Structure And Representation Of Infinite Dimensional Lie (Super)Algebras

The extended Schr¨odinger-Virasoro algebra s v and Lie algebra W are two Lie alge-bras related to the Virasoro algebra and have ample connection with it. In this thesis, weintroduce the so-called double extended Schro¨dinger-Virasoro algebrad sv and W [G] bygeneralizingd sv and Lie algebra W respectively and then discuss their derivation algebra,automorphism group and Verma module. Toroidal Lie (super)algebra is natural general-ization of afne Kac-Moody (super)algebra and they show many similar and interestingproperties. In this thesis, we construct representations of2-toroidal Lie (super)algebraof classical types by using bosonic and fermioic fields which generalize the correspondingresults for classical afne Lie (super)algebra in [25]and [59].This thesis consists of seven chapters. The content of each chapter are described asfollows:In chapter1, we recall some main results of Lie algebras related to Virasoro algebraand toroidal Lie (super)algebras obtained in the last two decades. In additional, weintroduce the main work done in this thesis.In chapter2, all basic knowledge needed in the subsequent chapters are collectedwhich include the basic notion of general Lie (supera)algebra, Kac-Moody Lie (super)algebras,Toroidal Lie (superalgebras) and techniques of formal calculus used in the the represen-tation theory of infinite dimensional Lie (super)algebras.In chapter3, we introduce the so-called double extended Schr¨odinger-Virasoro alge-brad sv, then study its derivation algebra and automorphism group.In chapter4, Lie algebra W [G] associated to group G is defined, then its automor-phism group and Verma module are investigated.Chapter5is the keystone of the later two chapters. We recall the procedure of con-structing free field representation for classical afne Lie (super)algebra by using bosonicor fermionic fields in [25] and [59] in detail and make preparations for the latter twochapters.In chapter6, we first recall the Moody-Rao-Yokonuma（MRY)-presentation for2-Toroidal Lie algebra and then construct the bosonic and fermionic representation for the2-Toroidal Lie algebra of classical types.In Chapter7, we give and prove a MRY-like presentation for the2-Toroidal Lie su-peralgebra of type A(m, n), B(m, n), C(n) and D(m, n) and then construct the bosonicand fermionic representation for these Lie superalgebras. In additional, we give a ver-tex representation of2-toroidal Lie superalgebra of type A(m, n) by using both vertex operators and Weyl bosoinic fields.For convenience, we list all extended Cartan matrix of Lie algebras of classical typesand extended distinguished Cartan matrix of Lie superalgebras of type A(m, n), B(m, n),C(n) and D(m, n).