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 | References | Similar Articles | Additional Information

Abstract: Consider the classification problem for irreducible, normal, algebraic monoids with unit group . We obtain complete results for the groups , and . If is one of these groups let denote the set of isomorphy types of normal, algebraic monoids with zero element and unit group . Our main result establishes a canonical one-to-one correspondence , where 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 . That done, we show that any other 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 , for some , which is ultimately an isomorphism because the monoid is regular.

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

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1985-0768719-7

Keywords:
Algebraic monoid,
reductive,
regular,
-monoid

Article copyright:
© Copyright 1985
American Mathematical Society