Infinite Dimensional Lie Algebras And Generalized Vertex Algebras

The main objects of study in this dissextation are some infinite-dimensional Lie algebras and generalized vertex algebras. These Lie algebras and generalized vertex algebras are directly related to affinc Lie algebras. This (lissertation consists of five chapters including the introduction as the first chapter.In chapter two, we discuss the regular locally finite elements and Cartan subalgebtas of Kac-Moody algebras. In lKac-Moody algel)ras the s1)ht Cartan suhalgebias play the siixular roles that Carta]i sit lxdgel)ras l)laY in the finiteimensional sexnisiinple Lie algebras. As the main esult of t;his chapter, we prove that a sub?algebra ~ of g(A) is a split Cartati subalgebra if and only if there exists a regular locally finite element x such that. tj o (a.dx). This result generalizes l)artially the corresponding result; about; the finiteiiueiis~oual seimsimple Lie algebras. We give also a description of all regular locally finite elenients in g(A).In chapter three, we study toroidal Lie algebras T[.riij. Toroidal Lie algebras were introduced in 1990 by R. V. Moody, S. Eswara Rao, and T. Yokonurna in [MEY]. They are natural generalizations of affine Lie algebras and when rn 1 they are precisely the untwisted affine Lie algebras. This chapter consists of four sections. In section 3.1, we recall the definition and some basic properties of a toroidal Lie algebra from [MEN. In section 3.2, we obtain some properties of T1m3 about the ideals, generators and derivations. We prove that the first cohomology group of TIm] with coefficients in is m2-climensional. In section 3.3, we give the classification and realization of irreducible finite-dimensional representations of iterated loop algebras. Finally in section 3.4, we construct a class of modules for T~rn~ by using the modules for affine Lie algebras.In chapter four, we discuss complete Lie algebias. A Lie algebra g is called a complete Lie algebra, if n satisfies the two conditions:0;In section 4.1, we give a sufficient and necessary condition for a solvable Lie algebra with 1-step nilpotent radical of maximal rank to be complete. In fact, we give a method to construct all these Lie algebras. In section 4.2 and section 4.3 we construct two class of new infinite-dimensional complete Lie algebras by using the modules foi Kac-Moocly algebras.In chapter five, we (lefine a notion of generalized ~ei tex prelgebx a of paialermnion operators on a vector space aIl(i we pi Ow?that any geiteilized vertex pvclgebra ona vectoi' sl>a<:e VV gene1'at'1s lIl il certa1n cil.l1()l1i(:al \vz1y a gelle1'alized vertex a1gel)rawith I4/ as a nat,uraI II1o<111le. This chaPteI' ('o11sists of three sectiofl, In sectioll5.:l 5 ut(, l.(\(t;1,1l H()11l(1 I))l,Hl(' (l(1fit1it,i()llH il.Il(l l.f\H1l1t,s f\ ()lll [Dl;,21. Tll(1 l1l;l,i11 r(is1llt ()fthis chixpter is givell in section 5.2. lll sectio11 5.:l, w(3 sl1ow t,he existence of certain(:"ll()lll('il.l g('ll(lt'ii,liz(t(1 v(\ll,(\x n,1g(il)l'as r(il;l1)('(I t() il,ffjll(! Li() a1g{il,l'a lTl()(llll()s of ti,nyllo11X()r() 1(1v(iI.