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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The ring of regular functions of an algebraic monoid


Authors: Lex Renner and Alvaro Rittatore
Journal: Trans. Amer. Math. Soc. 363 (2011), 6671-6683
MSC (2010): Primary 20M32; Secondary 14L30
Published electronically: May 17, 2011
MathSciNet review: 2833572
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Abstract: Let $ M$ be an irreducible normal algebraic monoid with unit group $ G$. It is known that $ G$ admits a Rosenlicht decomposition, $ G=G_{\operatorname{ant}}G_{\operatorname{aff}} \cong (G_{\operatorname{ant}} \times G_{\operatorname{aff}})/G_{\operatorname{aff}}\cap G_{\operatorname{ant}}$, where $ G_{\operatorname{ant}}$ is the maximal anti-affine subgroup of $ G$, and $ G_{\operatorname{aff}}$ the maximal normal connected affine subgroup of $ G$. In this paper we show that this decomposition extends to a decomposition $ M=G_{\operatorname{ant}}M_{\operatorname{aff}} \cong G_{\operatorname{ant}}*_{G_{\operatorname{aff}}\cap G_{\operatorname{ant}}}M_{\operatorname{aff}}$, where $ M_{\operatorname{aff}}$ is the affine submonoid $ M_{\operatorname{aff}}=\overline{G_{\operatorname{aff}}}$. We then use this decomposition to calculate $ \mathcal{O}(M)$ in terms of $ \mathcal{O}(M_{\operatorname{aff}})$ and $ G_{\operatorname{aff}}, G_{\operatorname{ant}}\subset G$. In particular, we determine when $ M$ is an anti-affine monoid, that is $ \mathcal{O}(M)=\Bbbk$.


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Additional Information

Lex Renner
Affiliation: Department of Mathematics, University of Western Ontario, London, Ontario, Canada N6A 5B7
Email: lex@uwo.ca

Alvaro Rittatore
Affiliation: Facultad de Ciencias, Universidad de la República, Uguá 4225, 11400 Montevideo, Uruguay
Email: alvaro@cmat.edu.uy

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05335-3
PII: S 0002-9947(2011)05335-3
Received by editor(s): February 12, 2009
Received by editor(s) in revised form: February 25, 2010
Published electronically: May 17, 2011
Additional Notes: The first author was partially supported by a grant from NSERC.
The second author was partially supported by grants from SNI-ANII grant, IMU/CDE, NSERC and PDT/54-02 research project
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.