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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Regular linear algebraic monoids


Author: Mohan S. Putcha
Journal: Trans. Amer. Math. Soc. 290 (1985), 615-626
MSC: Primary 20M10; Secondary 20G99, 20M17
MathSciNet review: 792815
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Abstract: In this paper we study connected regular linear algebraic monoids. If $ \phi :{G_0} \to {\text{GL}}(n,K)$ is a representation of a reductive group $ {G_0}$, then the Zariski closure of $ K\phi ({G_0})$ in $ {\mathcal{M}_n}(K)$ is a connected regular linear algebraic monoid with zero. In $ \S2$ we study abstract semigroup theoretic properties of a connected regular linear algebraic monoid with zero. We show that the principal right ideals form a relatively complemented lattice, that the idempotents satisfy a certain connectedness condition, and that these monoids are $ V$-regular. In $ \S3$ we show that when the ideals are linearly ordered, the group of units is nearly simple of type $ {A_l},{B_l},{C_l},{F_4}\;{\text{or}}\;{G_2}$. In $ \S4$, conjugacy classes are studied by first reducing the problem to nilpotent elements. It is shown that the number of conjugacy classes of minimal nilpotent elements is always finite.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1985-0792815-1
PII: S 0002-9947(1985)0792815-1
Keywords: Matrix semigroups, algebraic groups, idempotents, nilpotents, conjugacy classes, regular semigroups
Article copyright: © Copyright 1985 American Mathematical Society