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Tensor structures arising from affine Lie algebras. I, II. (English) Zbl 0786.17017

Let \(\widehat {\mathfrak g}\) be an affine Lie algebra of type \(A\), \(D\) or \(E\). The subject of interest in this paper is a category \({\mathcal O}_ k\) of \(\widehat {\mathfrak g}\)-modules with central charge \(k-h\) (where \(k\in C- Q_{\geq 0}\) and \(h\) is the Coxeter number), of finite length and whose composition factors are simple highest weight modules corresponding to weights which are dominant in the direction of \({\mathfrak g}\). Various equivalent definitions of this category are given. The simple modules in \({\mathcal O}_ k\) are in 1-1 correspondence with the simple finite- dimensional \({\mathfrak g}\)-modules.
The main result is that \({\mathcal O}_ k\) is a rigid braided tensor category. Existence of such a structure on smaller categories of integrable (in the sense of Kac) modules (which are nonzero only for non- negative integer central charge) has been realized in works of Belavin, Polyakov, Knizhnik and Zamolodchikov, Moore and Seiberg, etc. The authors announce results about the equivalence between \({\mathcal O}_ k\) and the category of finite-dimensional integrable representations of a quantum group with parameter \(e^{\sqrt{-1} \pi/k}\). The most interesting case is \(k\in Q_{<0}\), when the categories are not semisimple.
It was shown by Drinfeld that a tensor structure on the category of finite-dimensional representations of quantized enveloping algebras can be obtained from the Knizhnik-Zamolodchikov equations. Integrable connections on certain modules of coinvariants appearing in this paper should be regarded as generalizations of the Knizhnik-Zamolodchikov equations.

MSC:

17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
17B37 Quantum groups (quantized enveloping algebras) and related deformations
19D23 Symmetric monoidal categories
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