The Structures And Representations Of Some Infinite-Dimensional Lie (Bi)Algebras Related To The Virasoro Algebra

In this paper, we investigate the structures and representations of some infinite-dimensional Lie (bi)algebras that are related to the Virasoro algebra. We determine the Lie bialgebra structure of a class of infinite Lie algebras B, investigate the derivation algebra and automorphism group of the twisted Schro|¨dinger-Virasoro algebra and discuss the 2-cocycles of deformative Schro|¨dinger algebras. Finally, indecomposable modules of the intermediate series over the Schro|¨dinger-Virasoro algebra are classified.It is well known that the three-dimensional simple Lie algebra sl(2, F) over any closed field F of characteristic 0 is a powerful tool in investigating the structure and representation theory of finite-dimensional Lie algebras. Meanwhile, the Witt algebra and also its central extension, i.e., the Virasoro algebra, as the simplest examples of infinite-dimensional Lie algebras, have attracted the attentions of many mathematicians and physicists from their appearance. Many interesting results on these type algebras appeared. As the structure and representation theory of them become more and more complete, the results, methods and also techniques on these algebras are helpful to investigate infinite-dimensional Lie algebras of other types, particularly to the Lie algebras related to the Virasoro algebra. Then many papers on higher rank Virasoro Lie algebras and also generalized Virasoro Lie algebras appeared (cf. [1]-[6] etc), and many interesting results were generalized to algebras of these types. However, these generalizations were not trivial with some techniques. It is known that one can construct larger Lie algebras using Lie algebras themselves and their modules. Then many algebras based on the Virasoro algebra appeared such as the q-deformed Virasoro algebra, (generalized)Block type algebras, generalized Witt type algebras, (generalized)Virasoro-like Lie algebras and its q analogue, the Virasoro-toroidal Lie algebra, the (generalized) (twisted)Heisenberg-Virasoro Lie algebra, N = 2 superconformal algebras and Schro|¨dinger-Virasoro type Lie algebras. Naturally, there appeared many papers on these algebras (cf. [7]-[22] etc).We investigate Lie bialgebras of a class of infinite-dimensional Lie algebras B related to the Virasoro Lie algebra in Chapter f. We prove that every Lie bialgebra on the Lie algebra B is a triangular coboundary Lie bialgebra. The notion of Lie bialgebras was introduced by Drinfel'd in 1983. The Lie bialgebras of Virasoro type, generalized type and generalized Virasoro-like type were respectively investigated in [23, 24, 25]. These types algebras were quantized in [26, 27, 28] respectively. Considering the facts that there are no general methods to investigate Lie bialgebras, and there are different difficulties to algebras of different types corresponding to different Hopf algebra structures.In Chapter 2 we consider the 2-cocycles over the deformative Schro|¨dinger-Virasoro Lie algebras and the derivation algbra and the automorphism groups over the twisted Schro|¨dinger-Virasoro algebra with central extensions. We completely describe the second cohomology group of the deformative Schro|¨dinger-Virasoro algebras and list all the generators under different conditions. And it is proved that its derivation algebra is spanned by its inner derivation algebra and three outer derivations. The original Schro|¨dinger-Virasoro Lie algebra was introduced in [29], in the context of non-equilibrium statistical physics. Nowadays, there are not many references investigating Lie algebras of these types, but these types Lie algebras have been paid much attention recently (cf. [29]-[33]). The vertex representations were investigated in [32].In Chapter 3 we prove that there are no irreducible mixed modules over the twisted Schro|¨dinger-Virasoro Lie algebras without central extension either and classify the modules of intermediate series over the Schro|¨dinger-Virasoro Lie algebras with central extensions. The conjecture that there are no irreducible mixed modules over the Virasoro Lie algebras, given in [37], was proved in [38]. Afterwards, it was also proved that there are no irreducible mixed modules over the twisted Heisenberg-Virasoro Lie algebras and the W algebra W(2,2) respectively in [21] and [11]. Also there are many papers that have ever investigated the modules of intermediate series (e.g., [3, 5, 19, 39, 40, 42, 43, 44] etc).