Weyl Images of Kantor Pairs

Allison, Bruce ; Faulkner, John ; Smirnov, Oleg

Canadian journal of mathematics, Tome 69 (2017) no. 4, pp. 721-766

Résumé

Kantor pairs arise naturally in the study of 5-graded Lie algebras. In this article, we introduce and study Kantor pairs with short Peirce gradings and relate themto Lie algebras graded by the root system of type $\text{B}{{\text{C}}_{2}}$ . This relationship allows us to define so-called Weyl images of short Peirce graded Kantor pairs. We use Weyl images to construct new examples of Kantor pairs, including a class of infinite dimensional central simple Kantor pairs over a field of characteristic $\ne$ 2 or 3, as well as a family of forms of a split Kantor pair of type ${{\text{E}}_{6}}$ .

Détail

DOI : 10.4153/CJM-2016-047-1

Mots-clés : 17B60, 17B70, 17C99, 17B65, Kantor pair, graded Lie algebra, Jordan pair

Bibliographie

[A] [A] Allison, B.,A class of nonassociative algebras with involution containing the class of Jordan algebras. Math. Ann. 237(1978), 133–156. http://dx.doi.Org/10.1007/BF01351677 Google Scholar

[ABG] [ABG] Allison, B., Benkart, G., and Gao, Y., Lie algebras graded by the root systems BC, r ≥ 2. Mem. Amer. Math. Soc. 158(2002), no. 751. Google Scholar

[AF1] [AF1] Allison, B. and Faulkner, J., Elementary groups and invertibility for Kantor pairs. Comm. Algebra 27(1999), 519–556. Google Scholar | DOI

[AF2] [AF2] Allison, B. and Faulkner, J., Dynkin diagrams and short Peirce gradings of Kantor pairs. arxiv:1703.04055v1 Google Scholar

[ADG] [ADG] Aranda-Orna, D., Draper, C., and Guido, V., Weyl groups of some fine gradings on e. J. Algebra 417(2014), 333–390. http://dx.doi.Org/10.1016/j.jalgebra.2O14.07.001 Google Scholar

[At] [At] Atsuyama, K., On the algebraic structures of graded Lie algebras of second order. Kodai Math. J. 5(1982), 225–229. Google Scholar | DOI

[BN] [BN] Benkart, G. and Neher, E., The centroid of extended affme and root graded Lie algebras. J. Pure Appl. Algebra 205(2006), 117–145. http://dx.doi.Org/10.101 6/j.jpaa.2005.06.007 Google Scholar

[BS] [BS] Benkart, G. and Smirnov, O., Lie algebras graded by the root system BC. J. Lie Theory 13(2003), 91–132. Google Scholar

[BDDRE] [BDDRE] Borsten, L., Dahanayake, D., Duff, M. J., Rubens, W., and Ebrahim, H., Freudenthal triple classification of three-qubit entanglement, Phys. Rev. A 80(2009), 032326. Google Scholar | DOI

[Bl] [Bl] Bourbaki, N., Elements of mathematics. Algebra, Part I. Translated from the French. Hermann, Paris, 1974. Google Scholar

[B2] [B2] Bourbaki, N., Commutative algebra. Chapters 1-7. Translated from the French. Springer-Verlag, Berlin, 1998. Google Scholar

[B3] [B3] Bourbaki, N., Elements of Mathematics. Lie groups and Lie algebras. Chapters 4-6. Translated from the French. Springer-Verlag, Berlin, 2002. Google Scholar | DOI

[C] [C] Cartan, É., Sur la structure des groups de transformations finis et continues. Thése, Paris, Nony 1894. In: Œuvres complétes, Partie I, Groupes de Lie. Gauthier-Villars, Paris, 1952, pp. 137–287. Google Scholar

[D] [D] Djokovic, D., Classification ofL-graded real semisimple Lie algebras. J. Algebra 76(1982), 367–382. http://dx.doi.Org/10.1016/0021-8693(82)90220-4 Google Scholar

[E] [E] Elduque, A., The magie square and symmetric compositions. II Rev. Mat. Iberoam. 23(2007), 57–84. Google Scholar | DOI

[EK] [EK] Elduque, A. and Kochetov, M., Gradings on simple Lie algebras. Mathematical Surveys and Monographs, 189. American Mathematical Society, Providence, RI, 2013. Google Scholar

[EO] [EO] Elduque, A. and Okubo, S., Special Freudenthal-Kantor triple systems and Lie algebras with dicyclic symmetry. Proc. Roy. Soc. Edinburgh Sect. A 141(2011), 1225–1262. http://dx.doi.Org/10.1017/S0308210510000569 Google Scholar

[EKO] [EKO] Elduque, A., Kamiya, N. and Okubo, S., Left unital Kantor triple systems and structurable algebras. Linear Multilinear Algebra 62(2014), no. 10,1293–1313. Google Scholar | DOI

[F] [F] Faulkner, J., Some forms of exceptional Lie algebras. Comm. Algebra, 42(2014), 4854–4873. Google Scholar | DOI

[FF] [FF] Faulkner, J. and Ferrar, J., On the structure of symplectic ternary algebras. Nederl. Akad. Wetensch. Proc. Ser. A 75=Indag. Math. 34(1972), 247–256. http://dx.doi.Org/10.1016/1385-7258(72)90062-5 Google Scholar

[GLN] [GLN] Garcia, E., Gomez Lozano, M., and Neher, E., Nondegeneracy for Lie triple systems and Kantor pairs. Canad. Math. Bull. 54 (2011), 442–455. http://dx.doi.Org/10.4153/CMB-2O11-023-9 Google Scholar

[GN] [GN] Garcia, E. and Neher, E., Tits-Kantor-Koecher superalgebras of Jordan superpairs covered by grids. Comm. Algebra 31(2003), 3335–3375. Google Scholar | DOI

[G] [G] Günaydin, M., Extended superconformai symmetry, Freudenthal triple systems and gauged WZW models. Springer Lecture Notes in Physics 447(1995), 54–69. http://dx.doi.Org/10.1007/3-540-59163-X_255 Google Scholar

[GOV] [GOV] V. V.Gorbatsevich, , Onischhik, A. L., and Vinberg, E. B., Structure of Lie groups and Lie algebras. In: Lie groups and Lie algebras III, Encyclopaedia Math. Sci., 41. Springer-Verlag, Berlin, 1994. Google Scholar | DOI

[H] [H] Humphreys, J. E., Introduction to Lie algebras and representation theory. Graduate Texts in Mathematics, 9. Springer-Verlag, New York, 1972. Google Scholar

[Jl] [Jl] Jacobson, N., General representation theory of Jordan algebras. Trans. Amer. Math. Soc. 70(1951), 509–530. Google Scholar | DOI

[J2] [J2] Jacobson, N., Structure of rings. American Mathematical Society, Colloquium Publications, 37. American Mathematical Society, Providence. RI, 1956. Google Scholar

[J3] [J3] Jacobson, N., Lie algebras. Interscience Publishers, New York, 1962. Google Scholar

[Kl] [Kl] Kantor, I. L., Certain generalizations of Jordan algebras (Russian). Trudy Sem. Vektor. Tenzor. Anal. 16(1972), 407–499. Google Scholar

[K2] [K2] Kantor, I. L., Models of exceptional Lie algebras. Soviet Math. Dokl. 14(1973), 254–258. Google Scholar

[KS] [KS] Kantor, I.L. and Skopec, I. M., Freudenthal trilinear operations (Russian). Trudy. Sem. Vektor. Tenzor. Anal. 18(1978), 250–263 Google Scholar

[L-G] [L-G] Leedham-Green, C. R., The class group of Dedekind domains. Trans. Amer. Math. Soc. 163(1972), 493–500. Google Scholar | DOI

[L] [L] Loos, O., Jordan pairs. Lecture Notes in Mathematics, 460. Springer-Verlag, Berlin, 1975. Google Scholar

[LN] [LN] Loos, O. and Neher, E., Steinberg groups for Jordan pairs. C. R. Math. Acad. Sci. Paris 348(2010), 839–842. http://dx.doi.Org/10.1016/j.crma.2O10.07.012 Google Scholar

[LB] [LB] Fernandez Lopez, A. and Tocon Barroso, M., Strongly prime Jordan pairs with nonzero socle. ManuscriptaMath. 111(2003), 321–340. Google Scholar | DOI

[MQSTZ] [MQSTZ] Marrani, A., Qiu, C., Shih, S. D., Tagliaferro, A., and Zumino, B., Freudenthal gauge theory. J. High Energy Phys. (2013). Google Scholar

[Mc] [Mc] McCrimmon, K, A taste of Jordan algebras. Springer-Verlag, New York, 2004. Google Scholar

[M] [M] Meyberg, K., Eine Theorie der Freudenthalschen Tripelsysteme. I, II. Nederl. Akad. Wetensch. Proc. Ser. A 71=Indag. Math. 30(1968), 162-174, 175–190. Google Scholar | DOI

[Na] [Na] Narkiewicz, W., Elementary and analytic theory of algebraic numbers. Third edition. Springer-Verlag, Berlin, 2004. Google Scholar | DOI

[Nl] [Nl] Neher, E., Involutive gradings in Jordan structures, Comm. Algebra 9(1981), 575–599. http://dx.doi.Org/10.1080/0092 7878108822603 Google Scholar

[N2] [N2] Neher, E., Lie algebras graded by 3-graded root systems and Jordan pairs covered by grids. J. Algebra 140(1991), 284–329. Google Scholar

[Sel] [Sel] Seligman, G. B., Modular Lie algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, 40.Springer-Verlag,New York 1967. Google Scholar

[Ser] [Ser] Serre, J.P., Galois cohomology. Springer-Verlag, Berlin 2002. Google Scholar

[Sm] [Sm] Smirnov, O., Simple and semisimple structurable algebras. Algebra and Logic 29(1990), 377–394. Google Scholar

[St] [St] Steinberg, R., Automorphisms of classical Lie algebras. Pacific J. Math. 11(1961), 1119–1129. http://dx.doi.Org/10.2140/pjm.1961.11.1119 Google Scholar

[YA] [YA] Yamaguti, K. and Asano, H., On the Freudenthal*s construction of exceptional Lie algebras. Proc. Japan Acad. 51(1975), 253–258. http://dx.doi.Org/10.3792/pja/1195518629 Google Scholar

[YO] [YO] Yamaguti, K. and Ono, A., On representations of Freudenthal-Kantor triple systems U(∊,δ). Bull. Fac. School Ed. Hiroshima Univ. Part II 7(1984), 43–51. Google Scholar

[Zl] [Zl] Zelmanov, E., Primary Jordan triple systems II. Sibirskii Math. Zh. 25(1984), 50–61. Google Scholar | DOI

[Z2] [Z2] Zelmanov, E., Lie algebras with a finite grading, Math. USSR-Sb. 52(1985), 347–385. Google Scholar