The Study Of Structure And Representation For The Virasoro Algebra And Extended Affine Lie Algebras Of Type A

It’s well-known that Virasoro algebra is an important infinite-dimensional Lie algebras and the simplest example of infinite-dimensional Lie algebras in their structure and representation theories. Since their structure and representation theories are widely used in the Lie theory and theory physics, it is extensively studied by the majority of mathematicians and physicists. Another infinite Lie algebras which are intensively studied are Kac-Moody algebras and the class of them breaks up into three subclasses: finite, affine and indefinite type. Extended affine Lie algebras are a natural generalization of finite and affine Kac-Moody algebras. Bruce N. Allison, Saeid Azam, Stephen Berman, Yun Gao and Arturo Pianzola introduce the concept of semilattice to describe the extended affine root systems of extended affine Lie algebras, and define a Jordan algebra from a semilattice, then construct an extended affine Lie algebras of type A₁ which is coordinated by the Jordan algebra, i.e. TKK algebra. In this paper, we also discuss some properties in the Virasoro algebra and extended affine Lie algebras of Type A. Their further details are as follows:The first chapter introduces the background, the present research status and significance of the study about the Virasoro algebra and extended affine Lie algebras.Firstly, we recall the Virasoro algebra’s definition and main results in the first chapter,then investigate their representation theories. Finally, we rectify the Irving Kaplansky’s work in detail.In the third chapter, we recall the basic definition and natural construction of TKK algebra in the beginning, then consider maximal algebra of sl₂（?） in the next section. Eventually we put emphasis on making a study about maximal subalgebras of （?） containing the Cartan subalgebra and how to classify into the subalgebras. And we discovery that maximal subalgebra’s structure is closely related to semilattice of TKK Lie algebra. So we hope the discussions about maximal subalgebra can promote semilattice’s structures and are applied to other extended affine Lie algebras’ researches.