Lie Algebras

Schrodinger-Virasoro Lie algebra is a algebraic structure,introduced by M.Henkel in the study of the invariance of free Schrodinger equation in 1994.It is a natural extension of Virasoro algebra,playing an important role in the field of mathematics and physics.The repre-sentation theory of Schrodinger-Virasoro algebra has been studying by numerous scholars,in which the weight modules structure with finite-dimensional weight spaces has been completely solved.In recent years,the study of non-weight modules of Lie algebra has been paid more attentionThe main content of the first part in the paper is to study a class of non-weight modules of Schrodinger-Virasoro Lie algebra.Denoted by su(s)a Schrodinger-Virasoro Lie algebra,for s=0 or 1/2.Here the Lie algebra su(1/2)is called "original Schrodinger-Virasoro algebra",while the Lie algebra su(0)is called "twisted Schrodinger-Virasoro algebra".Firstly,we studied a class of non-weight modules of twisted Schrodinger-Virasoro alge-bra su(0).We constructed the free module of rank 1 when restricted to U(C0(?)CY0(?)CM0),which is completely reducible.Besides,we classified the class of non-weight U(su(0))-modules.Secondly,we studied the class of non-weight modules of original Schrodinger-Virasoro algebra sv(1/2),which is free of rank 1 when restricted to U(CL0(?)CM0).We proved that this class of U(su(1/2))-modules is nonexistent.The main content of the second part in the paper is to study the crossed modules of a certain five-dimensional Lie algebra L=sl2(?)Cc1(?)Cc2,where c1,c2 are the center of L,and determine the condition of its equivalent class.It is well-known that there exists a bijection between the set of equivalence classes of crossed modules of Lie algebras and the third relative cohomology group of Lie algebras.Furthermore,it is shown that the third relative cohomology group of the five-dimensional Lie algebra is not trivial.