Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-14T09:29:42.196Z Has data issue: false hasContentIssue false
  • English
  • Français

On Polynomial Invariants of Exceptional Simple Algebraic Groups

Published online by Cambridge University Press:  20 November 2018

A. Elduque  and
A. V. Iltyakov
A. Elduque
Affiliation:
Departamento de Matemáticas y Computación, Universidad de la Rioja, 26004 Logroño, Spain
A. V. Iltyakov
Affiliation:
Institute of Mathematics, Siberian Branch of Russian Academy of Sciences Novosibirsk, 630090, Russia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study polynomial invariants of systems of vectors with respect to exceptional simple algebraic groups in their minimal linear representations. For each type we prove that the algebra of invariants is integral over the subalgebra of trace polynomials for a suitable algebraic system $\left( cf.\,[27],\,[28],\,[13] \right)$.

Keywords

15A7217C20
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 51 , Issue 3 , 01 June 1999 , pp. 506 - 522
Copyright
Copyright © Canadian Mathematical Society 1999

References

[1] Allison, B. N., Models of isotropic simple Lie algebras. Comm. Algebra 7 (1979), 1835–1875.Google Scholar
[2] Allison, B. N. and Hein, W., Isotopies of some nonassociative algebras with involution. J. Algebra 69 (1981), 120–142.Google Scholar
[3] Allison, B. N. and Faulkner, J. R., A Cayley-Dickson process for a class of structurable algebras. Trans. Amer. Math. Soc. (1) 283 (1984), 185–210.Google Scholar
[4] Allison, B. N. and Schafer, R. D., Trace forms for structurable algebras. J. Algebra 121 (1989), 68–69.Google Scholar
[5] Brown, R. B., Groups of type E7. J. Reine Angew.Math. 236 (1969), 79–102.Google Scholar
[6] Chevalley, C., Théorie des Groupes de Lie. Tome 2. Paris, Hermann, 1951.Google Scholar
[7] Freudenthal, H., Sur le groupe exceptionnel E7. Nederl. Akad.Wetencsch. Proc. Ser. A 57 (1954), 218–230.Google Scholar
[8] Hilbert, D., Ü ber die vollen Invarianten-systeme. Math. Ann. 42 (1893), 313–373.Google Scholar
[9] Howe, R. E., The first fundamental theorem on invariant theory and spherical subgroups. Proc. Sym. Pure Math. (1) 56 (1994), 333–346.Google Scholar
[10] Humphreys, J. E., Introduction to Lie algebras and representation theory. Graduate Texts inMath. 9, Springer Verlag, New York, 1972.Google Scholar
[11] Humphreys, J. E., Linear algebraic groups. Graduate Texts in Math. 21, Springer Verlag, New York, 1975.Google Scholar
[12] Iltyakov, A. V., Trace polynomials and Invariant Theory. Geom. Dedicata 58 (1995), 327–333.Google Scholar
[13] Iltyakov, A. V., On invariants of the group of automorphisms of Albert algebras. Comm. Algebra (11) 23 (1995), 4047–4060.Google Scholar
[14] Jacobson, N., Some groups of transformation defined by Jordan algebras III, Groups of type E6. J. ReineAngew. Math. 207 (1961), 61–85.Google Scholar
[15] Jacobson, N., Structure and representation of Jordan algebras. Amer. Math. Soc. Colloq. Publ. 39, Providence, 1968.Google Scholar
[16] Jacobson, N., Lie algebras. 1962.Google Scholar
[17] Loos, Ottmar, Jordan Pairs. Springer-Verlag, 460, 1975.Google Scholar
[18] Polikarpov, S. V. and Shestakov, I. P., Nonassociative affine algebras. Algebra i Logika 29 (1990), 709–723.Google Scholar
[19] Popov, V. L. and Vinberg, E. B., Invariant Theory. Encyclopaedia of Mathematical Science: Algebraic geometry IV. 55, Springer Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1993.Google Scholar
[20] Popov, V. L., Analogue of M.Artin's conjecture on invariants for non-associative algebras. Lie groups and Lie algebras: E. B. Dynkin Seminar (Eds. Gindikin, S. G., Vinberg, E. B.), Amer. Math. Soc., Providence, R.I., 1995, 121–143.Google Scholar
[21] Rowen, L. H., Polynomial Identities in Ring Theory. Academic Press, New Jersey, 1980.Google Scholar
[22] Richardson, R. W., Conjugacy classes of n-tuples in Lie algebras and algebraic groups. DukeMath. J. 57 (1988), 1–33.Google Scholar
[23] Schafer, R. D., Introduction to non-associative algebras. Academic Press, New York, 1966.Google Scholar
[24] Schafer, R. D., Structurable bimodules. J. Algebra. , 96 (1985), 479–494.Google Scholar
[25] Schafer, R. D., Nilpotence of the radical of a structurable algebra. J. Algebra 99 (1986), 355–358.Google Scholar
[26] Schafer, R. D., Structurable algebras. Proc. Inter. Conf. Algebra, Part 2, (Novosibirsk, 1989), 135-148; Contemp. Math. 131 Part 2, Amer.Math. Soc., Rhode Island, 1992.Google Scholar
[27] Schwarz, G. W., Invariant Theory of G2. Bull. Amer. Math. Soc. 9 (1988), 335–338.Google Scholar
[28] Schwarz, G. W., Invariant theory of G2 an. Spin7. Comment.Math. Helv.. 63 (1988), 624–663.Google Scholar
[29] Weyl, H., The Classical Groups. Princeton Univ. Press, Princeton, 1946.Google Scholar