A completion of the quantized enveloping algebra of a Kac-Moody algebra
type: Item
instance of: scholarly article
title: A completion of the quantized enveloping algebra of a Kac-Moody algebra
zbMATH Open document ID: 1042.17014
published in: Journal of Algebra
publication date: 1999-01-01T00:00:00Z
review text: 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 \(\mathfr…
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 17B67
Mathematics Subject Classification ID: 16S30
zbMATH DE Number: 1275263
zbMATH Keywords: Kac-Moody algebra
zbMATH Keywords: symmetrizable algebra
zbMATH Keywords: quantized enveloping algebra
zbMATH Keywords: completion of enveloping algebra
author: Anthony Joseph
reviewed by: Vesselin Drensky
MaRDI profile type: Publication
full work available at URL: jabr.1998.7677
OpenAlex ID: W2023518665
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DOI: 10.1006/JABR.1998.7677
description: scientific article; zbMATH DE number 1275263