Voir la notice de l'article provenant de la source Math-Net.Ru
@article{AA_2015_27_6_a4,
author = {N. A. Vavilov and A. Yu. Luzgarev},
title = {Normaliser of the {Chevalley} group of type~$\mathrm E_7$},
journal = {Algebra i analiz},
pages = {57--88},
publisher = {mathdoc},
volume = {27},
number = {6},
year = {2015},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/AA_2015_27_6_a4/}
}
N. A. Vavilov; A. Yu. Luzgarev. Normaliser of the Chevalley group of type~$\mathrm E_7$. Algebra i analiz, Tome 27 (2015) no. 6, pp. 57-88. http://geodesic.mathdoc.fr/item/AA_2015_27_6_a4/
[1] Borel A., “Svoistva i lineinye predstavleniya grupp Shevalle”, Seminar po algebraicheskim gruppam, Mir, M., 1973, 9–59 | MR
[2] Bunina E. I., “Avtomorfizmy grupp Shevalle tipov $\mathrm A_l$, $\mathrm D_l$, $\mathrm E_l$ nad lokalnymi koltsami s $1/2$”, Fundam. i prikl. mat., 15:2 (2009), 35–59 | MR
[3] Burbaki N., Gruppy i algebry Li, Gl. IV–VI, Mir, M., 1972 | MR
[4] Vavilov N. A., “Vesovye elementy grupp Shevalle”, Dokl. AN SSSR, 298:3 (1988), 524–527 | Zbl
[5] Vavilov N. A., “Podgruppy grupp Shevalle, soderzhaschie maksimalnyi tor”, Tr. Leningr. mat. o-va, 1, 1990, 64–109 | MR
[6] Vavilov N. A., “Kak uvidet znaki strukturnykh konstant?”, Algebra i analiz, 19:4 (2007), 34–68 | MR | Zbl
[7] Vavilov N. A., “Vesovye elementy grupp Shevalle”, Algebra i analiz, 20:1 (2008), 34–85 | MR | Zbl
[8] Vavilov N. A., “Numerologiya kvadratnykh uravnenii”, Algebra i analiz, 20:5 (2008), 9–40 | MR | Zbl
[9] Vavilov N.|A., “Esche nemnogo isklyuchitelnoi numerologii”, Zap. nauch. semin. POMI, 375, 2010, 22–31 | MR | Zbl
[10] Vavilov N. A., Luzgarev A. Yu., “Normalizator gruppy Shevalle tipa $\mathrm E_6$”, Algebra i analiz, 19:5 (2007), 37–64 | MR | Zbl
[11] Vavilov N. A., Luzgarev A. Yu., “Gruppa Shevalle tipa $\mathrm E_7$ v $56$-mernom predstavlenii”, Zap. nauch. semin. POMI, 386, 2011, 5–99
[12] Vavilov N. A., Perelman E. Ya., “Polivektornye predstavleniya $\mathrm{GL}_n$”, Zap. nauch. semin. POMI, 338, 2006, 69–97 | MR | Zbl
[13] Vavilov N. A., Petrov V. A., “O nadgruppakh $Ep(2l,R)$”, Algebra i analiz, 15:4 (2003), 72–114 | MR | Zbl
[14] Vavilov N. A., Petrov V. A., “O nadgruppakh $\mathrm{EO}(n,R)$”, Algebra i analiz, 19:2 (2007), 10–51 | MR | Zbl
[15] Dynkin E. B., “Maksimalnye podgruppy klassicheskikh grupp”, Tr. Mosk. mat. o-va, 1, 1952, 39–166 | MR | Zbl
[16] Luzgarev A. Yu., “O nadgruppakh $E(\mathrm E_6,R)$ i $E(\mathrm E_7,R)$ v minimalnykh predstavleniyakh”, Zap. nauch. semin. POMI, 319, 2004, 216–243 | MR | Zbl
[17] Luzgarev A. Yu., “Opisanie nadgrupp $\mathrm F_4$ v $\mathrm E_6$ nad kommutativnym koltsom”, Algebra i analiz, 20:6 (2008), 148–185 | MR | Zbl
[18] Luzgarev A. Yu., Nadgruppy isklyuchitelnykh grupp, Kand. dis., SPbGU, 2008, 106 pp.
[19] Luzgarev A. Yu., “Ne zavisyaschie ot kharakteristiki invarianty chetvertoi stepeni dlya $G(\mathrm E_7,R)$”, Vestn. S.-Peterburg. un-ta. Ser. 1, 2013, no. 1, 44–51 | MR | Zbl
[20] Khamfri Dzh., Lineinye algebraicheskie gruppy, Nauka, M., 1980 | MR
[21] Shevalle K., “O nekotorykh prostykh gruppakh”, Matematika, 2:1 (1958), 3–58 | MR
[22] Aschbacher M., “Some multilinear forms with large isometry groups”, Geom. Dedicata, 25:1–3 (1988), 417–465 | MR | Zbl
[23] Berman S., Moody R. V., “Extensions of Chevalley groups”, Israel J. Math., 22:1 (1975), 42–51 | DOI | MR | Zbl
[24] Brown R. B., “Groups of type $\mathrm E_7$”, J. Reine Angew. Math., 236 (1969), 79–102 | MR | Zbl
[25] Cooperstein B. N., “The fifty-six-dimensional module for $\mathrm E_7$. I. A four form for $\mathrm E_7$”, J. Algebra, 173:2 (1995), 361–389 | DOI | MR | Zbl
[26] Dixon J. D., “Rigid embedding of simple groups in the general linear group”, Canad. J. Math., 29:2 (1977), 384–391 | DOI | MR | Zbl
[27] Faulkner J. R., Ferrar J. C., “Exceptional Lie algebras and related algebraic and geometric structures”, Bull. London Math. Soc., 9:1 (1977), 1–35 | DOI | MR | Zbl
[28] Freudenthal H., “Sur le groupe exceptionnel $\mathrm E_7$”, Proc. Nederl. Akad. Wetensch. Ser. A, 56 (1953), 81–89 | DOI | MR | Zbl
[29] Freudenthal H., “Beziehungen der $\mathrm E_7$ und $\mathrm E_8$ zur Oktavenebene. I”, Proc. Nederl. Akad. Wetensch. Ser. A, 57 (1954), 218–230 ; “II”, Proc. Nederl. Akad. Wetensch. Ser. A, 57 (1954), 363–368 ; “III”, Proc. Nederl. Akad. Wetensch. Ser. A, 58 (1955), 151–157 ; “IV”, Proc. Nederl. Akad. Wetensch. Ser. A, 58 (1955), 277–285 ; “V”, Proc. Nederl. Akad. Wetensch. Ser. A, 62 (1959), 165–179 ; “VI”, Proc. Nederl. Akad. Wetensch. Ser. A, 62 (1959), 180–191 ; “VII”, Proc. Nederl. Akad. Wetensch. Ser. A, 62 (1959), 192–201 ; “VIII”, Proc. Nederl. Akad. Wetensch. Ser. A, 62 (1959), 447–465 ; “IX”, Proc. Nederl. Akad. Wetensch. Ser. A, 62 (1959), 466–474 ; “X”, Proc. Nederl. Akad. Wetensch. Ser. A, 66 (1963), 457–471 ; “XI”, Proc. Nederl. Akad. Wetensch. Ser. A, 66 (1963), 472–487 | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | DOI | Zbl | DOI | DOI | DOI | DOI | DOI | MR | Zbl | DOI
[30] Haris S. J., “Some irreducible representations of exceptional algebraic groups”, Amer. J. Math., 93:1 (1971), 75–106 | DOI | MR | Zbl
[31] Hazrat R., Vavilov N., “$K_1$ of Chevalley groups are nilpotent”, J. Pure Appl. Algebra, 179:1–2 (2003), 99–116 | DOI | MR | Zbl
[32] Jacobson N., Structure and representations of Jordan algebras, Amer. Math. Soc. Colloq. Publ., 39, Amer. Math. Soc., Providence, RI, 1968 | MR | Zbl
[33] Lichtenstein W., “A system of quadrics describing the orbit of the highest weight vector”, Proc. Amer. Math. Soc., 84:4 (1982), 605–608 | DOI | MR | Zbl
[34] Liebeck M. W., Seitz G. M., “On the subgroup structure of exceptional groups of Lie type”, Trans. Amer. Math. Soc., 350:9 (1998), 3409–3482 | DOI | MR | Zbl
[35] Lurie J., “On simply laced Lie algebras and their minuscule representations”, Comment. Math. Helv., 166:3 (2001), 515–575 | DOI | MR
[36] Luzgarev A., Petrov V., Vavilov N., Explicit equations on orbit of the highest weight vector (to appear)
[37] Luzgarev A., Equations determining the orbit of the highest weight vector in the adjoint representation, arXiv: 1401.0849
[38] Matsumoto H., “Sur les sous-groupes arithmétiques des groupes semi-simples déployés”, Ann. Sci. École Norm. Sup. (4), 2 (1969), 1–62 | MR | Zbl
[39] Petrov V., “Overgroups of unitary groups”, $K$-Theory, 29:3 (2003), 147–174 | DOI | MR | Zbl
[40] Plotkin E. B., Semenov A. A., Vavilov N. A., “Visual basic representations: an atlas”, Internat. J. Algebra Comput., 8:1 (1998), 61–95 | DOI | MR | Zbl
[41] Seitz G. M., The maximal subgroups of classical algebraic groups, Mem. Amer. Math. Soc., 67, no. 365, 1987 | MR
[42] Stein M. R., “Generators, relations and coverings of Chevalley groups over commutative rings”, Amer. J. Math., 93 (1971), 965–1004 | DOI | MR | Zbl
[43] Stein M. R., “Stability theorems for $K_1$, $K_2$ and related functors modeled on Chevalley groups”, Japan. J. Math. (N.S.), 4:1 (1978), 77–108 | MR | Zbl
[44] Vavilov N. A., “Structure of Chevalley groups over commutative rings”, Nonassociative Algebras and Related Topics (Hiroshima, 1990), World Sci. Publ., River Edge, NJ, 1991, 219–335 | MR | Zbl
[45] Vavilov N. A., “A third look at weight diagrams”, Rend. Sem. Mat. Univ. Padova, 104 (2000), 201–250 | MR | Zbl
[46] Vavilov N. A., “An $\mathrm A_3$-proof of structure theorems for Chevalley groups of types $\mathrm E_6$ and $\mathrm E_7$”, Internat. J. Algebra Comput., 17:5–6 (2007), 1283–1298 | DOI | MR | Zbl
[47] Vavilov N. A., Plotkin E. B., “Chevalley groups over commutative rings. I. Elementary calculations”, Acta Appl. Math., 45:1 (1996), 73–113 | DOI | MR | Zbl
[48] Waterhouse W. C., “Automorphisms of $\mathrm{det}(X_{ij})$: the group scheme approach”, Adv. in Math., 65:2 (1987), 171–203 | DOI | MR | Zbl