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  3. Algebraic Groups
  • Cited by 166
Publisher:
Cambridge University Press
Online publication date:
October 2017
Print publication year:
2017
Online ISBN:
9781316711736
DOI:
https://doi.org/10.1017/9781316711736
Subjects:
Mathematics (general), Mathematics, Geometry and Topology, Algebra
Series:
Cambridge Studies in Advanced Mathematics (170)
Information

Book description

Algebraic groups play much the same role for algebraists as Lie groups play for analysts. This book is the first comprehensive introduction to the theory of algebraic group schemes over fields that includes the structure theory of semisimple algebraic groups, and is written in the language of modern algebraic geometry. The first eight chapters study general algebraic group schemes over a field and culminate in a proof of the Barsotti–Chevalley theorem, realizing every algebraic group as an extension of an abelian variety by an affine group. After a review of the Tannakian philosophy, the author provides short accounts of Lie algebras and finite group schemes. The later chapters treat reductive algebraic groups over arbitrary fields, including the Borel–Chevalley structure theory. Solvable algebraic groups are studied in detail. Prerequisites have also been kept to a minimum so that the book is accessible to non-specialists in algebraic geometry.

Reviews

'All together, this excellent text fills a long-standing gap in the textbook literature on algebraic groups. It presents the modern theory of group schemes in a very comprehensive, systematic, detailed and lucid manner, with numerous illustrating examples and exercises. It is fair to say that this reader-friendly textbook on algebraic groups is the long-desired modern successor to the old, venerable standard primers …'

Werner Kleinert Source: zbMath

'The author invests quite a lot to make difficult things understandable, and as a result, it is a real pleasure to read the book. All in all, with no doubt, Milne's new book will remain for decades an indispensable source for everybody interested in algebraic groups.'

Boris È. Kunyavskiĭ Source: MathSciNet

‘… fulfills the dual purpose of providing an updated account of the theory of reductive groups while at the same time serving as an accessible entry point into the general theory of reductive group schemes.’

Igor A. Rapinchuk Source: Bulletin of the American Mathematical Society

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Contents

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Contents

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References
References
Adams, J. 2013 Google Scholar. Galois cohomology of real groups. arxiv:1310.7917.
Adams, J. F. 1969. Lectures on Lie groups. W. A. Benjamin, Inc., New York and Amsterdam Google Scholar.
Allcock, D. 2009 CrossRef | Google Scholar. A new approach to rank one linear algebraic groups. J. Algebra 321:2540–2544.
Balaji, V., Deligne, P., and Parameswaran, A. J. 2016 Google Scholar. On complete reducibility in characteristic p. arxiv:1607.08564.
Barsotti, I. 1953. A note on abelian varieties. Rend. Circ. Mat. Palermo (2) 2 CrossRef | Google Scholar:236–257.
Barsotti, I. 1955. Un teorema di struttura per le variet`a gruppali. Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8) 18 Google Scholar:43–50.
Bate, M., Martin, B., Röhrle, G., and Tange, R. 2010. Complete reducibility and separability. Trans. Amer. Math. Soc. 362 CrossRef | Google Scholar:4283–4311.
Bergman, G. M. 1978. The diamond lemma for ring theory. Adv. in Math. 29 CrossRef | Google Scholar:178–218.
Berrick, A. J. and Keating, M. E. 2000. An introduction to rings and modules with K-theory in view. Cambridge Studies in Advanced Mathematics, Vol. 65. Cambridge University Press, Cambridge Google Scholar.
BiałYnicki-Birula, A. 1973. Some theorems on actions of algebraic groups. Ann. of Math. (2) 98 CrossRef | Google Scholar:480–497.
BiałYnicki-Birula, A. 1976. Some properties of the decompositions of algebraic varieties determined by actions of a torus. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 24 Google Scholar:667–674.
Birkhoff, G. 1937. Representability of Lie algebras and Lie groups by matrices. Ann. of Math. (2) 38 CrossRef | Google Scholar:526–532.
Borel, A. 1956. Groupes linéaires algébriques. Ann. of Math. (2) 64 CrossRef | Google Scholar:20–82.
Borel, A. 1970. Properties and linear representations of Chevalley groups, pp. 1– 55. In Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, NJ, 1968/69), Lecture Notes in Mathematics, Vol. 131. Springer-Verlag, Berlin Google Scholar.
Borel, A. 1975. Linear representations of semi-simple algebraic groups, pp. 421–440. In Algebraic geometry (Proceedings of Symposia in Pure Mathematics, Vol. 29, Humboldt State University, Arcata, Calif., 1974 Google Scholar). American Mathematical Society, Providence, RI.
Borel, A. 1985. On affine algebraic homogeneous spaces. Arch. Math. (Basel) 45 CrossRef | Google Scholar:74–78.
Borel, A. 1991. Linear algebraic groups. Graduate Texts in Mathematics, Vol. 126. Springer-Verlag, Berlin Google Scholar.
Borel, A. and Harder, G. 1978. Existence of discrete cocompact subgroups of reductive groups over local fields. J. Reine Angew. Math. 298 Google Scholar:53–64.
Borel, A. and Springer, T. A. 1966. Rationality properties of linear algebraic groups, pp. 26–32. In Algebraic Groups and Discontinuous Subgroups (Proceedings of Symposia in Pure Mathematics, Boulder, CO, 1965 Google Scholar). American Mathematical Society, Providence, RI.
Borel, A. and Springer, T. A. 1968. Rationality properties of linear algebraic groups. II. Tˆohoku Math. J. (2) 20 CrossRef | Google Scholar:443–497.
Borel, A. and Tits, J. 1965 CrossRef | Google Scholar. Groupes reductifs. Inst. Hautes Etudes Sci. Publ. Math. 27:55–150.
Borel, A. and Tits, J. 1972 CrossRef | Google Scholar. Compléments à l'article: “Groupes réductifs”. Inst. Hautes Études Sci. Publ. Math. 41:253–276.
Borel, A. and Tits, J. 1978. Théorèmes de structure et de conjugaison pour les groupes algébriques linéaires. C. R. Acad. Sci. Paris Sér. A-B 287 Google Scholar:A55–A57.
Borovoi, M. 2014 Google Scholar. Galois cohomology of reductive algebraic groups over the field of real numbers. arxiv:1401.5913.
Borovoi, M. and Timashev, D. A. 2015 Google Scholar. Galois cohomology of real semisimple groups. arxiv:1506.06252.
Bosch, S., Lütkebohmert, W., and Raynaud, M. 1990. Néron models. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Vol. 21. Springer-Verlag, Berlin Google Scholar.
Bourbaki, N. 1958. Algèbre. Chapitre 8: Modules et anneaux semi-simples. Hermann Google Scholar, Paris.
Bourbaki, N. 1968. Groupes et algèbres de Lie. Chapitres 4, 5 et 6. Hermann Google Scholar, Paris.
Bourbaki, N. 1972. Groupes et algèbres de Lie. Chapitres 2 et 3. Hermann Google Scholar, Paris.
Bourbaki, N. 1975. Groupes et algèbres de Lie. Chapitres 7 et 8. Hermann Google Scholar, Paris.
Brion, M. 2009. Anti-affine algebraic groups. J. Algebra 321 CrossRef | Google Scholar:934–952.
Brion, M. 2015a. On extensions of algebraic groups with finite quotient. Pacific J. Math. 279 Google Scholar:135–153.
Brion, M. 2015b Google Scholar. Some structure theorems for algebraic groups. arxiv:1509.03059.
Brion, M. 2016 Google Scholar. Epimorphic subgroups of algebraic groups. arxiv:1605.07769.
Brion, M., Samuel, P., and Uma, V. 2013. Lectures on the structure of algebraic groups and geometric applications. CMI Lecture Series in Mathematics, Vol 1. Hindustan Book Agency, New Delhi Google Scholar.
Brochard, S. 2014. Topologies de Grothendieck, descente, quotients, pp. 1–62. In Autour des schémas en groupes Vol. I, Panoramas et Synthèses, Vol. 42/43. Société Mathématique de France, Paris Google Scholar.
Brosnan, P. 2005 CrossRef | Google Scholar. On motivic decompositions arising from the method of Białynicki- Birula. Invent. Math. 161:91–111.
Bruhat, F. and Tits, J. 1987. Groupes algébriques sur un corps local. Chapitre III. Compléments et applications à la cohomologie galoisienne. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 Google Scholar:671–698.
Cartier, P. 1962. Groupes algébriques et groupes formels, pp. 87–111. In Colloque sur la Théorie des Groupes Algébriques (Bruxelles, 1962). Librairie Universitaire, Louvain Google Scholar.
Cartier, P. 2005. Sur les problèmes de classification des groups, pp. 266–274. In Classification des groupes algébriques semi-simples (Collected works, Vol. 3). Springer- Verlag, Berlin Google Scholar.
Casselman, B. 2015. On Chevalley's formula for structure constants. J. Lie Theory 25 Google Scholar:431–441.
Cayley, A. 1846. Sur quelques propriétés des déterminants gauches. J. Reine Angew. Math. 32 CrossRef | Google Scholar:119–123.
Chernousov, V.I. 1989. The Hasse principle for groups of type E8. Dokl. Akad. Nauk SSSR 306 Google Scholar:1059–1063.
Chevalley, C.C. 1955a. Sur certains groupes simples. Tôhoku Math. J. (2) 7 Google Scholar:14–66.
Chevalley, C.C. 1955b. Théorie des groupes de Lie. Tome III. Théorèmes généraux sur les algèbres de Lie. Actualités Scientifiques et Industrielles no. 1226. Hermann Google Scholar, Paris.
Chevalley, C.C. 1956–58. Classification des groupes de Lie algébriques, Seminaire ENS, Paris. mimeographed. Reprinted by Springer-Verlag, Berlin Google Scholar, 2005.
Chevalley, C.C. 1960. Une démonstration d'un théorème sur les groupes algébriques. J. Math. Pures Appl. (9) 39 Google Scholar:307–317.
Chevalley, C.C. 1961. Certains schémas de groupes semi-simples, pp. 219–234. In Séminaire Bourbaki, Vol. 6 Exp. No. 219. Société Mathématique de France, Paris Google Scholar.
Conrad, B. 2014. Reductive group schemes, pp. 93–444. In Autour des schémas en groupes. Vol. I, Panoramas et Synthèses, Vol. 42/43. Société Mathématique de France, Paris Google Scholar.
Conrad, B., Gabber, O., and Prasad, G. 2015. Pseudo-reductive groups. New Mathematical Monographs, Vol. 26. Cambridge University Press, Cambridge Google Scholar, second edition.
De Graaf, W. A. 2007. Classification of 6-dimensional nilpotent Lie algebras over fields of characteristic not 2. J. Algebra 309 CrossRef | Google Scholar:640–653.
Deligne, P. 1990. Catégories tannakiennes, pp. 111–195. In The Grothendieck Festschrift, Vol. II, Progress in Mathematics. Birkhäuser Boston, Boston Google Scholar, MA.
Deligne, P. 2014 CrossRef | Google Scholar. Semi-simplicité de produits tensoriels en caractéristique p. Invent. Math. 197:587–611.
Deligne, P. and Lusztig, G. 1976. Representations of reductive groups over finite fields. Ann. of Math. (2) 103 CrossRef | Google Scholar:103–161.
Deligne, P. and Milne, J.S. 1982. Tannakian categories, pp. 101–228. In Hodge cycles, motives, and Shimura varieties, Lecture Notes in Mathematics, Vol. 900. Springer-Verlag, Berlin Google Scholar.
Demazure, M. 1965. Schémas en groupes réductifs. Bull. Soc. Math. France 93 CrossRef | Google Scholar:369– 413.
Demazure, M. 1972. Lectures on p-divisible groups. Lecture Notes in Mathematics, Vol. 302. Springer-Verlag, Berlin Google Scholar.
Demazure, M. and Gabriel, P. 1966. Séminaire Heidelberg–Strasbourg 1965–66 (Groupes Algébriques) Google Scholar. Multigraphié par l'Institut de Mathématique de Strasbourg, 406 pages. Cited as SHS.
Demazure, M. and Gabriel, P. 1970. Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs. Masson, Paris Google Scholar. Cited as DG.
Demazure, M. and Grothendieck, A. 1964. Schémas en groupes (SGA 3). Séminaire de Géométrie Algébrique du Bois Marie 1962–64. Dirigé par M., Demazure et A., Grothendieck. Multigraphié par I.H.E.S; reprinted by Springer-Verlag, Berlin: Lecture Notes in Mathematics, Vols 151 Google Scholar, 152, 153 (1970).
Demazure, M. and Grothendieck, A. 2011. Schémas en groupes (SGA 3). Séminaire de Géométrie Algébrique du Bois Marie 1962–64. Dirigé par M., Demazure et A., Grothendieck. Documents Mathématiques, Vol. 7,8. Société Mathématique de France, Paris. Revised edition of Demazure and Grothendieck 1964. Annotated and edited by Gille, P., and Polo, P. Google Scholar. Cited as SGA 3.
DokovíC, D.Z. 1988. An elementary proof of the structure theorem for connected solvable affine algebraic groups. Enseign. Math. (2) 34 Google Scholar:269–273.
Fogarty, J. 1973. Fixed point schemes. Amer. J. Math. 95 CrossRef | Google Scholar:35–51.
Fogarty, J. and Norman, P. 1977. A fixed-point characterization of linearly reductive groups, pp. 151–155. In Contributions to algebra (collection of papers dedicated to Ellis Kolchin). Academic Press, New York Google Scholar.
Fossum, R. and Iversen, B. 1973. On Picard groups of algebraic fibre spaces. J. Pure Appl. Algebra 3 CrossRef | Google Scholar:269–280.
Garibaldi, R.S. 1998. Isotropic trialitarian algebraic groups. J. Algebra 210 CrossRef | Google Scholar:385–418.
Garibaldi, R.S. 2001. Groups of type E7 over arbitrary fields. Comm. Algebra 29 CrossRef | Google Scholar:2689–2710.
Garibaldi, S. 2016. E8, the most exceptional group. Bull. Amer. Math. Soc. (N.S.) 53 CrossRef | Google Scholar:643–671.
Geck, M. 2016 Google Scholar. On the construction of semisimple Lie algebras and Chevalley groups. arxiv:1602.04583.
Grothendieck, A. 1967 Google Scholar. Eléments de géométrie algébrique. Publ. Math. IHES 4, 8, 11, 17, 20, 24, 28, 32. (1960–67) En collaboration avec J. Dieudonné. Cited as EGA.
Grothendieck, A. 1972. Groupes de monodromie en géométrie algébrique. I. Lecture Notes in Mathematics, Vol. 288. Springer-Verlag, Berlin Google Scholar. Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 I).
Harder, G. 1965. Über einen Satz von E. Cartan. Abh. Math. Sem. Univ. Hamburg 28 CrossRef | Google Scholar:208–214.
Harder, G. 1966. Über die Galoiskohomologie halbeinfacher Matrizengruppen. II. Math. Z. 92 CrossRef | Google Scholar:396–415.
Harder, G. 1975. Über die Galoiskohomologie halbeinfacher algebraischer Gruppen. III. J. Reine Angew. Math. 274 Google Scholar/275:125–138.
Harebov, A. and Vavilov, N. 1996. On the lattice of subgroups of Chevalley groups containing a split maximal torus. Comm. Algebra 24 CrossRef | Google Scholar:109–133.
Hartshorne, R. 1977. Algebraic geometry. Graduate Texts in Mathematics, Vol. 52. Springer-Verlag, Berlin Google Scholar.
Herpel, S. 2013. On the smoothness of centralizers in reductive groups. Trans. Amer. Math. Soc. 365 CrossRef | Google Scholar:3753–3774.
Hesselink, W.H. 1981. Concentration under actions of algebraic groups, pp. 55–89. In Paul Dubreil and Marie-Paule Malliavin Algebra Seminar, 33rd Year (Paris, 1980), Lecture Notes in Mathematics, Vol. 867. Springer-Verlag, Berlin Google Scholar.
Hochschild, G.P. 1981. Basic theory of algebraic groups and Lie algebras. Graduate Texts in Mathematics, Vol. 75. Springer-Verlag, Berlin Google Scholar.
Humphreys, J.E. 1972. Introduction to Lie algebras and representation theory. Graduate Texts in Mathematics, Vol. 9. Springer-Verlag, Berlin Google Scholar.
Humphreys, J.E. 1975. Linear algebraic groups. Graduate Texts in Mathematics, No. 21. Springer-Verlag, Berlin Google Scholar.
Humphreys, J.E. 1990. Reflection groups and Coxeter groups. Cambridge Studies in Advanced Mathematics, Vol. 29. Cambridge University Press, Cambridge Google Scholar.
Humphreys, J.E. 1995. Conjugacy classes in semisimple algebraic groups. Mathematical Surveys and Monographs, Vol. 43. American Mathematical Society, Providence Google Scholar, RI.
Iversen, B. 1972 CrossRef | Google Scholar. A fixed point formula for action of tori on algebraic varieties. Invent. Math. 16:229–236.
Iversen, B. 1976. The geometry of algebraic groups. Advances in Math. 20 CrossRef | Google Scholar:57–85.
Jacobson, N. 1962. Lie algebras. Interscience Tracts in Pure and Applied Mathematics, No. 10. Interscience Publishers, Inc., New York and London Google Scholar. Reprinted by Dover, New York, 1979.
Jacobson, N. 1985. Basic algebra. I. W. H. Freeman and Company, New York Google Scholar, second edition.
Jacobson, N. 1989. Basic algebra. II. W. H. Freeman and Company, New York Google Scholar, second edition.
Jantzen, J.C. 1997. Low-dimensional representations of reductive groups are semisimple, pp. 255–266. In Algebraic groups and Lie groups, Australian Mathematical Society Lecture Series., Vol. 9. Cambridge University Press, Cambridge Google Scholar.
Jantzen, J.C. 2003. Representations of algebraic groups. Mathematical Surveys and Monographs, Vol. 107. American Mathematical Society, Providence Google Scholar, RI, second edition.
Kambayashi, T., Miyanishi, M., and Takeuchi, M. 1974. Unipotent algebraic groups. Lecture Notes in Mathematics, Vol. 414. Springer-Verlag, Berlin Google Scholar.
Kambayashi, T. and Russell, P. 1982. On linearizing algebraic torus actions. J. Pure Appl. Algebra 23 CrossRef | Google Scholar:243–250.
Kempf, G.R. 1976. Linear systems on homogeneous spaces. Ann. of Math. (2) 103 CrossRef | Google Scholar:557– 591.
Kneser, M. 1969. Lectures on Galois cohomology of classical groups. Tata Institute of Fundamental Research, Bombay Google Scholar.
Knus, M.-A., Merkurjev, A., Rost, M., and Tignol, J.-P. 1998. The book of involutions. American Mathematical Society Colloquium Publications, Vol. 44. American Mathematical Society, Providence Google Scholar, RI.
Kohls, M. 2011. A user friendly proof of Nagata's characterization of linearly reductive groups in positive characteristics. Linear Multilinear Algebra 59 CrossRef | Google Scholar:271–278.
Kolchin, E.R. 1948. Algebraic matric groups and the Picard–Vessiot theory of homogeneous linear ordinary differential equations. Ann. of Math. (2) 49 CrossRef | Google Scholar:1–42.
Kostant, B. 1966. Groups over Z, pp. 90–98. In Algebraic Groups and Discontinuous Subgroups (Proceedings of Symposia in Pure Mathematics, Vol. 9, Boulder, CO, 1965). American Mathematical Society, Providence Google Scholar, RI.
Lang, S. 1956. Algebraic groups over finite fields. Amer. J. Math. 78 CrossRef | Google Scholar:555–563.
Lang, S. 2002. Algebra. Graduate Texts in Mathematics, Vol. 211. Springer-Verlag, Berlin Google Scholar.
Lazard, M. 1955. Sur les groupes de Lie formels à un paramètre. Bull. Soc. Math. France 83 CrossRef | Google Scholar:251–274.
Lemire, N., Popov, V.L., and Reichstein, Z. 2006. Cayley groups. J. Amer. Math. Soc. 19 CrossRef | Google Scholar:921–967.
Luna, D. 1999. Retour sur un théorème de Chevalley. Enseign. Math. (2) 45 Google Scholar:317–320.
Lusztig, G. 2009. Study of a Z-form of the coordinate ring of a reductive group. J. Amer. Math. Soc. 22 CrossRef | Google Scholar:739–769.
Mac Lane, S. 1969. One universe as a foundation for category theory, pp. 192–200. In Reports of the Midwest Category Seminar. III, Lecture Notes in Mathematics, Vol. 195. Springer-Verlag, Berlin Google Scholar.
Mac Lane, S. 1971. Categories for the working mathematician. Graduate Texts in Mathematics, Vol. 5. Springer-Verlag, Berlin Google Scholar.
Magid, A. 2011. The Hochschild–Mostow group. Notices Amer. Math. Soc. 58 Google Scholar:1089– 1090.
Malle, G. and Testerman, D. 2011. Linear algebraic groups and finite groups of Lie type. Cambridge Studies in Advanced Mathematics, Vol. 133. Cambridge University Press, Cambridge Google Scholar.
Matsumura, H. 1986. Commutative ring theory. Cambridge Studies in Advanced Mathematics, Vol. 8. Cambridge University Press, Cambridge Google Scholar.
Matsusaka, T. 1953. Some theorems on Abelian varieties. Nat. Sci. Rep. Ochanomizu Univ. 4 Google Scholar:22–35.
Mcninch, G. 2013. On the descent of Levi factors. Arch. Math. (Basel) 100 CrossRef | Google Scholar:7–24.
Mcninch, G.J. 1998. Dimensional criteria for semisimplicity of representations. Proc. London Math. Soc. (3) 76 CrossRef | Google Scholar:95–149.
Mcninch, G.J. 2005. Optimal SL.2/-homomorphisms. Comment. Math. Helv. 80 CrossRef | Google Scholar:391– 426.
Mcninch, G.J. 2010. Levi decompositions of a linear algebraic group. Transform. Groups 15 CrossRef | Google Scholar:937–964.
Mcninch, G.J. 2014a. Levi factors of the special fiber of a parahoric group scheme and tame ramification. Algebr. Represent. Theory 17 Google Scholar:469–479.
Mcninch, G.J. 2014b. Linearity for actions on vector groups. J. Algebra 397 Google Scholar:666–688.
Milne, J.S. 1986. Abelian varieties, pp. 103–150. In Arithmetic geometry (Storrs, Conn., 1984). Springer-Verlag, Berlin Google Scholar.
Milne, J.S. 2007 Google Scholar. Semisimple algebraic groups in characteristic zero. arxiv:0705.1348.
Milne, J.S. 2013 Google Scholar. A proof of the Barsotti–Chevalley theorem on algebraic groups. arxiv:1311.6060.
Milne, J.S. 2017 Google Scholar. A primer of commutative algebra, v4.02. Available at www.jmilne.org/math and http://hdl.handle.net/2027.42/136228. Cited as CA.
Milne, J.S. and Shih, K.-Y. 1982. Conjugates of Shimura varieties, pp. 280–356. In Hodge cycles, motives, and Shimura varieties, Lecture Notes in Mathematics, Vol. 900. Springer-Verlag, Berlin Google Scholar.
MirkovĆ, I. and Vilonen, K. 2007. Geometric Langlands duality and representations of algebraic groups over commutative rings. Ann. of Math. (2) 166 CrossRef | Google Scholar:95–143.
Müller, P. 2003. Algebraic groups over finite fields, a quick proof of Lang's theorem. Proc. Amer. Math. Soc. 131 CrossRef | Google Scholar:369–370.
Mumford, D. 1970. Abelian varieties. Tata Institute of Fundamental Research Studies in Mathematics, Vol. 5. Published for the Tata Institute of Fundamental Research, Bombay; Oxford University Press, London Google Scholar.
Mumford, D., Fogarty, J., and Kirwan, F. 1994. Geometric invariant theory. Ergebnisse der Mathematik und ihrer Grenzgebiete (2), Vol. 34. Springer-Verlag, Berlin Google Scholar, third edition.
Nagata, M. 1960. On the fourteenth problem of Hilbert, pp. 459–462. In Proceedings International Congress of Mathematics 1958. Cambridge University Press, Cambridge Google Scholar.
Nagata, M. 1961/1962. Complete reducibility of rational representations of a matric group. J. Math. Kyoto Univ. 1 Google Scholar:87–99.
Nielsen, H.A. 1974. Diagonalizably linearized coherent sheaves. Bull. Soc. Math. France 102 CrossRef | Google Scholar:85–97.
Nori, M.V. 1987 CrossRef | Google Scholar. On subgroups of GLn.Fp/. Invent. Math. 88:257–275.
Oesterĺe, J. 1984 CrossRef | Google Scholar. Nombres de Tamagawa et groupes unipotents en caractéristique p. Invent. Math. 78:13–88.
Oort, F. 1966 CrossRef | Google Scholar. Algebraic group schemes in characteristic zero are reduced. Invent. Math. 2:79–80.
Pink, R. 2004. On Weil restriction of reductive groups and a theorem of Prasad. Math. Z. 248 CrossRef | Google Scholar:449–457.
Platonov, V. and Rapinchuk, A. 1994. Algebraic groups and number theory. Pure and Applied Mathematics, Vol. 139. Academic Press Inc., Boston Google Scholar, MA.
Popov, V.L. 2015. On the equations defining affine algebraic groups. Pacific J. Math. 279 CrossRef | Google Scholar:423–446.
Prasad, G. and Rapinchuk, A.S. 2006. On the existence of isotropic forms of semi-simple algebraic groups over number fields with prescribed local behavior. Adv. Math. 207 CrossRef | Google Scholar:646–660.
Prasad, G. and Yu, J.-K. 2006. On quasi-reductive group schemes. J. Algebraic Geom. 15 CrossRef | Google Scholar:507–549. With an appendix by Brian Conrad.
Raghunathan, M.S. 2015 CrossRef | Google Scholar. On Chevalley's Z-form. Indian J. Pure Appl. Math. 46:695–700.
Raynaud, M. 1970. Faisceaux amples sur les schémas en groupes et les espaces homogènes. Lecture Notes in Mathematics, Vol. 119. Springer-Verlag, Berlin Google Scholar.
Ree, R. 1964. Commutators in semi-simple algebraic groups. Proc. Amer. Math. Soc. 15 CrossRef | Google Scholar:457–460.
Richardson, R.W. 1977. Affine coset spaces of reductive algebraic groups. Bull. London Math. Soc. 9 CrossRef | Google Scholar:38–41.
Rosenlicht, M. 1956. Some basic theorems on algebraic groups. Amer. J. Math. 78 CrossRef | Google Scholar:401–443.
Rosenlicht, M. 1957. Some rationality questions on algebraic groups. Ann. Mat. Pura Appl. (4) 43 CrossRef | Google Scholar:25–50.
Rosenlicht, M. 1961. Toroidal algebraic groups. Proc. Amer. Math. Soc. 12 CrossRef | Google Scholar:984–988.
Rosenlicht, M. 1963. Questions of rationality for solvable algebraic groups over nonperfect fields. Ann. Mat. Pura Appl. (4) 61 CrossRef | Google Scholar:97–120.
Russell, P. 1970. Forms of the affine line and its additive group. Pacific J. Math. 32 CrossRef | Google Scholar:527–539.
Saavedra Rivano, N. 1972. Catégories Tannakiennes. Lecture Notes in Mathematics, Vol. 265. Springer-Verlag, Berlin Google Scholar.
Sancho De Salas, C. 2001. Grupos algebraicos y teoŕıa de invariantes. Aportaciones Matemáticas: Textos, Vol. 16. Sociedad Matemática Mexicana, México Google Scholar.
Sancho De Salas, C. and Sancho De Salas, F. 2009. Principal bundles, quasiabelian varieties and structure of algebraic groups. J. Algebra 322 CrossRef | Google Scholar:2751–2772.
Satake, I. 1963. On the theory of reductive algebraic groups over a perfect field. J. Math. Soc. Japan 15 CrossRef | Google Scholar:210–235.
Satake, I. 1967. Symplectic representations of algebraic groups satisfying a certain analyticity condition. Acta Math. 117 CrossRef | Google Scholar:215–279.
Satake, I. 1971. Classification theory of semi-simple algebraic groups. Lecture Notes in Pure and Applied Mathematics, Vol. 3. Marcel Dekker, Inc., New York Google Scholar. With an appendix by M. Sugiura.
Scharlau, W. 1985. Quadratic and Hermitian forms. Grundlehren der Mathematischen Wissenschaften, Vol. 270. Springer-Verlag, Berlin Google Scholar.
Selbach, M. 1976. Klassifikationstheorie halbeinfacher algebraischer Gruppen. Mathematisches Institut der Universität Bonn, Bonn. Diplomarbeit, Universität Bonn, Bonn Google Scholar, 1973, Bonner Mathematische Schriften, Nr. 83.
Serre, J.-P. 1959. Groupes algébriques et corps de classes. Publications de l'institut de mathématique de l'université de Nancago, VII. Hermann, Paris. Translated as Algebraic groups and class fields, Springer-Verlag, Berlin Google Scholar, 1988.
Serre, J.-P. 1962. Corps locaux. Publications de l'institut de mathématique de l'université de Nancago, VIII. Hermann, Paris. Translated as Local fields, Springer- Verlag, Berlin Google Scholar, 1979.
Serre, J.-P. 1966. Algèbres de Lie semi-simples complexes. W. A., Benjamin, Inc., New York and Amsterdam. Translated as Complex semisimple Lie algebras, Springer-Verlag, Berlin Google Scholar, 1987.
Serre, J.-P. 1970. Cours d'arithmétique. Collection SUP: “Le Mathématicien”, Vol. 2. Presses Universitaires de France, Paris. Translated as A Course in Arithmetic, Springer- Verlag, Berlin Google Scholar, 1973.
Serre, J.-P. 1993. Gèbres. Enseign. Math. (2) 39 Google Scholar:33–85.
Serre, J.-P. 1994 CrossRef | Google Scholar. Sur la semi-simplicité des produits tensoriels de représentations de groupes. Invent. Math. 116:513–530.
Serre, J.-P. 1997. Galois cohomology. Springer-Verlag, Berlin Google Scholar. Translation of Cohomologie Galoisienne; revised by the author.
Sopkina, E. 2009. Classification of all connected subgroup schemes of a reductive group containing a split maximal torus. J. K-Theory 3 CrossRef | Google Scholar:103–122.
Springer, T.A. 1977. Invariant theory. Lecture Notes in Mathematics, Vol. 585. Springer-Verlag, Berlin Google Scholar.
Springer, T.A. 1979. Reductive groups, pp. 3–27. In Automorphic forms, representations and L-functions (Proceedings of Symposia in Pure Mathematics, Vol. 33, Part 1, Oregon State University, Corvallis, OR, 1977). American Mathematical Society, Providence Google Scholar, RI.
Springer, T.A. 1994. Linear algebraic groups, pp. 1–121. In Algebraic geometry. IV, Encyclopaedia of Mathematical Sciences, Vol. 55. Springer-Verlag, Berlin Google Scholar. (Translation of Algebraicheskaya geometriya 4, VNINITI, Moscow, 1989).
Springer, T.A. 1998. Linear algebraic groups. Progress in Mathematics, Vol. 9. Birkhäuser Boston, Boston Google Scholar, MA.
Springer, T.A. and Veldkamp, F.D. 2000. Octonions, Jordan algebras and exceptional groups. Springer Monographs in Mathematics. Springer-Verlag, Berlin Google Scholar.
Steinberg, R. 1965 Google Scholar. Regular elements of semisimple algebraic groups. Inst. Hautes Etudes Sci. Publ. Math. pp. 49–80. Reprinted in Serre 1997.
Steinberg, R. 1967. Lectures on Chevalley groups Google Scholar. Department of Mathematics, Yale University. mimeographed notes (reprinted by the American Mathematical Society, 2016).
Steinberg, R. 1968. Endomorphisms of linear algebraic groups. Memoirs of the American Mathematical Society, No. 80. American Mathematical Society, Providence Google Scholar, RI.
Steinberg, R. 1999. The isomorphism and isogeny theorems for reductive algebraic groups. J. Algebra 216 CrossRef | Google Scholar:366–383.
Suzuki, K. 1971. A note on a theorem of E. Cartan. Tôhoku Math. J. (2) 23 CrossRef | Google Scholar:17–20.
Sweedler, M.E. 1967. Hopf algebras with one grouplike element. Trans. Amer. Math. Soc. 127 CrossRef | Google Scholar:515–526.
Sweedler, M.E. 1969. Hopf algebras. Mathematics Lecture Note Series. W. A. Benjamin, Inc., New York and Amsterdam Google Scholar.
Takeuchi, M. 1972. A correspondence between Hopf ideals and sub-Hopf algebras. Manuscripta Math. 7 CrossRef | Google Scholar:251–270.
Takeuchi, M. 1983. A hyperalgebraic proof of the isomorphism and isogeny theorems for reductive groups. J. Algebra 85 CrossRef | Google Scholar:179–196.
Tate, J. 1997. Finite flat group schemes, pp. 121–154. In Modular forms and Fermat's last theorem (Boston, MA, 1995). Springer-Verlag, Berlin Google Scholar.
Tate, J. and Oort, F. 1970. Group schemes of prime order. Ann. Sci. École Norm. Sup. (4) 3 CrossRef | Google Scholar:1–21.
Thăńg, N. 2008. On Galois cohomology of semisimple groups over local and global fields of positive characteristic. Math. Z. 259 CrossRef | Google Scholar:457–467.
ThĂńg, N. 2012. On Galois cohomology of semisimple groups over local and global fields of positive characteristic, II. Math. Z. 270 CrossRef | Google Scholar:1057–1065.
Tits, J. 1966. Classification of algebraic semisimple groups, pp. 33–62. In Algebraic Groups and Discontinuous Subgroups (Proceedings of Symposia in Pure Mathematics, Boulder, CO, 1965). American Mathematical Society, Providence Google Scholar, RI.
Tits, J. 1968 Google Scholar. Lectures on Algebraic Groups, notes by P. André and D. Winter, fall term 1966–1967, Yale University.
Tits, J. 1971. Représentations linéaires irréductibles d'un groupe réductif sur un corps quelconque. J. Reine Angew. Math. 247 Google Scholar:196–220.
Tits, J. 1992 Google Scholar. Théorie des groupes, Annuaire du Collège de France 1991-92. Reprinted in Tits 2013.
Tits, J. 1993 Google Scholar. Théorie des groupes, Annuaire du Collège de France 1992-93. Reprinted in Tits 2013.
Tits, J. 2013. Résumés des cours au Collège de France 1973–2000. Documents Mathématiques (Paris), Vol. 12. Société Mathématique de France, Paris Google Scholar.
Totaro, B. 2008. Hilbert's 14th problem over finite fields and a conjecture on the cone of curves. Compos. Math. 144 CrossRef | Google Scholar:1176–1198.
Totaro, B. 2013. Pseudo-abelian varieties. Ann. Sci. École Norm. Sup. (4) 46 CrossRef | Google Scholar:693–721.
VoskresenskiĬ, V.E. 1998. Algebraic groups and their birational invariants. Translations of Mathematical Monographs, Vol. 179. American Mathematical Society, Providence Google Scholar, RI.
Waterhouse, W.C. 1979. Introduction to affine group schemes. Graduate Texts in Mathematics, Vol. 66. Springer-Verlag, Berlin Google Scholar.
Weil, A. 1946. Foundations of Algebraic Geometry. American Mathematical Society Colloquium Publications, Vol. 29. American Mathematical Society, New York Google Scholar.
Weil, A. 1957. On the projective embedding of Abelian varieties, pp. 177–181. In Algebraic geometry and topology. A symposium in honor of S., Lefschetz. Princeton University Press, Princeton Google Scholar, NJ.
Weil, A. 1982. Adeles and algebraic groups. Progress in Mathematics, Vol. 23 Google Scholar. Birkhäuser Boston. Based on lectures at IAS, 1959–1960.
Wu, X.L. 1986. On the extensions of abelian varieties by affine group schemes, pp. 361–387. In Group theory, Beijing 1984, Lecture Notes in Mathematics, Vol. 1185. Springer-Verlag, Berlin Google Scholar.
Zibrowius, M. 2015. Symmetric representation rings are rings. New York J. Math. 21 Google Scholar:1055–1092.

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