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Classification of Reductive Monoid Spaces over an Arbitrary Field

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Advances in Algebra (SRAC 2017)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 277))

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Abstract

In this semi-expository paper, we review the notion of a spherical space. In particular, we present some recent results of Wedhorn on the classification of spherical spaces over arbitrary fields. As an application, we introduce and classify reductive monoid spaces over an arbitrary field.

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Notes

  1. 1.

    This book is one of the few if not the only book in algebraic geometry that acknowledges monoid schemes as part of the theory of group schemes.

References

  1. Borel, A.: Linear Algebraic Groups, 2nd edn. Graduate Texts in Mathematics, vol. 126. Springer-Verlag, New York (1991)

    Book  Google Scholar 

  2. Bravi, P., Pezzini, G.: Primitive wonderful varieties. Math. Z. 282(3–4), 1067–1096 (2016)

    Article  MathSciNet  Google Scholar 

  3. Brion, M.: Quelques propriétés des espaces homogènes sphériques. Manuscr. Math. 55(2), 191–198 (1986)

    Article  Google Scholar 

  4. Brion, M.: Groupe de Picard et nombres caractéristiques des variétés sphériques. Duke Math. J. 58(2), 397–424 (1989)

    Article  MathSciNet  Google Scholar 

  5. Brion, M.: On algebraic semigroups and monoids. In: Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics. Fields Institute Communications, vol. 71, pp. 1–54. Springer, New York (2014)

    Google Scholar 

  6. Brion, M.: Some structure theorems for algebraic groups. In: Algebraic Groups: Structure and Actions. Proceedings of Symposia in Pure Mathematics, vol. 94, pp. 53–126. American Mathematical Society, Providence, RI (2017)

    Google Scholar 

  7. Bröcker, T., tom Dieck, T.: Representations of Compact Lie Groups. Graduate Texts in Mathematics, vol. 98. Springer-Verlag, New York (1995). Translated from the German manuscript, Corrected reprint of the 1985 translation

    Google Scholar 

  8. Chevalley, C.: Theory of Lie Groups I. Princeton Mathematical Series, vol. 8. Fifteenth printing, Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ (1999)

    Google Scholar 

  9. Conrad, B.: Reductive group schemes. In: Autour des Schémas en Groupes, vol. I. Panor. Synthèses, vol. 42/43, pp. 93–444. Société mathématique de France, Paris (2014)

    Google Scholar 

  10. Danilov, V.I.: The geometry of toric varieties. Uspekhi Mat. Nauk 33(2), 85–134, 247 (1978)

    Article  MathSciNet  Google Scholar 

  11. De Concini, C.: Normality and non normality of certain semigroups and orbit closures. In: Algebraic Transformation Groups and Algebraic Varieties. Encylopaedia of Mathematical Sciences, vol. 132, pp. 15–35. Springer, Berlin (2004)

    Google Scholar 

  12. Demazure, M.: Sous-groupes algébriques de rang maximum du groupe de Cremona. Ann. Sci. École Norm. Sup. 4(3), 507–588 (1970)

    Article  Google Scholar 

  13. Demazure, M., Gabriel, P.: Introduction to Algebraic Geometry and Algebraic Groups. North-Holland Mathematics Studies, vol. 39. North-Holland Publishing Co., Amsterdam-New York (1980). Translated from the French by J. Bell

    Google Scholar 

  14. Helgason, S.: Groups and Geometric Analysis. Mathematical Surveys and Monographs, vol. 83. American Mathematical Society, Providence, RI (2000). Integral geometry, invariant differential operators, and spherical functions, Corrected reprint of the 1984 original

    Chapter  Google Scholar 

  15. Huruguen, M.: Toric varieties and spherical embeddings over an arbitrary field. J. Algebra 342, 212–234 (2011)

    Article  MathSciNet  Google Scholar 

  16. Knop, F.: The Luna-Vust theory of spherical embeddings. In: Proceedings of the Hyderabad Conference on Algebraic Groups, Hyderabad, pp. 225–249 (1989). Manoj Prakashan, Madras (1991)

    Google Scholar 

  17. Knutson, D.: Algebraic Spaces. Lecture Notes in Mathematics, vol. 203. Springer-Verlag, Berlin-New York (1971)

    Chapter  Google Scholar 

  18. Legendre, A.M.: Recherches sur l’attraction des spheroides homogenes. In: Mémoires de mathématique et de physique : prés. à l’Académie Royale des Sciences, par divers savans, et lûs dans ses assemblées, vol. 1785, pp. 411–434 (2007)

    Google Scholar 

  19. Li, Z., Putcha, M.: Types of reductive monoids. J. Algebra 221(1), 102–116 (1999)

    Article  MathSciNet  Google Scholar 

  20. Luna, D., Vust, T.: Plongements d’espaces homogènes. Comment. Math. Helv. 58(2), 186–245 (1983)

    Article  MathSciNet  Google Scholar 

  21. Mumford, D.: Abelian Varieties. Tata Institute of Fundamental Research Studies in Mathematics, vol. 5. Published for the Tata Institute of Fundamental Research, Bombay; by Hindustan Book Agency, New Delhi (2008). With appendices by C.P. Ramanujam and Y. Manin, Corrected reprint of the second edition (1974)

    Google Scholar 

  22. Onishchik, A.L., Vinberg, È.B.: Lie Groups and Algebraic Groups. Springer Series in Soviet Mathematics. Springer-Verlag, Berlin (1990). Translated from the Russian and with a preface by D.A. Leites

    Chapter  Google Scholar 

  23. Ono, T.: On the field of definition of Borel subgroups of semi-simple algebraic groups. J. Math. Soc. Jpn. 15, 392–395 (1963)

    Article  MathSciNet  Google Scholar 

  24. Popov, V.L.: Contractions of actions of reductive algebraic groups. Mat. Sb. (N.S.) 130(172)(3(7)), 310–334, 431 (1986)

    Google Scholar 

  25. Putcha, M.S.: Linear Algebraic Monoids. London Mathematical Society Lecture Note Series, vol. 133. Cambridge University Press, Cambridge (1988)

    Google Scholar 

  26. Renner, L.E.: Reductive monoids are von Neumann regular. J. Algebra 93(2), 237–245 (1985)

    Article  MathSciNet  Google Scholar 

  27. Renner, L.E.: Classification of semisimple varieties. J. Algebra 122(2), 275–287 (1989)

    Article  MathSciNet  Google Scholar 

  28. Renner, L.E.: Linear Algebraic Monoids. Encyclopaedia of Mathematical Sciences, vol. 134. Springer-Verlag, Berlin (2005). Invariant Theory and Algebraic Transformation Groups, V

    Google Scholar 

  29. Rittatore, A.: Algebraic monoids and group embeddings. Trans. Groups 3(4), 375–396 (1998)

    Article  MathSciNet  Google Scholar 

  30. The Stacks Project Authors. Stacks project (2017). http://stacks.math.columbia.edu

  31. Timashev, D.A.: Homogeneous Spaces and Equivariant Embeddings. Encyclopaedia of Mathematical Sciences, vol. 138. Springer, Heidelberg (2011). Invariant Theory and Algebraic Transformation Groups, 8

    Chapter  Google Scholar 

  32. Vinberg, È.B.: Complexity of actions of reductive groups. Funktsional. Anal. i Prilozhen. 20(1), 1–13, 96 (1986)

    Google Scholar 

  33. Vinberg, È.B.: On reductive algebraic semigroups. In: Lie Groups and Lie Algebras: E. B. Dynkin’s Seminar. American Mathematical Society Translations: Series 2, vol. 169, pp. 145–182. American Mathematical Society, Providence, RI (1995)

    Google Scholar 

  34. Wedhorn, T.: Spherical spaces. Ann. l’Institut Fourier 68(1), 229–256 (2018)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

I thank the organizers of the 2017 Southern Regional Algebra Conference: Laxmi Chataut, Jörg Feldvoss, Lauren Grimley, Drew Lewis, Andrei Pavelescu, and Cornelius Pillen. I am grateful to Jörg Feldvoss, Lex Renner, Soumya Dipta Banerjee, and to the anonymous referee for their very careful reading of the paper and for their suggestions, which improved the quality of the article. This work was partially supported by a grant from the Louisiana Board of Regents.

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Authors and Affiliations

  1. Mathematics Department, Tulane University, New Orleans, LA, 70118, USA

    Mahir Bilen Can

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  1. Mahir Bilen Can
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Correspondence to Mahir Bilen Can .

Editor information

Editors and Affiliations

  1. Department of Mathematics and Statistics, University of South Alabama, Mobile, AL, USA

    Jörg Feldvoss

  2. Department of Mathematics, Spring Hill College, Mobile, AL, USA

    Lauren Grimley

  3. Department of Mathematics and Statistics, University of South Alabama, Mobile, AL, USA

    Drew Lewis

  4. Department of Mathematics and Statistics, University of South Alabama, Mobile, AL, USA

    Andrei Pavelescu

  5. Department of Mathematics and Statistics, University of South Alabama, Mobile, AL, USA

    Cornelius Pillen

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Can, M.B. (2019). Classification of Reductive Monoid Spaces over an Arbitrary Field. In: Feldvoss, J., Grimley, L., Lewis, D., Pavelescu, A., Pillen, C. (eds) Advances in Algebra. SRAC 2017. Springer Proceedings in Mathematics & Statistics, vol 277. Springer, Cham. https://doi.org/10.1007/978-3-030-11521-0_4

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