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Classification of semisimple rank one monoids


Author: Lex E. Renner
Journal: Trans. Amer. Math. Soc. 287 (1985), 457-473
MSC: Primary 20G99; Secondary 20M99
DOI: https://doi.org/10.1090/S0002-9947-1985-0768719-7
MathSciNet review: 768719
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Abstract: Consider the classification problem for irreducible, normal, algebraic monoids with unit group $ G$. We obtain complete results for the groups $ \operatorname{Sl}_2(K) \times {K^\ast}$, $ \operatorname{Gl}_2(K)$ and $ \operatorname{PGl}_2(K) \times {K^\ast}$. If $ G$ is one of these groups let $ \mathcal{E}(G)$ denote the set of isomorphy types of normal, algebraic monoids with zero element and unit group $ G$. Our main result establishes a canonical one-to-one correspondence $ \mathcal{E}(G) \cong {{\mathbf{Q}}^ + }$, where $ {{\mathbf{Q}}^ + }$ is the set of positive rational numbers.

The classification is achieved in two steps. First, we construct a class of monoids from linear representations of $ G$. That done, we show that any other $ E$ must already be one of those constructed. To do this, we devise an extension principle analogous to the big cell construction of algebraic group theory. This yields a birational comparison morphism $ \varphi :{E_r} \to E$, for some $ r \in {{\mathbf{Q}}^ + }$, which is ultimately an isomorphism because the monoid $ {E_r} \in \mathcal{E}(G)$ is regular.

The relatively insignificant classification problem for normal monoids with group $ G$ and no zero element is also solved. For each $ G$ there is only one such $ E$ with $ G \subsetneqq E$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1985-0768719-7
Keywords: Algebraic monoid, reductive, regular, $ D$-monoid
Article copyright: © Copyright 1985 American Mathematical Society

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