Universal central extensions of twisted and untwisted Lie algebras extended over commutative rings
type: Item
instance of: scholarly article
title: Universal central extensions of twisted and untwisted Lie algebras extended over commutative rings
zbMATH Open document ID: 0852.17020
author: Stephen Berman
author: Yaroslav Krylyuk
published in: Journal of Algebra
publication date: 1996-12-12T00:00:00Z
review text: Let \(\Gamma\) be an abelian group, \({\mathcal L} = \oplus_{\alpha \in \Gamma} {\mathcal L}_\alpha\) a \(\Gamma\)-graded Lie algebra over a field \(k\) of characteristic \(p \neq 2\), and \(A = \oplus_{\alpha \in \Gamma} A_\alpha\) a \(\Gamma\)-graded commutative, associative \(k\)-algebra. The universal central extensions of the Lie algebra \(({\mathcal L}, A, \Gamma) = \oplus_{\alpha \in \Gamma} ({\mathcal L}_\alpha \otimes A_\alpha)\) under the assumptions \([{\mathcal L}_0, {\mathcal L}_\alpha] = {\mathcal L}_\alpha\), \(H^1 ({\mathcal L}_0, {\mathcal L}^*_\alpha) = 0\), \(\alpha \in \Gamma\) are studied. The situation considered generalizes most of the cases which have been dealt with by many authors. In the main theorem under the above assumptions the construction of the universal central extension is given. Then the authors consider the untwisted case when \(\Gamma = (0)\) to obtain the structure of universal central extensions in terms of Kähler differentials covering the number of well known results. Under extra assumptions the suggested general construction of universal central extensions is studied in the twisted case when \(\Gamma\) is a cyclic group of order 2 or 3. Various applications of the general construction are presented including classical finite-dimensional simple Lie algebras \({\mathcal L}\) and Lie algebras of Cartan type of characteristic \(p \geq 0\). The results on universal central extensions are used to answer a very interesting question on whether or not the character of the Lie algebra \(L^+_p (A)\) given by generators \(X_0, \dots, X_l\) and defining relations \((\text{ad} X_i)^{-a_{ij} + 1} X_j = 0\), where \(A\) is an affine Cartan matrix, depends on the characteristic \(p\) of the ground field \(k\). It seems to be surprising that it does.
reviewed by: Michael Kuznetsov
Mathematics Subject Classification ID: 17B65
Mathematics Subject Classification ID: 17B67
Mathematics Subject Classification ID: 17B70
Mathematics Subject Classification ID: 17B56
Mathematics Subject Classification ID: 17B66
Mathematics Subject Classification ID: 17B50
zbMATH DE Number: 763947
zbMATH Keywords: twisted Lie algebras
zbMATH Keywords: untwisted Lie algebras
zbMATH Keywords: universal central extensions
Wikidata QID: Q115396206
MaRDI profile type: Publication
full work available at URL: jabr.1995.1090
OpenAlex ID: W2066080250
DOI: 10.1006/JABR.1995.1090
description: scientific article; zbMATH DE number 763947