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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Conjugacy classes in algebraic monoids


Author: Mohan S. Putcha
Journal: Trans. Amer. Math. Soc. 303 (1987), 529-540
MSC: Primary 20G99; Secondary 20M10
DOI: https://doi.org/10.1090/S0002-9947-1987-0902783-9
MathSciNet review: 902783
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Abstract: Let $M$ be a connected linear algebraic monoid with zero and a reductive group of units $G$. The following theorem is established. Theorem. There exist affine subsets ${M_1}, \ldots ,{M_k}$ of $M$, reductive groups ${G_1}, \ldots ,{G_k}$ with antiautomorphisms $^{\ast }$, surjective morphisms ${\theta _i}:{M_i} \to {G_i}$, such that: (1) Every element of $M$ is conjugate to an element of some ${M_i}$, and (2) Two elements $a$, $b$ in ${M_i}$ are conjugate in $M$ if and only if there exists $x \in {G_i}$ such that $x{\theta _i}(a){x^{\ast }} = {\theta _i}(b)$. As a consequence, it is shown that $M$ is a union of its inverse submonoids.


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Article copyright: © Copyright 1987 American Mathematical Society