Classification of semisimple varieties
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 algebraic monoids with the same group G of units and with embeddings \(j_ n: G\to M_ n\). Then the varieties \((G/k^*,(M_ n\setminus \{0\})/k^*,j_ n)\), \(n=1,2\) are isomorphic iff the associated polyhedral root systems \((X,\Phi,C_ n)\), \(n=1,2\) are equivariant.
scientific article; zbMATH DE number 4105228
scientific article