The Structure And The Representation Of Lie Algebra Gca

As we all know,the Virasoro algebra and the Schrodinger-Virasoro algebra are two im-portant classes of Lie algebras,whose representations have been attracted a lot of attention from mathematicians.Motivated by those algebras,we mainly discuss the structure and the representation of the Lie algebra gca in this paper.There are six chapters,as follows:In Chapter 1,we briefly review the history of Lie algebra,and present some basic notions.In Chapter 2,we give the definition of the Lie algebra gca and introduce simply impor-tant conclusions which appeared in the article.In Chapter 3,we study the derivations,the central extensions and the automorphism group of the Lie algebra gca.We show that the center of gca is zero and all derivations of gca are inner derivation,which implies that gca is a complete Lie algebra in Section 1.We give the universal central extension of gca in Section 2.We show that there is no non-zero symmetric invariant bilinear form on gca,which implies the universal central extension of gca in the category of Leibniz algebras is the same as that in the category of Lie algebras in Section 3.We determine the automorphism group of gca in Section 4.In Chapter 4,we investigate the unitary representation over the Lie algebra gca along with the central element c.In Section 1,we introduce the definition of conjugate-linear anti-involutions and show that there are only two types of conjugate-linear anti-involutions over the Lie algebra gca.In Section 2,we prove that a unitary weight module over the Lie algebra gca is completely determined by a unitary weight module over the Virasoro algebra.Thus we can classify the unitary modules over the Lie algebra gca.In Chapter 5,we study the automorphism group and Verma module over the gener-alized Lie algebra ggca[G].Firstly,isomorphisms between two generalized Lie algebras ggca[G]and ggca[G]are determined.Next,we discuss the automorphisms of ggca[G].Fi-nally,the irreducibility of Verma module V(c,h)over the generalized Lie algebra ggca[G]is discussed.In Chapter 6,we mainly summarize the conclusions.