CHAPTER 1 I N T R O D U C T I O N This article contains a complete description of the set of all unitarizable highest weight modules of classical Lie superalgebras. The algebras are over C, and it is a part of the classification to determine which real forms, defined by anti-linear anti-involutions, may occur. As far as we know, this is the first systematic study of the problem but there have been many investigations for some special superalgebras, [2], [3], [4], [6], [7], [8], [10]. For a recent contribution, see [21]. Superalgebras and supergroups have now for some time been attracting a lot of attention for various reasons. Our investigation only deals with superalgebras, and our main references are the articles of V. Kac ([16], [17]). We try to give all the relevant background information in the first two chapters. For further background we refer to the articles by C. Fronsdal, M. Flato, and T. Hirai, collected in [5]. This latter, as well as references cited therein, also provides an introduction to the physical theory. One of the main reasons for our own interest in the subject was our generel interest in unitarity, in particular of highest weight modules. Several years ago we completely solved the analogous question for simple Lie algebras ([14]), after previously having solved the question for special series of algebras (cf. [12] for su(p,q), and [13] for sp(n,E)). Superalgebras are a natural setting for both applying and extending these results. In fact, it has for long been known to us that the main idea behind the proof for the case of simple Lie algebras, "unitarity at the last possible place" ([13]), once properly extended, again would be the key that could open up to the whole classification. Probably the best known finite-dimensional Lie superalgebra is sl(m,n) = (i.i) {I , j | a e sl(m),d e s/(n), b,c arbitrary, and tra = trd}. For m ^ m, this is in fact a basic classical Lie superalgebra which we, to conform to the notation of V. Kac ([16]), will denote by A(m,n) (A(n,n) is the Received by editor 3 March, 1992 1

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