Algebraic monoids whose nonunits are products of idempotents
HTML articles powered by AMS MathViewer
- by Mohan S. Putcha PDF
- Proc. Amer. Math. Soc. 103 (1988), 38-40 Request permission
Abstract:
Let $M$ be a connected regular linear algebraic monoid with zero and group of units $G$. Suppose $G$ is nearly simple, i.e. the center of $G$ is one dimensional and the derived group $G’$ is a simple algebraic group. Then it is shown that $S = M\backslash G$ is an idempotent generated semigroup. If $M$ has a unique nonzero minimal ideal, the converse is also proved. It follows that if ${G_0}$ is any simple algebraic group defined over an algebraically closed field $K$ and if $\Phi :{G_0} \to GL(n,K)$ is any representation of ${G_0}$, then the nonunits of the monoid $M(\Phi ) = \overline {K\Phi ({G_0})}$ form an idempotent generated semigroup.References
- Roger W. Carter, Finite groups of Lie type, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1985. Conjugacy classes and complex characters; A Wiley-Interscience Publication. MR 794307
- R. J. H. Dawlings, On idempotent affine mappings, Proc. Roy. Soc. Edinburgh Sect. A 93 (1982/83), no. 3-4, 345–348. MR 688796, DOI 10.1017/S0308210500016024
- J. A. Erdos, On products of idempotent matrices, Glasgow Math. J. 8 (1967), 118–122. MR 220751, DOI 10.1017/S0017089500000173
- J. M. Howie, An introduction to semigroup theory, L. M. S. Monographs, No. 7, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. MR 0466355
- Mohan S. Putcha, Green’s relations on a connected algebraic monoid, Linear and Multilinear Algebra 12 (1982/83), no. 3, 205–214. MR 678826, DOI 10.1080/03081088208817484
- Mohan S. Putcha, Connected algebraic monoids, Trans. Amer. Math. Soc. 272 (1982), no. 2, 693–709. MR 662061, DOI 10.1090/S0002-9947-1982-0662061-8
- Mohan S. Putcha, A semigroup approach to linear algebraic groups, J. Algebra 80 (1983), no. 1, 164–185. MR 690712, DOI 10.1016/0021-8693(83)90026-1
- Mohan S. Putcha, Reductive groups and regular semigroups, Semigroup Forum 30 (1984), no. 3, 253–261. MR 765495, DOI 10.1007/BF02573457
- Mohan S. Putcha, Regular linear algebraic monoids, Trans. Amer. Math. Soc. 290 (1985), no. 2, 615–626. MR 792815, DOI 10.1090/S0002-9947-1985-0792815-1
- Lex E. Renner, Reductive monoids are von Neumann regular, J. Algebra 93 (1985), no. 2, 237–245. MR 786751, DOI 10.1016/0021-8693(85)90157-7
- Lex E. Renner, Classification of semisimple algebraic monoids, Trans. Amer. Math. Soc. 292 (1985), no. 1, 193–223. MR 805960, DOI 10.1090/S0002-9947-1985-0805960-9
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 103 (1988), 38-40
- MSC: Primary 20M10
- DOI: https://doi.org/10.1090/S0002-9939-1988-0938640-8
- MathSciNet review: 938640