×

zbMATH — the first resource for mathematics

\(E_8\), the most exceptional group. (English) Zbl 1398.20062
Summary: The five exceptional simple Lie algebras over the complex number are included one within the other as \(\mathfrak{g}_2 \subset \mathfrak{f}_4 \subset \mathfrak{e}_6 \subset \mathfrak{e}_7 \subset \mathfrak{e}_8\). The biggest one, \(\mathfrak{e}_8\), is in many ways the most mysterious. This article surveys what is known about it, including many recent results, and it focuses on the point of view of Lie algebras and algebraic groups over fields.

MSC:
20G41 Exceptional groups
17B25 Exceptional (super)algebras
20G15 Linear algebraic groups over arbitrary fields
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Adams, J. F., Lectures on exceptional Lie groups, Chicago Lectures in Mathematics, xiv+122 pp., (1996), University of Chicago Press, Chicago, IL · Zbl 0866.22008
[2] AFS:Weyl B. Allison, J. Faulkner, and O. Smirnov, \emph Weyl images of Kantor pairs, arxiv:1404.3339, 2014.
[3] Allison, B. N., Models of isotropic simple Lie algebras, Comm. Algebra, 7, 17, 1835-1875, (1979) · Zbl 0422.17006
[4] A:survey B. N. Allison, \emph Structurable algebras and the construction of simple Lie algebras, Jordan algebras (Berlin) (W. Kaup, K. McCrimmon, and H.P. Petersson, eds.), de Gruyter, 1994, (Proceedings of a conference at Oberwolfach, 1992).
[5] Auel, Asher; Brussel, Eric; Garibaldi, Skip; Vishne, Uzi, Open problems on central simple algebras, Transform. Groups, 16, 1, 219-264, (2011) · Zbl 1230.16016
[6] Azad, H.; Barry, M.; Seitz, G., On the structure of parabolic subgroups, Comm. Algebra, 18, 2, 551-562, (1990) · Zbl 0717.20029
[7] Baez, John; Huerta, John, The algebra of grand unified theories, Bull. Amer. Math. Soc. (N.S.), 47, 3, 483-552, (2010) · Zbl 1196.81252
[8] Baez, John C., The octonions, Bull. Amer. Math. Soc. (N.S.), 39, 2, 145-205, (2002) · Zbl 1026.17001
[9] Bannai, Eiichi; Sloane, N. J. A., Uniqueness of certain spherical codes, Canad. J. Math., 33, 2, 437-449, (1981) · Zbl 0457.05017
[10] Bayer-Fluckiger, E.; Parimala, R., Galois cohomology of the classical groups over fields of cohomological dimension \(≤ 2\), Invent. Math., 122, 2, 195-229, (1995) · Zbl 0851.11024
[11] Belitskii, Genrich; Lipyanski, Ruvim; Sergeichuk, Vladimir, Problems of classifying associative or Lie algebras and triples of symmetric or skew-symmetric matrices are wild, Linear Algebra Appl., 407, 249-262, (2005) · Zbl 1159.17304
[12] Blinstein, Sam; Merkurjev, Alexander, Cohomological invariants of algebraic tori, Algebra Number Theory, 7, 7, 1643-1684, (2013) · Zbl 1368.11034
[13] Block, Richard E., Trace forms on Lie algebras, Canad. J. Math., 14, 553-564, (1962) · Zbl 0111.03903
[14] Block, Richard E.; Zassenhaus, Hans, The Lie algebras with a nondegenerate trace form, Illinois J. Math., 8, 543-549, (1964) · Zbl 0131.27102
[15] Boelaert, Lien; De Medts, Tom, A new construction of Moufang quadrangles of type \(E_6\), \(E_7\) and \(E_8\), Trans. Amer. Math. Soc., 367, 5, 3447-3480, (2015) · Zbl 1360.51004
[16] Borel, A.; De Siebenthal, J., Les sous-groupes ferm\'es de rang maximum des groupes de Lie clos, Comment. Math. Helv., 23, 200-221, (1949) · Zbl 0034.30701
[17] Borel, Armand; Tits, Jacques, Groupes r\'eductifs, Inst. Hautes \'Etudes Sci. Publ. Math., 27, 55-150, (1965) · Zbl 0145.17402
[18] Borthwick, David; Garibaldi, Skip, Did a \(1\)-dimensional magnet detect a \(248\)-dimensional Lie algebra?, Notices Amer. Math. Soc., 58, 8, 1055-1066, (2011) · Zbl 1267.82002
[19] Bourbaki, Nicolas, Lie groups and Lie algebras. Chapters 4–6, Elements of Mathematics (Berlin), xii+300 pp., (2002), Springer-Verlag, Berlin · Zbl 0983.17001
[20] Bou:g7 Nicolas Bourbaki, \emph Lie groups and Lie algebras: Chapters 7–9, Springer-Verlag, Berlin, 2005.
[21] Bruhat, F.; Tits, J., Groupes alg\'ebriques sur un corps local. Chapitre III. Compl\'ements et applications \`a la cohomologie galoisienne, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 34, 3, 671-698, (1987) · Zbl 0657.20040
[22] Brylinski, Ranee; Kostant, Bertram, Minimal representations of \(E_6\), \(E_7\), and \(E_8\) and the generalized Capelli identity, Proc. Nat. Acad. Sci. U.S.A., 91, 7, 2469-2472, (1994) · Zbl 0812.22009
[23] Ca:th E. Cartan, \emph Sur la structure des groupes de transformations finis et continus, thesis, Paris, 1894; 2nd edition, Vuibert, 1933 (= Oe., part 1, vol. 1 (1952), 137–287).
[24] Cartan:real E. Cartan, \emph Les groupes r\'eels simples finis et continus, Ann. Sci. \'Ecole Norm. Sup. (3) <span class=”textbf”>3</span>1 (1914), 265–355.
[25] \bibCarter:simplebook author=Carter, Roger W., title=Simple groups of Lie type, series=Wiley Classics Library, pages=x+335, publisher=John Wiley & Sons, Inc., New York, date=1989, note=Reprint of the 1972 original; A Wiley-Interscience Publication, isbn=0-471-50683-4, review=\MR 1013112,
[26] Cederwall, Martin; Palmkvist, Jakob, The octic \(E_8\) invariant, J. Math. Phys., 48, 7, 073505, 7 pp., (2007) · Zbl 1144.81322
[27] Chernousov, V. I., The Hasse principle for groups of type \(E_8\), Dokl. Akad. Nauk SSSR. Soviet Math. Dokl., 306 39, 3, 592-596, (1989) · Zbl 0703.20040
[28] Chernousov, V. I., A remark on the \(({\rm mod}\, 5)\)-invariant of Serre for groups of type \(E_8\), Mat. Zametki. Math. Notes, 56 56, 1-2, 730-733 (1995), (1994) · Zbl 0835.20059
[29] Ch:mod3 V. I. Chernousov, \emph On the kernel of the Rost invariant for \(E_8\) modulo \(3\), Quadratic forms, linear algebraic groups, and cohomology (J.-L. Colliot-Th\'el\`ene, S. Garibaldi, R. Sujatha, and V. Suresh, eds.), Developments in Mathematics, vol. 18, Springer, 2010, pp. 199–214.
[30] Chernousov, Vladimir; Serre, Jean-Pierre, Lower bounds for essential dimensions via orthogonal representations, J. Algebra, 305, 2, 1055-1070, (2006) · Zbl 1181.20042
[31] Chevalley, Claude C., The algebraic theory of spinors, viii+131 pp., (1954), Springer, Berlin, 1997, reprint of 1954 edition · Zbl 0057.25901
[32] Chevalley, Claude; Schafer, R. D., The exceptional simple Lie algebras \(F_4\) and \(E_6\), Proc. Nat. Acad. Sci. U. S. A., 36, 137-141, (1950) · Zbl 0037.02003
[33] Cohen, Arjeh M.; Wales, David B., Finite simple subgroups of semisimple complex Lie groups—a survey. Groups of Lie type and their geometries, Como, 1993, London Math. Soc. Lecture Note Ser. 207, 77-96, (1995), Cambridge Univ. Press, Cambridge · Zbl 0849.20010
[34] Coldea R. Coldea, D. A. Tennant, E. M. Wheeler, E. Wawrzynska, D. Prabhakaran, M. Telling, K. Habicht, P. Smibidl, and K. Kiefer, \emph Quantum criticality in an Ising chain: experimental evidence for emergent \(E_8\) symmetry, Science <span class=”textbf”>3</span>27 (2010), 177–180.
[35] Coleman, A. J., The greatest mathematical paper of all time, Math. Intelligencer, 11, 3, 29-38, (1989) · Zbl 0683.01007
[36] Conrad:Z B. Conrad, \emph Non-split reductive groups over \(\mathbf Z\), Autour des Sch\'emas en Groupes II, Panoramas et synth\`eses, vol. 46, Soci\'et\'e Math\'ematique de France, 2014, pp. 193–253.
[37] Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A., Atlas of finite groups, xxxiv+252 pp., (1985), Oxford University Press, Eynsham
[38] ConwaySloane J. H. Conway and N. J. A. Sloane, \emph Sphere packings, lattices, and groups, 2nd ed., Springer, 2013.
[39] Conway, John H.; Smith, Derek A., On quaternions and octonions: their geometry, arithmetic, and symmetry, xii+159 pp., (2003), A K Peters, Ltd., Natick, MA · Zbl 1098.17001
[40] de Jong, A. J.; He, Xuhua; Starr, Jason Michael, Families of rationally simply connected varieties over surfaces and torsors for semisimple groups, Publ. Math. Inst. Hautes \'Etudes Sci., 114, 1-85, (2011) · Zbl 1285.14053
[41] Delfino, G.; Mussardo, G., The spin-spin correlation function in the two-dimensional Ising model in a magnetic field at \(T=T_{\rm c}\), Nuclear Phys. B, 455, 3, 724-758, (1995) · Zbl 0925.82042
[42] Deligne, Pierre; Gross, Benedict H., On the exceptional series, and its descendants, C. R. Math. Acad. Sci. Paris, 335, 11, 877-881, (2002) · Zbl 1017.22008
[43] Demazure, M., Automorphismes et d\'eformations des vari\'et\'es de Borel, Invent. Math., 39, 2, 179-186, (1977) · Zbl 0406.14030
[44] SGA3 M. Demazure and A. Grothendieck, \emph SGA3: Sch\'emas en groupes, Lecture Notes in Mathematics, vol. 151–153, Springer, 1970.
[45] Diaconescu, Duiliu-Emanuel; Moore, Gregory; Witten, Edward, \(E_8\) gauge theory, and a derivation of \(K\)-theory from M-theory, Adv. Theor. Math. Phys., 6, 6, 1031-1134 (2003), (2002)
[46] Diaconescu, Emanuel; Moore, Gregory; Freed, Daniel S., The M-theory \(3\)-form and \(E_8\) gauge theory. Elliptic cohomology, London Math. Soc. Lecture Note Ser. 342, 44-88, (2007), Cambridge Univ. Press, Cambridge · Zbl 1232.58016
[47] Distler, Jacques; Garibaldi, Skip, There is no “theory of everything” inside \(E_8\), Comm. Math. Phys., 298, 2, 419-436, (2010) · Zbl 1201.22005
[48] Dynkin, E. B., Semisimple subalgebras of semisimple Lie algebras, Mat. Sbornik N.S., 30(72), 349-462 (3 plates), (1952) · Zbl 0048.01701
[49] Ebbinghaus, H.-D.; Hermes, H.; Hirzebruch, F.; Koecher, M.; Mainzer, K.; Neukirch, J.; Prestel, A.; Remmert, R., Numbers, Readings in Mathematics, Graduate Texts in Mathematics 123, xviii+395 pp., (1991), Springer-Verlag, New York · Zbl 0705.00001
[50] Eguchi, Tohru; Yang, Sung-Kil, Deformations of conformal field theories and soliton equations, Phys. Lett. B, 224, 4, 373-378, (1989)
[51] Elduque, Alberto, Jordan gradings on exceptional simple Lie algebras, Proc. Amer. Math. Soc., 137, 12, 4007-4017, (2009) · Zbl 1229.17009
[52] Elkies, Noam D., Lattices, linear codes, and invariants. I, Notices Amer. Math. Soc., 47, 10, 1238-1245, (2000) · Zbl 0992.11041
[53] Elman, Richard; Karpenko, Nikita; Merkurjev, Alexander, The algebraic and geometric theory of quadratic forms, American Mathematical Society Colloquium Publications 56, viii+435 pp., (2008), American Mathematical Society, Providence, RI · Zbl 1165.11042
[54] Fateev, V. A.; Zamolodchikov, A. B., Conformal field theory and purely elastic \(S\)-matrices, Internat. J. Modern Phys. A, 5, 6, 1025-1048, (1990) · Zbl 0737.17014
[55] Faulkner, John R., Some forms of exceptional Lie algebras, Comm. Algebra, 42, 11, 4854-4873, (2014) · Zbl 1377.17015
[56] Faulkner, J. R.; Ferrar, J. C., Exceptional Lie algebras and related algebraic and geometric structures, Bull. London Math. Soc., 9, 1, 1-35, (1977) · Zbl 0349.17004
[57] Feingold, Alex J.; Frenkel, Igor B.; Ries, John F. X., Spinor construction of vertex operator algebras, triality, and \(E^{(1)}_8\), Contemporary Mathematics 121, x+146 pp., (1991), American Mathematical Society, Providence, RI · Zbl 0743.17029
[58] Figueroa-O’Farrill, Jos\'e, A geometric construction of the exceptional Lie algebras \(F_4\) and \(E_8\), Comm. Math. Phys., 283, 3, 663-674, (2008) · Zbl 1157.17003
[59] Frenkel, Igor; Lepowsky, James; Meurman, Arne, Vertex operator algebras and the Monster, Pure and Applied Mathematics 134, liv+508 pp., (1988), Academic Press, Inc., Boston, MA · Zbl 0674.17001
[60] Frenkel, I. B.; Kac, V. G., Basic representations of affine Lie algebras and dual resonance models, Invent. Math., 62, 1, 23-66, (1980/81) · Zbl 0493.17010
[61] Freudenthal, Hans, Sur le groupe exceptionnel \(E_7\), Nederl. Akad. Wetensch. Proc. Ser. A. 56=Indagationes Math., 15, 81-89, (1953) · Zbl 0052.02404
[62] Freudenthal, Hans, Sur le groupe exceptionnel \(E_8\), Nederl. Akad. Wetensch. Proc. Ser. A. 56=Indagationes Math., 15, 95-98, (1953) · Zbl 0051.25905
[63] Freudenthal, Hans, Oktaven, Ausnahmegruppen und Oktavengeometrie, Geom. Dedicata, 19, 1, 7-63, (1985) · Zbl 0573.51004
[64] Fulton, William; Harris, Joe, Representation theory, Readings in Mathematics, Graduate Texts in Mathematics 129, xvi+551 pp., (1991), Springer-Verlag, New York · Zbl 0744.22001
[65] Garibaldi, Ryan Skip, The Rost invariant has trivial kernel for quasi-split groups of low rank, Comment. Math. Helv., 76, 4, 684-711, (2001) · Zbl 1001.20042
[66] Garibaldi, Skip, Cohomological invariants: exceptional groups and spin groups, Mem. Amer. Math. Soc., 200, 937, xii+81 pp., (2009) · Zbl 1191.11009
[67] Garibaldi, Skip, Orthogonal involutions on algebras of degree \(16\) and the Killing form of \(E_8\). Quadratic forms—algebra, arithmetic, and geometry, Contemp. Math. 493, 131-162, (2009), Amer. Math. Soc., Providence, RI · Zbl 1229.11063
[68] Garibaldi, Skip, Vanishing of trace forms in low characteristics, Algebra Number Theory, 3, 5, 543-566, (2009) · Zbl 1282.20052
[69] Garibaldi, Skip; Gille, Philippe, Algebraic groups with few subgroups, J. Lond. Math. Soc. (2), 80, 2, 405-430, (2009) · Zbl 1183.20053
[70] GG:edp S. Garibaldi and R. M. Guralnick, \emph Essential dimension of algebraic groups, including bad characteristic, Arch. Math. (Basel), to appear. DOI 10.1007/S00013-016-0925-Z · Zbl 1369.11029
[71] GG:simple S. Garibaldi and R. M. Guralnick, \emph Simple groups stabilizing polynomials, Forum of Mathematics: Pi <span class=”textbf”>3</span> (2015), e3 (41 pages). DOI 10.1017/fmp.2015.3
[72] GGN S. Garibaldi, R. M. Guralnick, and D. K. Nakano, \emph Globally irredicible Weyl modules, arxiv:1604.08911, 2016.
[73] Garibaldi, Skip; Merkurjev, Alexander; Serre, Jean-Pierre, Cohomological invariants in Galois cohomology, University Lecture Series 28, viii+168 pp., (2003), American Mathematical Society, Providence, RI · Zbl 1159.12311
[74] Garibaldi, Skip; Petersson, Holger P., Groups of outer type \(E_6\) with trivial Tits algebras, Transform. Groups, 12, 3, 443-474, (2007) · Zbl 1139.17004
[75] Garibaldi, S.; Petrov, V.; Semenov, N., Shells of twisted flag varieties and the Rost invariant, Duke Math. J., 165, 2, 285-339, (2016) · Zbl 1344.14004
[76] Garibaldi, S.; Qu\'eguiner-Mathieu, A., Restricting the Rost invariant to the center, Algebra i Analiz. St. Petersburg Math. J., 19 19, 2, 197-213, (2008) · Zbl 1209.11045
[77] Garibaldi, Skip; Semenov, Nikita, Degree \(5\) invariant of \(E_8\), Int. Math. Res. Not. IMRN, 19, 3746-3762, (2010) · Zbl 1204.22019
[78] Geck M. Geck, \emph On the construction of semisimple Lie algebras and Chevalley groups, arxiv:1602.04583, 2016. · Zbl 1419.17018
[79] Gille, Philippe, Cohomologie galoisienne des groupes alg\'ebriques quasi-d\'eploy\'es sur des corps de dimension cohomologique \(≤ 2\), Compositio Math., 125, 3, 283-325, (2001) · Zbl 1017.11019
[80] Gille, Philippe, Alg\`ebres simples centrales de degr\'e \(5\) et \(E_8\), Canad. Math. Bull., 45, 3, 388-398, (2002) · Zbl 1021.16008
[81] Gille:KT Philippe Gille, \emph Le probl\`eme de Kneser-Tits, Ast\'erisque <span class=”textbf”>3</span>26 (2009), 39–82.
[82] Gille:scsurv Philippe Gille, \emph Serre’s Conjecture II: a survey, Quadratic forms, linear algebraic groups, and cohomology (J.-L. Colliot-Th\'el\`ene, S. Garibaldi, R. Sujatha, and V. Suresh, eds.), Developments in Mathematics, vol. 18, Springer, 2010, pp. 41–56. · Zbl 1239.11046
[83] Gille, Philippe; Qu\'eguiner-Mathieu, Anne, Formules pour l’invariant de Rost, Algebra Number Theory, 5, 1, 1-35, (2011) · Zbl 1262.11050
[84] Griess, Robert L., Jr., Elementary abelian \(p\)-subgroups of algebraic groups, Geom. Dedicata, 39, 3, 253-305, (1991) · Zbl 0733.20023
[85] Griess, Robert L., Jr.; Ryba, A. J. E., Finite simple groups which projectively embed in an exceptional Lie group are classified!, Bull. Amer. Math. Soc. (N.S.), 36, 1, 75-93, (1999) · Zbl 0916.22008
[86] Gross, Benedict H., Groups over \(\textbf{Z}\), Invent. Math., 124, 1-3, 263-279, (1996) · Zbl 0846.20049
[87] Gross, Benedict H.; Wallach, Nolan R., A distinguished family of unitary representations for the exceptional groups of real rank \(=4\). Lie theory and geometry, Progr. Math. 123, 289-304, (1994), Birkh\"auser Boston, Boston, MA · Zbl 0839.22006
[88] Gross, David J.; Harvey, Jeffrey A.; Martinec, Emil; Rohm, Ryan, Heterotic string, Phys. Rev. Lett., 54, 6, 502-505, (1985)
[89] Groth:tor A. Grothendieck, \emph Torsion homologique et sections rationnelles, S\'eminaire Claude Chevalley <span class=”textbf”>3</span> (1958), no. 5, 1–29.
[90] G\"unaydin, M.; Koepsell, K.; Nicolai, H., Conformal and quasiconformal realizations of exceptional Lie groups, Comm. Math. Phys., 221, 1, 57-76, (2001) · Zbl 0992.22016
[91] Guralnick, Robert; Malle, Gunter, Rational rigidity for \(E_8(p)\), Compos. Math., 150, 10, 1679-1702, (2014) · Zbl 1328.12007
[92] Harder, G\`‘unter, \'’Uber die Galoiskohomologie halbeinfacher Matrizengruppen. I, Math. Z., 90, 404-428, (1965) · Zbl 0152.00903
[93] Harder, G\`‘unter, \'’Uber die Galoiskohomologie halbeinfacher Matrizengruppen. II, Math. Z., 92, 396-415, (1966) · Zbl 0152.01001
[94] Harder, G., Chevalley groups over function fields and automorphic forms, Ann. of Math. (2), 100, 249-306, (1974) · Zbl 0309.14041
[95] Hawkins, Thomas, Emergence of the theory of Lie groups, Sources and Studies in the History of Mathematics and Physical Sciences, xiv+564 pp., (2000), Springer-Verlag, New York · Zbl 0965.01001
[96] HeMcKay Y.-H. He and J. McKay, \emph Sporadic and exceptional, arxiv:1505.06742, 2015.
[97] Helgason, Sigurdur, A centennial: Wilhelm Killing and the exceptional groups, Math. Intelligencer, 12, 3, 54-57, (1990) · Zbl 0711.01033
[98] Hollowood, Timothy J.; Mansfield, Paul, Rational conformal field theories at, and away from, criticality as Toda field theories, Phys. Lett. B, 226, 1-2, 73-79, (1989)
[99] Hum:Lie J. E. Humphreys, \emph Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, vol. 9, Springer-Verlag, 1980, Third printing, revised.
[100] Jacobson, N., Exceptional Lie algebras, Lecture Notes in Pure and Applied Mathematics 1, v+125 pp. (loose errata) pp., (1971), Marcel Dekker, Inc., New York
[101] Jacobson, Nathan, Finite-dimensional division algebras over fields, viii+278 pp., (1996), Springer-Verlag, Berlin · Zbl 0874.16002
[102] Jantzen, Jens Carsten, Representations of algebraic groups, Mathematical Surveys and Monographs 107, xiv+576 pp., (2003), American Mathematical Society, Providence, RI · Zbl 1034.20041
[103] Kac:inf V. Kac, \emph Infinite dimensional Lie algebras, 3rd ed., Cambridge, 1995.
[104] Kazhdan, D.; Savin, G., The smallest representation of simply laced groups. Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part I, Ramat Aviv, 1989, Israel Math. Conf. Proc. 2, 209-223, (1990), Weizmann, Jerusalem · Zbl 0737.22008
[105] Killing2 W. Killing, \emph Die Zusammensetzung der stetigen endlichen Transformationsgruppen. Zweiter Theil, Math. Ann. <span class=”textbf”>3</span>3 (1889), 1–48.
[106] Klein:ico F. Klein, \emph Lectures on the ikosahedron, Tr\"ubner, London, 1888, translated by G.G. Morrice.
[107] Knus, Max-Albert; Merkurjev, Alexander; Rost, Markus; Tignol, Jean-Pierre, The book of involutions, American Mathematical Society Colloquium Publications 44, xxii+593 pp., (1998), American Mathematical Society, Providence, RI · Zbl 0955.16001
[108] Kostant, Bertram, Experimental evidence for the occurrence of \(E_8\) in nature and the radii of the Gosset circles, Selecta Math. (N.S.), 16, 3, 419-438, (2010) · Zbl 1204.22018
[109] LaackmanM D. Laackman and A. Merkurjev, \emph Degree three cohomological invariants of reductive groups, preprint, 2015.
[110] Lam, T. Y., Introduction to quadratic forms over fields, Graduate Studies in Mathematics 67, xxii+550 pp., (2005), American Mathematical Society, Providence, RI · Zbl 1068.11023
[111] Landsberg, J. M.; Manivel, L., Triality, exceptional Lie algebras and Deligne dimension formulas, Adv. Math., 171, 1, 59-85, (2002) · Zbl 1035.17016
[112] Lehalleur S. P. Lehalleur, \emph Subgroups of maximal rank of reductive groups, Autour des Sch\'emas en Groupes III, vol. 47, Soci\'et\'e Math\'ematique de France, 2016, pp. 147–172.
[113] Lemire, N., Essential dimension of algebraic groups and integral representations of Weyl groups, Transform. Groups, 9, 4, 337-379, (2004) · Zbl 1076.14060
[114] Loke:th H. Y. Loke, \emph Exceptional Lie groups and Lie algebras, Ph.D. thesis, Harvard University, 1997.
[115] LoetscherMacD R. L\"otscher and M. MacDonald, \emph The slice method for \(G\)-torsors, preprint, 2015.
[116] Lusztig, G., Homomorphisms of the alternating group \(\mathcal {A}_5\) into reductive groups, J. Algebra, 260, 1, 298-322, (2003) · Zbl 1074.20030
[117] Lusztig:E8 G. Lusztig, \emph On conjugacy classes in the Lie group \(E_8\), arxiv:1309.1382, 2013.
[118] Lusztig:can G. Lusztig, \emph The canonical basis of the quantum adjoint representation, arxiv:1602.07276, 2016.
[119] MacDonald, Mark L., Essential dimension of Albert algebras, Bull. Lond. Math. Soc., 46, 5, 906-914, (2014) · Zbl 1357.17029
[120] Marcus, Neil; Schwarz, John H., Three-dimensional supergravity theories, Nuclear Phys. B, 228, 1, 145-162, (1983)
[121] Solar I. McEwan, \emph Solar, Random House, 2010.
[122] Merkurjev, Alexander S., Essential dimension: a survey, Transform. Groups, 18, 2, 415-481, (2013) · Zbl 1278.14066
[123] Merkurjev, Alexander, Degree three cohomological invariants of semisimple groups, J. Eur. Math. Soc. (JEMS), 18, 3, 657-680, (2016) · Zbl 1367.12003
[124] Merkur\cprime ev, A. S.; Suslin, A. A., The group \(K_3\) for a field, Izv. Akad. Nauk SSSR Ser. Mat.. Math. USSR-Izv., 54 36, 3, 541-565, (1991)
[125] M\"uhlherr, Bernhard; Petersson, Holger P.; Weiss, Richard M., Descent in buildings, Annals of Mathematics Studies 190, xvi+336 pp., (2015), Princeton University Press, Princeton, NJ · Zbl 1338.51002
[126] Neher:3 E. Neher, \emph Lie algebras graded by \(3\)-graded root systems and Jordan pairs covered by grids, J. Algebra <span class=”textbf”>1</span>40 (1991), 284–329.
[127] Parimala, R.; Tignol, J.-P.; Weiss, R. M., The Kneser-Tits conjecture for groups with Tits-index \(\mathsf {E}_{8,2}^{66}\) over an arbitrary field, Transform. Groups, 17, 1, 209-231, (2012) · Zbl 1251.20045
[128] Petrov, Viktor; Semenov, Nikita; Zainoulline, Kirill, \(J\)-invariant of linear algebraic groups, Ann. Sci. \'Ec. Norm. Sup\'er. (4), 41, 6, 1023-1053, (2008) · Zbl 1206.14017
[129] Platonov, Vladimir; Rapinchuk, Andrei, Algebraic groups and number theory, Pure and Applied Mathematics 139, xii+614 pp., (1994), Academic Press, Inc., Boston, MA · Zbl 0841.20046
[130] Reichstein, Zinovy, Essential dimension. Proceedings of the International Congress of Mathematicians. Volume II, 162-188, (2010), Hindustan Book Agency, New Delhi · Zbl 1232.14030
[131] Reichstein, Zinovy, What is\(… \)essential dimension?, Notices Amer. Math. Soc., 59, 10, 1432-1434, (2012) · Zbl 1284.12003
[132] Reichstein, Zinovy; Youssin, Boris, Essential dimensions of algebraic groups and a resolution theorem for \(G\)-varieties, Canad. J. Math., 52, 5, 1018-1056, (2000) · Zbl 1044.14023
[133] Rost:H90 M. Rost, \emph Hilbert \(90\) for \(K_3\) for degree-two extensions, preprint, 1986.
[134] Rubenthaler, Hubert, Alg\`ebres de Lie et espaces pr\'ehomog\`enes, Travaux en Cours [Works in Progress] 44, viii+209 pp., (1992), Hermann \'Editeurs des Sciences et des Arts, Paris · Zbl 0840.17007
[135] Schafer, Richard D., An introduction to nonassociative algebras, x+166 pp., (1995), Dover Publications, Inc., New York · Zbl 1375.17001
[136] Seligman, G. B., Modular Lie algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 40, ix+165 pp., (1967), Springer-Verlag New York, Inc., New York · Zbl 0189.03201
[137] Semenov, Nikita, Motivic construction of cohomological invariants, Comment. Math. Helv., 91, 1, 163-202, (2016) · Zbl 1344.14005
[138] Serre, Jean-Pierre, Cohomologie galoisienne des groupes alg\'ebriques lin\'eaires. Colloq. Th\'eorie des Groupes Alg\'ebriques, Bruxelles, 1962, 53-68, (1962), Librairie Universitaire, Louvain; Gauthier-Villars, Paris
[139] Se9091 Jean-Pierre Serre, \emph R\'esum\'e des cours de 1990–91, Annuaire du Coll\`ege de France (1991), 111–121, (= Oe. 153).
[140] Selet Jean-Pierre Serre, \emph Letter to Markus Rost dated December 3rd, 1992.
[141] Serre, Jean-Pierre, Cohomologie galoisienne: progr\`es et probl\`emes, Ast\'erisque, 227, Exp. No. 783, 4, 229-257, (1995) · Zbl 0837.12003
[142] Se:plon Jean-Pierre Serre, \emph Exemples de plongements des groupes \(PSL_2(F_p)\) dans des groupes de Lie simples, Invent. Math. (1996), 525–562.
[143] Selet2 Jean-Pierre Serre, \emph Letter to Markus Rost dated June 23rd, 1999.
[144] Se:sgf Jean-Pierre Serre, \emph Sous-groupes finis des groupes de Lie, S\'eminaire Bourbaki, 51\`eme ann\'ee, 1998–99, no. 864, June 1999.
[145] Serre, Jean-Pierre, Galois cohomology, Springer Monographs in Mathematics, x+210 pp., (2002), Springer-Verlag, Berlin · Zbl 1004.12003
[146] Se:Kac Jean-Pierre Serre, \emph Coordonn\'ees de Kac, Oberwolfach Reports <span class=”textbf”>3</span> (2006), 1787–1790.
[147] Springer, T. A., Linear algebraic groups, Progress in Mathematics 9, xiv+334 pp., (1998), Birkh\"auser Boston, Inc., Boston, MA · Zbl 0927.20024
[148] Springer, Tonny A.; Veldkamp, Ferdinand D., Octonions, Jordan algebras and exceptional groups, Springer Monographs in Mathematics, viii+208 pp., (2000), Springer-Verlag, Berlin · Zbl 1087.17001
[149] Steinberg, Robert, Automorphisms of classical Lie algebras, Pacific J. Math., 11, 1119-1129, (1961) · Zbl 0104.02905
[150] Steinberg, Robert, Regular elements of semisimple algebraic groups, Inst. Hautes \'Etudes Sci. Publ. Math., 25, 49-80, (1965)
[151] Suslin, A. A., Algebraic \(K\)-theory and the norm residue homomorphism. Current problems in mathematics, Vol. 25, Itogi Nauki i Tekhniki, 115-207, (1984), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow
[152] TFV M. J. Tejwani, O. Ferreira, and O. E. Vilches, \emph Possible Ising transition in \(^4\)He monolayer adsorbed on Kr-plated graphite, Phys. Rev. Lett. <span class=”textbf”>4</span>4 (1980), no. 3, 152–155.
[153] Thakur, Maneesh, Automorphisms of Albert algebras and a conjecture of Tits and Weiss, Trans. Amer. Math. Soc., 365, 6, 3041-3068, (2013) · Zbl 1325.17014
[154] Tits, J., Alg\`ebres alternatives, alg\`ebres de Jordan et alg\`ebres de Lie exceptionnelles. I. Construction, Nederl. Akad. Wetensch. Proc. Ser. A 69 = Indag. Math., 28, 223-237, (1966) · Zbl 0139.03204
[155] Tits, J., Classification of algebraic semisimple groups. Algebraic Groups and Discontinuous Subgroups, Proc. Sympos. Pure Math., Boulder, Colo., 1965, 33-62, (1966), Amer. Math. Soc., Providence, R.I., 1966
[156] Ti:struct J. Tits, \emph Sur les constantes de structure et le th\'eor\`eme d’existence des alg\`ebres de Lie semi-simples, Inst. Hautes \'Etudes Sci. Publ. Math. (1966), no. 31, 21–58.
[157] Tits, Jacques, Sur les degr\'es des extensions de corps d\'eployant les groupes alg\'ebriques simples, C. R. Acad. Sci. Paris S\'er. I Math., 315, 11, 1131-1138, (1992) · Zbl 0823.20042
[158] Totaro, Burt, Splitting fields for \(E_8\)-torsors, Duke Math. J., 121, 3, 425-455, (2004) · Zbl 1048.11031
[159] Totaro, Burt, The torsion index of \(E_8\) and other groups, Duke Math. J., 129, 2, 219-248, (2005) · Zbl 1093.57011
[160] Totaro, Burt, The torsion index of the spin groups, Duke Math. J., 129, 2, 249-290, (2005) · Zbl 1094.57031
[161] van Leeuwen, Marc, Computing Kazhdan-Lusztig-Vogan polynomials for split \(E_8\), Nieuw Arch. Wiskd. (5), 9, 2, 113-116, (2008) · Zbl 1239.17005
[162] Vavilov, N. A., Do it yourself structure constants for Lie algebras of types \(E_l\), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI). J. Math. Sci. (N. Y.), 281 120, 4, 1513-1548, (2004) · Zbl 1062.17004
[163] Viazovska:E8 M. S. Viazovska, \emph The sphere packing problem in dimension \(8\), arxiv:1603.04246, March 2016.
[164] Vinberg:Weyl E. B. Vinberg, \emph The Weyl group of a graded Lie algebra, Math. USSR Izv. <span class=”textbf”>1</span>0 (1976), 463–495.
[165] Vinberg, \`E. B.; \`Ela\v svili, A. G., A classification of the three-vectors of nine-dimensional space, Trudy Sem. Vektor. Tenzor. Anal., 18, 197-233, (1978) · Zbl 0441.15010
[166] Vogan, David, The character table for \(E_8\), Notices Amer. Math. Soc., 54, 9, 1122-1134, (2007) · Zbl 1142.22009
[167] Walde R. E. Walde, \emph Composition algebras and exceptional Lie algebras, Ph.D. thesis, University of California-Berkeley, 1967.
[168] Waterhouse, William C., Introduction to affine group schemes, Graduate Texts in Mathematics 66, xi+164 pp., (1979), Springer-Verlag, New York-Berlin · Zbl 0442.14017
[169] Weiss, Richard M., Quadrangular algebras, Mathematical Notes 46, x+131 pp., (2006), Princeton University Press, Princeton, NJ · Zbl 1129.17001
[170] Yun, Zhiwei, Motives with exceptional Galois groups and the inverse Galois problem, Invent. Math., 196, 2, 267-337, (2014) · Zbl 1374.14013
[171] Zamolodchikov, A. B., Integrals of motion and \(S\)-matrix of the (scaled) \(T=T_c\) Ising model with magnetic field, Internat. J. Modern Phys. A, 4, 16, 4235-4248, (1989)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.