A Family Of Representations Of The General Linear Lie Superalgebra Coordinated By Certain Rings

In this thesis, we will construct a family representations of general linear Lie su-peralgebra glm|n(C) and extend this construction to general linear affine Lie superalge-bra0tTO|n(C), finally we construct similar representations for A{in■1, n■1)梘raded Lie superalgebra g(m(n (Cq)’.In1977, Kac in [42] classified all simple complex finite-dimensional Lie superal-gebras. Since then, these superalgebras, particularly of type A(m, n), have been found applications in various areas including quantum mechanics, nuclear physics, particle physics, and string theory. Understanding their module theory, however, has been a very difficult problem, even at the level of the finite-dimensional simple modules for the Lie superalgebras of type A(m,ri). As for the structure and representations of infinite-dimensional Lie superalgebras, these are understood only in particular cases.People have constructed a great number of various irreducible modules for affine Lie superalgebras from different contexts. For example, Iohara and Koga in [39] con-struct Wakimoto modules for basic affine Lie superalgebras of type A(m■1, n■1)(1) and D(2, l,a)^. It is certainly not possible to classify all irreducible modules for affine Lie superalgebras but people are able to classify irreducible modules which have certain good conditions and natural properties. For example, Rao and Zhao and others (see [72]) classified all irreducible integrable modules with finite dimensional weight spaces for the affine superalgebras which are not type A(m, n) or C(m), who proved that such modules comprise of irreducible integrable highest weight modules, irre-ducible integrable lowest weight modules and evaluations modules. Recently, Wu and Zhang (see [91]) classified all irreducible integrable modules for affine Lie superal-gebras of type A(m,n) and C(m), more precisely, there is a new class of integrable modules for affine Lie superalgebras of type A(m, n) and C{m) which must be of highest weight type, but are not necessarily evaluation modules.Wakimoto used free fields to construct representations for affine Kac-Moody alge-bra A^and Hermitian representations for some extended affine Lie algebras ([87,88]). Wakimoto’s free field construction provides a remarkable way to realize representation-s for some infinite-dimensional Lie algebras and Lie superalgebra (see [12,28,33,34]). Gao and Zeng in [33,92,93], used the free fields to construct Hermitian representa-tions for extended affine Lie algebras. Later, they also construct a class for irreducible modules of the affine Lie algebra gln([34,94]).Some researchers called these modules Wakimoto-like modules.In fact,the constructions are different from Wakimoto’s orig-inal construction.Recently,Bhargava,Chen and Gao in[12]extend this construction to Lie superalgebras g[12—1(Cq)on the exterior algebra with infinitely many vailabIes and they shOW the representation to be irreducible if and only if the parameter p is nonzero.so,a natural question for us is that what abOUt the Lie superalgebm g[mIn(C), g[mI.(c)and A(m一1,ln一1)一graded Lie superalgebra g[mIn(Cq)’cases.In this thesis we6rst construet representations Of general1inear Lie superalgebra g[mIn(C).secondly,we extend thjs construction to generallinear a伍ne Lie superalge. bra g[min(C).Finally,we constmct similar representations for A(仃L一1,n一1).graded "e superalgebra g[m}n(Cq)’.Our construction is consistent wim[34]when we restrict佗=O,and the same as[12]when we res仃ict m=1.In other words,we generalize their methOd.MoreoveL The results are well known when we restrict our construccjon to n=O(see[28,34,38,94]).S0,in this paper we only need to coilsider ln≥1.