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Transactions of the American Mathematical Society

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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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DOI: https://doi.org/10.1090/S0002-9947-1987-0902783-9
Article copyright: © Copyright 1987 American Mathematical Society

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