Representations of quantum algebras
type: Item
instance of: scholarly article
title: Representations of quantum algebras
zbMATH Open document ID: 0724.17012
DOI: 10.1007/BF01245066
author: P. Polo
author: Henning Haahr Andersen
author: Kexin Wen
published in: Inventiones Mathematicae
publication date: 1991-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143874
review text: Let U be the quantum group associated to a Cartan matrix \((a_{ij})\) of finite type. Thus, U is a Hopf algebra over the field \(A={\mathbb{Q}}(v)\) of rational functions in an indeterminate v. The authors study the representation theory of U by defining a kind of dual of U, the `coordinate algebra' \({\mathcal A}[U]\), and setting up a general theory of induction from suitable subalgebras. In the case of `generalized parabolic subalgebras', it is shown that this induction procedure has the standard properties, e.g. Frobenius reciprocity, transitivity and the tensor identity. In the case of a `Borel subalgebra', analogues of Serre's theorem, Grothendieck's theorem, Kempf's vanishing theorem for dominant characters and Demazure's character formula are obtained, and it is shown that the concepts and results about good and excellent filtrations extend to the quantum case. Let \(U_{\Gamma}\) be obtained from U by specializing \({\mathcal A}\) into a field \(\Gamma\). The authors develop a Borel-Weil-Bott theory for \(U_{\Gamma}\). If the image \(\zeta\) of v is not a root of unity, the theory is analogous to the classical case, while if \(\zeta\) is a root of unity (and \(char(\Gamma)=0)\) it is analogous to the classical modular theory. The authors establish a linkage principle and a translation principle for \(U_{\Gamma}\) which enables them to define a kind of ``Jantzen filtration'' and prove a sum formula. It is also shown that (whether or not \(\zeta\) is a root of unity) finite-dimensional \(U_{\Gamma}\)-modules are integrable.
reviewed by: Andrew N. Pressley
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 4193960
zbMATH Keywords: quantum group
zbMATH Keywords: representation
zbMATH Keywords: coordinate algebra
zbMATH Keywords: induction
zbMATH Keywords: generalized parabolic subalgebras
zbMATH Keywords: Borel-Weil-Bott theory
zbMATH Keywords: linkage principle
zbMATH Keywords: translation principle
zbMATH Keywords: Jantzen filtration
MaRDI profile type: Publication
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OpenAlex ID: W2081085222
description: scientific article