Automorphisms of Chevalley groups of types $A_l$, $D_l$, $E_l$ over local rings without~1/2

E. I. Bunina

Fundamentalʹnaâ i prikladnaâ matematika, Tome 15 (2009) no. 7, pp. 47-80.

Résumé

In the given paper, we prove that every automorphism of a Chevalley group of type $A_l$, $D_l$, or $E_l$, $l\geq3$, over a commutative local ring without 1/2 is standard, i.e., it is a composition of ring, inner, central, and graph automorphisms.

Bibliographie

[1] Bloshitsyn V. Ya., “Avtomorfizmy obschei lineinoi gruppy nad kommutativnym koltsom, ne porozhdaemym delitelyami nulya”, Algebra i logika, 17:6 (1978), 639–642 | MR

[2] Borel A., “Svoistva i lineinye predstavleniya grupp Shevalle”, Seminar po algebraicheskim gruppam, M., 1973, 9–59 | MR

[3] Bunina E. I., “Avtomorfizmy prisoedinënnykh grupp Shevalle tipov $B_2$ i $G_2$ nad lokalnymi koltsami”, Fundament. i prikl. mat., 13:4 (2007), 3–29 | MR | Zbl

[4] Bunina E. I., “Avtomorfizmy grupp Shevalle tipa $B_l$ nad lokalnymi koltsami s 1/2”, Fundament. i prikl. mat., 15:7 (2009), 3–46, arXiv: 0911.4243 | MR

[5] Bunina E. I., “Avtomorfizmy i normalizatory grupp Shevalle tipov $A_l$, $D_l$, $E_l$ nad lokalnymi koltsami s 1/2”, Fundament. i prikl. mat., 15:2 (2009), 35–59, arXiv: 0907.5595 | MR

[6] Bunina E. I., “Avtomorfizmy elementarnykh prisoedinënnykh grupp Shevalle tipov $A_l$, $D_l$, $E_l$ nad lokalnymi koltsami”, Algebra i logika, 48:4 (2009), 443–470, arXiv: 0702046 | MR | Zbl

[7] Vavilov N. A., “Parabolicheskie podgruppy grupp Shevalle nad kommutativnym koltsom”, Zap. nauch. sem. LOMI AN SSSR, 116, 1982, 20–43 | MR | Zbl

[8] Vavilov N. A., Gavrilovich M. R., “$A_2$-dokazatelstvo strukturnykh teorem dlya grupp Shevalle tipov $E_6$ i $E_7$”, Algebra i analiz, 116:4 (2004), 54–87 | MR | Zbl

[9] Vavilov N. A, Gavrilovich M. R., Nikolenko S. I., “Stroenie grupp Shevalle: dokazatelstvo iz knigi”, Zap. nauch. sem. LOMI AN SSSR, 330, 2006, 36–76 | MR | Zbl

[10] Vavilov N. A., Petrov V. A., “O nadgruppakh $\mathrm{Ep}(2l,R)$”, Algebra i analiz, 15:4 (2003), 72–114 | MR | Zbl

[11] Golubchik I. Z., Mikhalëv A. V., “Izomorfizmy obschei lineinoi gruppy nad assotsiativnym koltsom”, Vestn. Mosk. un-ta. Ser. 1. Matematika, mekhanika, 1983, no. 3, 61–72 | MR | Zbl

[12] Golubchik I. Z., Mikhalëv A. V., “Izomorfizmy unitarnykh grupp nad assotsiativnymi koltsami”, Zap. nauch. sem. LOMI AN SSSR, 132, 1983, 97–109 | MR

[13] Dedonne Zh., Geometriya klassicheskikh grupp, Mir, M., 1974 | MR

[14] Zalesskii A. E., “Lineinye gruppy”, Algebra – 4, Itogi nauki i tekhn. Ser. Fundamentalnye napravleniya, 37, VINITI, M., 1989, 114–228 | MR | Zbl

[15] Zelmanov E. I., “Izomorfizmy polnykh lineinykh grupp nad assotsiativnymi koltsami”, Sib. mat. zhurn., 26:4 (1985), 49–67 | MR | Zbl

[16] Petechuk V. M., “Avtomorfizmy grupp $\mathrm{SL}_n$, $\mathrm{GL}_n$ nad nekotorymi lokalnymi koltsami”, Mat. zametki, 28:2 (1980), 187–204 | MR | Zbl

[17] Petechuk V. M., “Avtomorfizmy grupp $\mathrm{SL}_3(K)$, $\mathrm{GL}_3(K)$”, Mat. zametki, 31:5 (1982), 657–668 | MR | Zbl

[18] Petechuk V. M., “Avtomorfizmy matrichnykh grupp nad kommutativnymi koltsami”, Mat. sb., 117(159):4 (1982), 534–547 | MR | Zbl

[19] Suslin A. A., “Ob odnoi teoreme Kona”, Zap. nauch. sem. LOMI AN SSSR, 64, 1976, 127–130 | MR | Zbl

[20] Shevalle K., “O nekotorykh prostykh gruppakh”, Matematika, 2:1 (1958), 3–58 | MR

[21] Abe E., “Chevalley groups over local rings”, Tôhoku Math. J., 21:3 (1969), 474–494 | DOI | MR | Zbl

[22] Abe E., “Chevalley groups over commutative rings”, Proc. Conf. Radical Theory, Sendai, 1988, 1–23 | MR | Zbl

[23] Abe E., “Automorphisms of Chevalley groups over commutative rings”, Alg. Anal., 5:2 (1993), 74–90 | MR | Zbl

[24] Abe E., “Normal subgroups of Chevalley groups over commutative rings”, Contemp. Math., 83 (1989), 1–17 | DOI | MR | Zbl

[25] Abe E., Hurley J., “Centers of Chevalley groups over commutative rings”, Commun. Algebra, 16:1 (1988), 57–74 | DOI | MR | Zbl

[26] Bak A., “Nonabelian K-theory: The nilpotent class of $K_1$ and general stability”, K-Theory, 4 (1991), 363–397 | DOI | MR | Zbl

[27] Bak A., “Vavilov normality of the elementary subgroup functors”, Math. Proc. Cambridge Philos. Soc., 118:1 (1995), 35–47 | DOI | MR | Zbl

[28] Borel A., Tits J., “Homomorphismes “abstraits” de groupes algébriques simples”, Ann. Math., 73 (1973), 499–571 | DOI | MR

[29] Bourbaki N., Groupes et algébres de Lie, Hermann, 1968 | MR

[30] Bunina E. I., “Automorphisms of Chevalley groups of type $F_4$ over local rings with 1/2”, J. Algebra (to appear) , arXiv: 0907.5592 | MR

[31] Carter R. W., Simple Groups of Lie Type, Wiley, London, 1989 | MR | Zbl

[32] Carter R. W., Chen Y., “Automorphisms of affine Kac–Moody groups and related Chevalley groups over rings”, J. Algebra, 155 (1993), 44–94 | DOI | MR | Zbl

[33] Chen Y., “Isomorphic Chevalley groups over integral domains”, Rend. Sem. Mat. Univ. Padova, 92 (1994), 231–237 | MR | Zbl

[34] Chen Y., “Automorphisms of simple Chevalley groups over $\mathbb Q$-algebras”, Tôhoku Math. J., 348 (1995), 81–97 | DOI | MR

[35] Chen Y., “On representations of elementary subgroups of Chevalley groups over algebras”, Proc. Amer. Math. Soc., 123:8 (1995), 2357–2361 | DOI | MR | Zbl

[36] Chen Y., “Isomorphisms of adjoint Chevalley groups over integral domains”, Trans. Amer. Math. Soc., 348:2 (1996), 1–19 | DOI | MR

[37] Chen Y., “Isomorphisms of Chevalley groups over algebras”, J. Algebra, 226 (2000), 719–741 | DOI | MR | Zbl

[38] Chevalley C., “Certains schémas de groupes semi-simples”, Sem. Bourbaki, 219 (1960–1961), 1–16 | MR

[39] Cohn P., “On the structure of the $\mathrm{GL}_2$ of a ring”, Inst. Hautes Études Sci. Publ. Math., 30 (1966), 365–413 | DOI | MR | Zbl

[40] Demazure M., Gabriel P., Groupes algébriques. I, North-Holland, Amsterdam, 1970 | MR | Zbl

[41] Demazure M., Grothendieck A., Schémas en groupes, v. I, Lect. Notes Math., 151, Springer, Berlin, 1971; v. II, 152 ; v. III, 153 | Zbl | Zbl

[42] Dieudonné J., On the Automorphisms of Classical Groups, Mem. Amer. Math. Soc., 2, Amer. Math. Soc., 1951 | MR | Zbl

[43] Golubchik I. Z., “Isomorphisms of the linear general group $\mathrm{GL}_n(R)$, $n\ge4$, over an associative ring”, Contemp. Math., 131:1 (1992), 123–136 | DOI | MR | Zbl

[44] Grothendieck A., “Éléments de géoméntrie algébrique (rédigés avec la collaboration de Jean Dieudonné). IV. Étude locale des schémas et des morphismes de schémas”, Inst. Hautes Études Sci. Publ. Math., 32 (1967), 5–361 | DOI | MR | Zbl

[45] Hahn A. J., O'Meara O. T., The Classical Groups and K-Theory, Springer, Berlin, 1989 | MR | Zbl

[46] Hazrat R., Vavilov N. A., “$K_1$ of Chevalley groups are nilpotent”, J. Pure Appl. Algebra, 179 (2003), 99–116 | DOI | MR | Zbl

[47] Hua L. K., Reiner I., “Automorphisms of unimodular groups”, Trans. Amer. Math. Soc., 71 (1951), 331–348 | DOI | MR | Zbl

[48] Humphreys J. E., Introduction to Lie Algebras and Representation Theory, Springer, New York, 1978 | MR | Zbl

[49] Humphreys J. F., “On the automorphisms of infinite Chevalley groups”, Can. J. Math., 21 (1969), 908–911 | DOI | MR | Zbl

[50] Jantzen J. C., Representations of Algebraic Groups, Academic Press, New York, 1987 | MR | Zbl

[51] Klyachko A. A., Automorphisms and isomorphisms of Chevalley groups and algebras, 2007, arXiv: 0708.2256v3 | MR

[52] Li F., Li Z., “The isomorphisms of $\mathrm{GL}_3$ over commutative rings”, Classical Groups and Related Topics, Proc. Conf. (Beijing/China 1987), Contemp. Math., 82, Amer. Math. Soc., 1989, 47–52 | DOI | MR

[53] McDonald B. R., “Automorphisms of $\mathrm{GL}_n(R)$”, Trans. Amer. Math. Soc., 215 (1976), 145–159 | MR | Zbl

[54] O'Meara O. T., “The automorphisms of linear groups over any integral domain”, J. Reine Angew. Math., 223 (1966), 56–100 | DOI | MR | Zbl

[55] Stein M. R., “Generators, relations and coverings of Chevalley groups over commutative rings”, Amer. J. Math., 93:4 (1971), 965–1004 | DOI | MR | Zbl

[56] Stein M. R., “Surjective stability in dimension 0 for $K_2$ and related functors”, Trans. Amer. Math. Soc., 178:1 (1973), 165–191 | MR | Zbl

[57] Stein M. R., “Stability theorems for $K_1$, $K_2$ and related functors modeled on Chevalley groups”, Japan J. Math., 4:1 (1978), 77–108 | MR | Zbl

[58] Steinberg R., “Automorphisms of finite linear groups”, Can. J. Math., 121 (1960), 606–615 | DOI | MR

[59] Steinberg R., Lectures on Chevalley Groups, Yale University, 1967 | MR

[60] Suzuki K., “On the automorphisms of Chevalley groups over $p$-adic integer rings”, Kumamoto J. Sci. Math., 16:1 (1984), 39–47 | MR | Zbl

[61] Swan R., “Generators and relations for certain special linear groups”, Adv. Math., 6 (1971), 1–77 | DOI | MR | Zbl

[62] Taddei G., “Normalité des groupes élémentaire dans les groupes de Chevalley sur un anneau”, Contemp. Math., 55 (1986), 693–710 | DOI | MR | Zbl

[63] Vaserstein L. N., “On normal subgroups of Chevalley groups over commutative rings”, Tôhoku Math. J., 36:5 (1986), 219–230 | DOI | MR

[64] Vavilov N. A., “Structure of Chevalley groups over commutative rings”, Proc. Conf. Non-Associative Algebras and Related Topics (Hiroshima-1990), World Scientific, London, 1991, 219–335 | MR | Zbl

[65] Vavilov N. A., “An $A_3$-proof of structure theorems for Chevalley groups of types $E_6$ and $E_7$”, Int. J. Algebra Comput., 17:5–6 (2007), 1283–1298 | DOI | MR | Zbl

[66] Vavilov N. A., Plotkin E. B., “Chevalley groups over commutative rings. I. Elementary calculations”, Acta Appl. Math., 45 (1996), 73–113 | DOI | MR | Zbl

[67] Waterhouse W. C., Introduction to Affine Group Schemes, Springer, Berlin, 1979 | MR | Zbl

[68] Waterhouse W. C., “Automorphisms of $\mathrm{GL}_n(R)$”, Proc. Amer. Math. Soc., 79 (1980), 347–351 | MR | Zbl

[69] Waterhouse W. C., “Automorphisms of quotients of $\prod\mathrm{GL}(n_i)$”, Pacific J. Math., 79 (1982), 221–233 | DOI | MR

[70] Waterhouse W. C., “Automorphisms of $\operatorname{det}(X_{ij})$: the group scheme approach”, Adv. Math., 65:2 (1987), 171–203 | DOI | MR | Zbl