Virasoro Algebra Class Vertex Lie Algebra

The structure and representations theory of Lie algebra play important roles instudying the in?nite dimensional Lie theory. Virasoro algebra is an important kindof Lie algebra. With the development of the Lie algebra, many Lie algebras, whichare related with the Virasoro algebra, emerge. They become one of the populardirections of researching.In this paper, it has four main parts. The ?rst chapter is the introduction. Itcontains the development of Lie algebra, In?nite dimensional Lie algebra, Vertexalgebra. It has three sections. We study the central extension of Triangular deriva-tion Lie algebra. In the Triangular derivation Lie algebra, it has at least d Wittalgebras, it is a class of central extension of Witt algebra. The ?rst section is thede?nition of the Triangular derivation Lie algebra; In the second section we give thecharacterization of the Triangular derivation Lie algebra; Then we give the centralextension of the Triangular derivation Lie algebra through the method of calculatingthe 2?cocycle of the Triangular derivation Lie algebra. The form of the central ex-tension of the Triangular derivation Lie algebra is the natural extension of Virasoroalgebra. The third chapter is the vertex algebra of the twisted Heisenberg-Virasoroalgebra. We get the vertex algebra of the twisted Heisenberg-Virasoro algebra bycharactering its restricted module. The fourth chapter is the conclusion.