The Structures And Representations Of Schr(?)dinger-Virasoro Lie Algebras And Non-Graded Virasoro-Like Lie Algebras

The structures and representations of infinite dimensional Lie algebras have drawn more and more attentions because of their great importance and applica-tions.Many non-graded infinite dimensional Lie algebras naturally appear in the fields of conformal field theory,statistic mechanics and Hamilton operator theory. In general,the structures of non-graded Lie algebras are complex,which makes it much more difficult and more challenging for us to study their structures and representations.In the first part of the paper,representations of the infinite dimensional non-graded Virasoro-like Lie algebra L are studied.Firstly,we prove that an irreducible module over L,which satisfies a natural condition,is a GHW module or uniformly bounded.Secondly,we give the complete classification of a class of indecomposable L-modules and prove there are seven cases:A_{α,λ,μ},A_0,λ,μ,A_1,λ,μ,A_{1,0,λ,μ},B_{1,0,λ,μ}, A_{0,1,λ,μ}and A_{0,1,λ,μ}.Finally,the central extension of the subalgebra W of L is determined.In 1994,M.Henkel[3]originally introduced the definition of Schr(?)dinger-Virasoro Lie algebra sb,which is of great application in the field of mathematical physics and statistical mechanics.Recently,J.Unterberger[4]defined an infinite-dimensional Lie algebra called the extended Schr(?)dinger-Virasoro Lie algebra sb_e. Few results have been obtained related to the structure and representation of sb_e. In the second part of the paper,we prove that an irreducible representation over the universal central extension of(?)is a highest weight module,or a lowest weight module or uniformly bounded.Secondly,we show sb_e has no outer derivation. Consequently,sb_e is an infinite dimensional complete Lie algebra.Thirdly,the universal central extension(?)of sb_e is determined.Fourthly,we prove there is no non-trivial invariant bilinear fbrm on sb_e,which suggests the universal central extension of sb_e in the category of Leibniz algebras is the same as that in the category of Lie algebras.Finally,the automorphism group of sb_e is given.In the last part,we study the structure of the semi-direct product of Witt algebra and its density tensor module,denote by W(a,b).The Lie algebras W(a,b) and their central extensions are important structures,which naturally emerge in super-string theory.W(a,b)include some well-known algebras,such as the twisted Heisenberg-Virasoro algebra which is the universal central extension of W(0,0). In this section,the derivation algebras,central extensions and the automorphism groups of W(a,b)are all determined.