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# Classification of Semisimple Algebraic Monoids

Lex E. Renner
Transactions of the American Mathematical Society
Vol. 292, No. 1 (Nov., 1985), pp. 193-223
DOI: 10.2307/2000177
Stable URL: http://www.jstor.org/stable/2000177
Page Count: 31
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## Abstract

Let X be a semisimple algebraic monoid with unit group G. Associated with E is its polyhedral root system (X, Φ, C), where X = X(T) is the character group of the maximal torus $T \subseteq G, \Phi \subseteq X(T)$ is the set of roots, and $C = X(\bar T)$ is the character monoid of $\bar T \subseteq E$ (Zariski closure). The correspondence E → (X, Φ, C) is a complete and discriminating invariant of the semisimple monoid E, extending the well-known classification of semisimple groups. In establishing this result, monoids with preassigned root data are first constructed from linear representations of G. That done, we then show that any other semisimple monoid must be isomorphic to one of those constructed. To do this we devise an extension principle based on a monoid analogue of the big cell construction of algebraic group theory. This, ultimately, yields the desired conclusions.

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