Abstract. In this paper we investigate Lie bialgebra structures on the Schrödinger-Virasoro algebra L. Surprisingly, we find out an interesting fact that not all Lie bialgebra structures on the Schrödinger-Virasoro algebra are triangular coboundary, which is different from the related known results of some Lie algebras related to the Virasoro algebra.

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A class of weight modules M(V,a) over the twisted Heisenberg-Virasoro H are constructed, which includes modules of intermediate series, where V is an H̄r,d-module and a is a complex number. We give the necessary and sufficient conditions under which these modules are simple and also determine all the equivalent simple modules in this class. Moreover, we show that simple modules in this class are new. Finally, we construct a class of new simple non-weight H-modules.

In this paper, we study the structure theory of a class of not-finitely graded Lie algebras related to generalized Virasoro algebras. In particular, the derivation algebras, the automorphism groups and the second cohomology groups of these Lie algebras are determined.

In this paper, we study a class of non-weight modules over the affine-Virasoro algebra of type A 1 , which are free modules of rank one when restricted to the Cartan subalgebra (modulo center). We give the classification of such modules. Moreover, the simplicity and the isomorphism classes of these modules are determined.