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Classification of semisimple algebraic monoids


Author: Lex E. Renner
Journal: Trans. Amer. Math. Soc. 292 (1985), 193-223
MSC: Primary 14M99; Secondary 20M99
DOI: https://doi.org/10.1090/S0002-9947-1985-0805960-9
MathSciNet review: 805960
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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,\Phi ,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(\overline T )$ is the character monoid of $ \overline T \subseteq E$ (Zariski closure). The correspondence $ E \to (X,\Phi ,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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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1985-0805960-9
Keywords: Semisimple monoid, polyhedral root system
Article copyright: © Copyright 1985 American Mathematical Society

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