0 Classification of Infinite Dimensional Weight Modules over the Lie

We give a complete classification of infinite dimensional indecomposable weight modules over the Lie superalgebra sl(2/1). §1. Introduction Among the basic-classical Lie superalgebras classified by Kac [3], the lowest dimensional of these is the Lie superalgebra B(0, 1) or osp(1, 2), while the lowest dimensional of these which has an isotropic odd simple root is the Lie superalgebra A(1, 0) or sl(2/1). The finite dimensional indecomposable B(0, 1)-modules are known to be simple by Kac [4]. Chmelev [1] classified finite dimensional indecomposable weight sl(2/1)-modules and Leites [5] generalized the result to sl(m/1). We found some indecomposable generalized weight sl(2/1)-modules in [7] (thanks are due to Germoni [2], who pointed out that the list in [7] is incomplete). More generally, [2] obtained the classification of indecomposable (weight or generalized weight) sl(m/n)-modules of singly atypical type and [2] proved that indecomposable sl(m/n)-modules of other types are wild and are unable to be classified. In [6], by using the technique employed in [8] by us in the classification of indecompos-able sl(2)-modules, we were able to give the classification of infinite dimensional indecom-posable B(0, 1)-modules. In this paper, we give the classification of infinite dimensional indecomposable weight sl(2/1)-modules (but not for generalized weight modules) and obtain our main result in Theorem 3.5. The problem of classifying infinite dimensional indecomposable modules of the basic-classical Lie superalgebras has so far received little attention in the literature. However, we believe that our classification of infinite dimensional indecomposable sl(2/1)-modules will certainly help us to better understand modules over (finite or infinite dimensional) Lie superalgebras, just as we have seen in [8] that the classification of infinite dimensional indecomposable sl(2)-modules has helped us to understand modules over the Virasoro algebra. This is our motivation to present the results here. §2. Notations and preliminary results First let us recall some basic concepts. Let G denote the Lie superalgebra sl(2/1),