Classification of semisimple varieties
type: Item
instance of: scholarly article
title: Classification of semisimple varieties
zbMATH Open document ID: 0675.20036
DOI: 10.1016/0021-8693(89)90216-0
author: Lex E. Renner
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: A projective equivariant completion j: \(G\to V\) of a semisimple algebraic group G over an algebraically closed field k of arbitrary characteristic is called a semisimple variety if it is further assumed that V is normal, and that equivariance refers to the two sided action of G. It is shown that there is an algebraic monoid M containing G as the group of units such that the semisimple varieties V and \((M\setminus \{0\})/k^*\) are isomorphic, where \(k^*\) is the identity component of the center of G. Semisimple algebraic monoids have been classified by the author [Trans. Am. Math. Soc. 287, 457-473 (1985; Zbl 0545.20054)] in terms of combinatorial data, the so-called polyhedral root systems (X,\(\Phi\),C), where X is the character group of a maximal torus T of G, where \(\Phi\) denotes the set of roots, and where C is the monoid of those characters which extend to the Zariski closure of T. Indeed the author established the existence of a bijection between the isomorphy classes of the M and the isomorphy classes of the (X,\(\Phi\),C). The polyhedral root systems occurring here are integral, i.e. there is a homomorphism \(\nu\) : \(X\to {\mathbb{Z}}\) mapping \(\Phi\) to \(\{\) \(0\}\) and the set \({\mathcal F}\) of fundamental generators (spanning a subgroup of finite index in X) to \(\{\) \(1\}\). Each fundamental generator \(\chi\in {\mathcal F}\) and each \(\rho\in X\) gives a self-map \(\sigma_{\chi,\rho}\) via \(\sigma_{\chi,\rho}(\chi)=\rho\) and \(\sigma_{\chi,\rho}(\alpha)=\alpha\) for \(\alpha\in \Phi\). Now two integral polyhedral root systems \((X,\Phi,C_ 1)\) and \((X,\Phi,C_ 2)\) are equivariant iff there is a bijection \(\gamma\) : \({\mathcal F}_ 1\to {\mathcal F}_ 2\) which preserves chambers and with which for each \(\chi\in {\mathcal F}_ 1\), the map \(\sigma_{\chi,\gamma (\chi)}\) restricts to an isomorphism of monoids \(C_ 1[1/\chi]\to C_ 2[1/\gamma (\chi)]\). The main result is now readily formulated: Let \(M_ n\), \(n=1,2\) be semisimple alg…
reviewed by: Karl H. Hofmann
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 20M99
Mathematics Subject Classification ID: 17B20
Mathematics Subject Classification ID: 14M99
Mathematics Subject Classification ID: 20M20
zbMATH DE Number: 4105228
zbMATH Keywords: projective equivariant completion
zbMATH Keywords: semisimple algebraic group
zbMATH Keywords: group of units
zbMATH Keywords: semisimple varieties
zbMATH Keywords: Semisimple algebraic monoids
zbMATH Keywords: polyhedral root systems
zbMATH Keywords: character group
zbMATH Keywords: maximal torus
zbMATH Keywords: fundamental generators
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90216-0
OpenAlex ID: W2052221756
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description: scientific article; zbMATH DE number 4105228