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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Regular linear algebraic monoids
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by Mohan S. Putcha PDF
Trans. Amer. Math. Soc. 290 (1985), 615-626 Request permission

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 $\S 2$ 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 $\S 3$ 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 $\S 4$, 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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Additional Information
  • © Copyright 1985 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 290 (1985), 615-626
  • MSC: Primary 20M10; Secondary 20G99, 20M17
  • DOI: https://doi.org/10.1090/S0002-9947-1985-0792815-1
  • MathSciNet review: 792815