The ring of regular functions of an algebraic monoid
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- by Lex Renner and Alvaro Rittatore PDF
- Trans. Amer. Math. Soc. 363 (2011), 6671-6683 Request permission
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 $\mathcal {O}(M)=\Bbbk$.References
- Andrzej Białynicki-Birula, On induced actions of algebraic groups, Ann. Inst. Fourier (Grenoble) 43 (1993), no. 2, 365–368 (English, with English and French summaries). MR 1220274
- A. Białynicki-Birula, G. Hochschild, and G. D. Mostow, Extensions of representations of algebraic linear groups, Amer. J. Math. 85 (1963), 131–144. MR 155938, DOI 10.2307/2373191
- Michel Brion, Log homogeneous varieties, Proceedings of the XVIth Latin American Algebra Colloquium (Spanish), Bibl. Rev. Mat. Iberoamericana, Rev. Mat. Iberoamericana, Madrid, 2007, pp. 1–39. MR 2500349
- Michel Brion, Local structure of algebraic monoids, Mosc. Math. J. 8 (2008), no. 4, 647–666, 846 (English, with English and Russian summaries). MR 2499350, DOI 10.17323/1609-4514-2008-8-4-647-666
- Michel Brion, Anti-affine algebraic groups, J. Algebra 321 (2009), no. 3, 934–952. MR 2488561, DOI 10.1016/j.jalgebra.2008.09.034
- Michel Brion and Alvaro Rittatore, The structure of normal algebraic monoids, Semigroup Forum 74 (2007), no. 3, 410–422. MR 2321575, DOI 10.1007/s00233-007-0701-2
- Brian Conrad, A modern proof of Chevalley’s theorem on algebraic groups, J. Ramanujan Math. Soc. 17 (2002), no. 1, 1–18. MR 1906417
- Michel Demazure and Pierre Gabriel, Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs, Masson & Cie, Éditeurs, Paris; North-Holland Publishing Co., Amsterdam, 1970 (French). Avec un appendice Corps de classes local par Michiel Hazewinkel. MR 0302656
- Walter Ferrer Santos and Alvaro Rittatore, Actions and invariants of algebraic groups, Pure and Applied Mathematics (Boca Raton), vol. 269, Chapman & Hall/CRC, Boca Raton, FL, 2005. MR 2138858, DOI 10.1201/9781420030792
- Frank Grosshans, Observable groups and Hilbert’s fourteenth problem, Amer. J. Math. 95 (1973), 229–253. MR 325628, DOI 10.2307/2373655
- Mohan S. Putcha, Linear algebraic monoids, London Mathematical Society Lecture Note Series, vol. 133, Cambridge University Press, Cambridge, 1988. MR 964690, DOI 10.1017/CBO9780511600661
- Lex E. Renner, Linear algebraic monoids, Encyclopaedia of Mathematical Sciences, vol. 134, Springer-Verlag, Berlin, 2005. Invariant Theory and Algebraic Transformation Groups, V. MR 2134980
- Lex Renner and Alvaro Rittatore, Observable actions of algebraic groups, Transform. Groups 14 (2009), no. 4, 985–999. MR 2577204, DOI 10.1007/s00031-009-9073-x
- A. Rittatore, Algebraic monoids and group embeddings, Transform. Groups 3 (1998), no. 4, 375–396. MR 1657536, DOI 10.1007/BF01234534
- Alvaro Rittatore, Algebraic monoids with affine unit group are affine, Transform. Groups 12 (2007), no. 3, 601–605. MR 2356324, DOI 10.1007/s00031-006-0049-9
- Maxwell Rosenlicht, Some basic theorems on algebraic groups, Amer. J. Math. 78 (1956), 401–443. MR 82183, DOI 10.2307/2372523
- J.-P. Serre, Morphismes universels et variété d’Albanese, Séminaire Chevalley (1958–1959), Exposé No. 10, Documents Mathématiques 1, Soc. Math. France, Paris, 2001.
- Louis Solomon, An introduction to reductive monoids, Semigroups, formal languages and groups (York, 1993) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 466, Kluwer Acad. Publ., Dordrecht, 1995, pp. 295–352. MR 1630625
- D. A. Timashev, Homogeneous spaces and equivariant embeddings, to appear in the Encyclopædia of Mathematical Sciences, subseries Invariant Theory and Algebraic Transformation Groups;arXiv: math.AG/0602228.
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
- 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 - © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 363 (2011), 6671-6683
- MSC (2010): Primary 20M32; Secondary 14L30
- DOI: https://doi.org/10.1090/S0002-9947-2011-05335-3
- MathSciNet review: 2833572