A completion of the quantized enveloping algebra of a Kac-Moody algebra
Let \(\mathfrak g\) be a symmetrizable Kac-Moody algebra, i.e. the associated Cartan matrix has diagonal entries equal to 2, non-negative integer off-diagonal entries and is symmetrizable. Then \(\mathfrak g\) admits a triangular decomposition \({\mathfrak g}={\mathfrak n}^-\oplus{\mathfrak h}\oplus{\mathfrak n}^+\) with Cartan subalgebra \(\mathfrak h\). If \(A\) is positive definite, then \(\mathfrak g\) is finite dimensional and semisimple. In this case the structure of the enveloping algebra \(U({\mathfrak g})\) is well understood. The aim of the paper under review is to establish a framework in which results assembling those for \(\mathfrak g\) semisimple hold for any symmetrizable \(\mathfrak g\). The general philosophy is to pass to some completion of \(U({\mathfrak g})\), then to determine its centre and as far as possible to describe its \(\text{ad}U({\mathfrak g})\)-module structure. A first step in the realization of this scheme is a rather surprising ``cancellation-of-terms'' theorem. Fixing a simple highest weight module \(V(\lambda)\), \(\lambda\in {\mathfrak h}^{\ast}\) a dominant integral weight, then each \(a\in\text{End} V(\lambda)\) can be lifted to an element \(l_{\lambda}(a)\) of a suitable completion of the ``simply connected'' Drinfeld-Jimbo quantization \(U_q({\mathfrak g})\) of \(U({\mathfrak g})\) and the map \(a\to l_{\lambda}(a)\) is \(\text{ad}U_q({\mathfrak g})\)-invariant. This completion admits a triangular decomposition and a Harish-Chandra projection \({\mathcal P}'\). For a weight vector \(a\) of weight zero, the author derives a remarkably simple formula for \({\mathcal P}'(l_{\lambda}(a))\) which is new and interesting even for \(\mathfrak g\) semisimple. In the general case the completion is too big to carry an algebra structure. One has to cut down this completion to recover an algebra structure and the author indicates possible ways to do this. There are some interesting phenomena and simplifications which arise when \(\mathfrak g\) is affine, i.e. when \(A\) is positive semidefinite. The author gives indications that some of the results carry over to \(U_q({\mathfrak g})\) and that passing to a completion of \(U_q({\mathfrak g})\) drastically changes the situation.
scientific article; zbMATH DE number 1275263