Representations Of The Affine Nappi-Witten Lie Algebras And Infinite Dimensional Lie Algebra (?)(α,β)

The Lie algebra H4and W(α,β) are in the profound physical backgrounds, which have close connections with many important areas of Lie theory, for instance, VOA, Virasoro algebra, K-M Lie algebra and so on. Therefore it is significant in physics to study their representation theory, meanwhile, can enrich structures and representations of the Lie algebra H4and W(α,β) in mathematics. In this thesis, the representation theories of the twisted Lie algebras and Lie algebra W(α,β) are mainly researched.Part1investigates the polynomial representations of the non-twisted Lie al-gebra H4and the classification of irreducible non-zero level quasifinite H4-modules. Firstly, the polynomial representations of the non-twisted Lie algebra H4on C[x0, xi,j, yi,j|i=1,2; j=1,2,...] are given. Secondly, the irreducible non-zero level quasifi-nite H4-modules are classified. The irreducible non-zero level quasifinite H4-module is the HW module or LW module.In part2, the representation theory of the twisted Lie algebra H4[τ1] is studied. Firstly, the Verma module MH4[τ1](k,l) of Lie algebra H4[τ1] is an irreducible mod-ule(?)k≠0. Besides, the irreducible quotient module and singular vectors of Lie algebra H4[τ1] are presented under the reducibility. Secondly, the vertex operator representations of Lie algebra H4[τ1] are constructed. Finally, the irreducible non-zero level quasifinite modules of Lie algebra H4[τ1] are determined. The irreducible non-zero level quasifinite H4[τ1]-module is the HW module or LW module.Part3mainly investigates the representation theory of the twisted Lie alge-bra H4[τ2]. The Verma module MH4[τ2](c, d,l) of Lie algebra H4[τ2] is irreducible (?)l≠0and c(?)(2Z+1)l, and all the linearly independent singular vectors of Lie algebra H4[τ2] under the reducibility are given. Then vertex operator repre- sentations of Lie algebra H4and H4[τ2] are constructed respectively. Finally, the irreducible non-zero level quasifinite modules of Lie algebra H4[τ2] are classified. That is, the irreducible non-zero level quasifinite H4[τ2]-module is the HW module or LW module.Part4is devoted to study the representation theory of the Lie algebra W(α,β). Firstly, MIS over W(α,β) for ai=0, bi(?){-1,0,1},i=1,2are determined, and one can see that the W(α,β)-module of the intermediate series is isomorphic to MIS over Virasoro algebra. Secondly, all irreducible H-C modules over W(α,β) for ai=0,bi≠1,i=1,2are classified, and the irreducible H-C module over W(α,β) is either the HW/LW module or UBM. Moreover, the irreducible weight modules of W(α,β) with at least one non-trivial finite dimensional weight space for ai=0,-1≤bi≠1if bi∈Z, i=1,2are classified. Finally, the Verma module M(c, h, hy) over a subalgebra of W(α,β) is irreducible (?)hV≠0.