Regular linear algebraic monoids
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- by Mohan S. Putcha PDF
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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 $\S 2$ 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.References
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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