Group varieties and group structures

V. L. Popov

Izvestiya. Mathematics , Tome 86 (2022) no. 5, pp. 903-924.

Résumé

We explore to what extent the group variety of a connected algebraic group or the group manifold of a real Lie group determines its group structure.

Détail

Mots-clés : algebraic group, Lie group.

Bibliographie

[1] V. L. Popov, On algebraic group varieties, arXiv: 2102.08032

[2] B. Kunyavski, Letter to V. L. Popov, January 18, 2021

[3] A. Weil, Variétés Abéliennes et courbes algébriques, Actualités Sci. Indust., 1064, Publ. Inst. Math. Univ. Strasbourg, 8(1946), Hermann Cie, Paris, 1948 | MR | Zbl

[4] C. Chevalley, Theory of Lie Groups. I., Princeton Mathematical Series, 8, Princeton University Press, Princeton, NJ, 1999 ; Transl. from the French C. Chevalley, Théorie des groupes de Lie, v. t. 3, Actualités Sci. Indust., 1226, Théorèmes généraux sur les algèbres de Lie, Hermann Cie, Paris, 1955 | MR | Zbl | MR | Zbl

[5] A. Borel, “Groupes linéaires algébriques”, Ann. of Math. (2), 64 (1956), 20–82 | DOI | MR | Zbl

[6] A. Grothendieck, “Torsion homologique et sections rationnelles”, Séminaire C. Chevalley, 2e année, v. t. 3, Anneaux de Chow et applications, Secrétariat mathématique, Paris, 1958, Exp. No. 5 | MR | Zbl

[7] M. Rosenlicht, “Toroidal algebraic groups”, Proc. Amer. Math. Soc., 12:6 (1961), 984–988 | DOI | MR | Zbl

[8] M. Lazard, “Sur la nilpotence de certains groupes algébriques”, C. R. Acad. Sci. Paris, 241 (1955), 1687–1689 | MR | Zbl

[9] N. Bourbaki, “Lie algebras”, Lie groups and algebras, Chapter I, Mir, Moscow, 1976, 7–123 ; Transl. from the French N. Bourbaki, Éléments de mathématique. Groupes et algébres de Lie, Ch. I. Algébres de Lie, Éléments de mathématique, Reprint of the 1972 original, Springer, Berlin, 2007 | MR | DOI | MR | Zbl

[10] V. L. Popov, Underlying varieties and group structures, arXiv: 2105.12861

[11] V. L. Popov, Multidimensional residues and tropical geometry, Conf. video (Internat. conf., Sochi, June 16, 2021)

[12] V. L. Popov, “Group structures on algebraic varieties”, Dokl. Akad. Nauk. Matem., inform., proc. upr., 500 (2021), 52–54 ; English transl. Dokl. Math., 104:2 (2021), 264–266 | DOI | Zbl | DOI

[13] V. L. Popov, Linear algebraic groups and related structures (Internat. conf., St. Petersburg, November 11, 2021)

[14] A. Borel, Linear algebraic groups, Grad. Texts in Math., 126, 2nd ed., Springer-Verlag, New York, 1991 | DOI | MR | Zbl

[15] J. E. Humphreys, Linear algebraic groups, Grad. Texts in Math., 21, Springer-Verlag, New York–Heidelberg, 1975 ; Russian transl. Nauka, Moscow, 1980 | DOI | MR | Zbl | MR | Zbl

[16] T. A. Springer, Linear algebraic groups, Progr. Math., 9, 2nd ed., Birkhäuser Boston, Inc., Boston, MA, 1998 | DOI | MR | Zbl

[17] V. L. Popov, “Variations on the theme of Zariski's cancellation problem”, Polynomial rings and affine algebraic geometry (Tokyo 2018), Springer Proc. Math. Stat., 319, Springer, Cham, 2020, 233–250 | DOI | MR | Zbl

[18] A. L. Onishchik and E. B. Vinberg, Seminar on Lie groups and algebraic groups, Nauka, Moscow, 1988 ; English transl. Lie groups and algebraic groups, Springer Ser. Soviet Math., Springer-Verlag, Berlin, 1990 | MR | Zbl | DOI | MR | Zbl

[19] J. Milnor, Morse theory, Based on lecture notes by M. Spivak and R. Wells, Ann. of Math. Stud., 51, Princeton Univ. Press, Princeton, NJ, 1963 ; Russian transl. Mir, Moscow, 1965 | DOI | MR | Zbl | MR

[20] V. I. Danilov, “Cohomology of algebraic varieties”, Algebraic geometry II, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr., 35, VINITI, Moscow, 1989, 5–130 ; English transl. V. I. Danilov, “Cohomology of algebraic varieties”, Algebraic geometry II, Encyclopaedia Math. Sci., 35, Springer, Berlin, 1996, 1–125 | MR | Zbl | DOI | MR | Zbl

[21] A. Andreotti and T. Frankel, “The Lefschetz theorem on hyperplane sections”, Ann. of Math. (2), 69:3 (1959), 713–717 | DOI | MR | Zbl

[22] A. L. Onishchik, Topology of transitive transformation groups, Fizmatlit, Moscow, 1995 ; English transl. Johann Ambrosius Barth Verlag GmbH, Leipzig, 1994 | MR | Zbl | MR | Zbl

[23] M. Brion, “Some structure theorems for algebraic groups”, Algebraic groups: structure and actions, Proc. Sympos. Pure Math., 94, Amer. Math. Soc., Providence, RI, 2017, 53–126 | DOI | MR | Zbl

[24] M. Rosenlicht, “Some basic theorems on algebraic groups”, Amer. J. Math., 78:2 (1956), 401–443 | DOI | MR | Zbl

[25] A. Białynicki-Birula, “Remarks on the action of an algebraic torus on $k^n$”, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 14:4 (1966), 177–181 | MR | Zbl

[26] G. A. Dill, On morphisms between connected commutative algebraic groups, arXiv: 2107.14667

[27] H. Scheerer, “Homotopieäquivalente kompakte Liesche Gruppen”, Topology, 7:3 (1968), 227–232 | DOI | MR | Zbl

[28] P. F. Baum and W. Browder, “The cohomology of quotients of classical groups”, Topology, 3:4 (1965), 305–336 | DOI | MR | Zbl

[29] A. Hatcher, Algebraic topology, Cambridge Univ. Press, Cambridge, 2002 ; Russian transl. MCCME, Moscow, 2011 | MR | Zbl

[30] A. Borel, “On affine algebraic homogeneous spaces”, Arch. Math. (Basel), 45:1 (1985), 74–78 | DOI | MR | Zbl