Gradings on the Albert algebra and on $\mathfrak{f}_4$
Cristina
Draper
and
Cándido
Martín
Source: Rev. Mat. Iberoamericana Volume 25, Number 3
(2009), 841-908.
Abstract
We study group gradings on the Albert algebra and on the exceptional simple Lie algebra $\frak{f}_4$ over algebraically closed fields of characteristic zero. The immediate precedent of this work is [Draper, C. and Martin, C.: Gradings on $\frak{g}_2$. Linear Algebra Appl. 418 (2006), no. 1, 85-111] where we described (up to equivalence) all the gradings on the exceptional simple Lie algebra $\frak{g}_2$. In the cases of the Albert algebra and $\frak{f}_4$, we look for the nontoral gradings finding that there are only eight nontoral nonequivalent gradings on the Albert algebra (three of them being fine) and nine on $\frak{f}_4$ (also three of them fine).
First Page:
Show
Hide
Full-text: Access denied (no subscription
detected)
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.rmi/1257258097
Zentralblatt MATH identifier: 05663366
Mathematical Reviews number (MathSciNet): MR2590049
References
Adams, J. F.: Lectures on Exceptional Lie Groups. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1996.
Mathematical Reviews (MathSciNet): MR1428422
Zentralblatt MATH: 0866.22008
Andersen, K. K. S.: Some computations for the exceptional groups relevant to the classification of $p$-compact groups. Ph.D. Thesis. Institute for Mathematical Sciences. University of Copenhagen, 2001.
Andersen, K. K. S., Grodal, J., Mœller, J. M. and Viruel, A.: The classification of $p$-compact groups for $p$ odd. Ann. of Math. (2) 167 (2008), no. 1, 95-210.
Mathematical Reviews (MathSciNet): MR2373153
Zentralblatt MATH: 1149.55011
Digital Object Identifier: doi:10.4007/annals.2008.167.95
Alekseevski, A. V.: Finite commutative Jordan subgroups of simple complex Lie groups. Functional Analysis Appl. 8 (1974), 277-279.
Mathematical Reviews (MathSciNet): MR379748
Allison, B., Berman, S., Faulkner, J. and Pianzola, A.: Realization of graded-simple algebras as loop algebras. Forum Math. 20 (2008), no. 3, 395-432.
Mathematical Reviews (MathSciNet): MR2418198
Zentralblatt MATH: 1157.17009
Digital Object Identifier: doi:10.1515/FORUM.2008.020
Bahturin, Y. A. and Shestakov, I. P.: Gradings of simple Jordan algebras and their relation to the gradings of simple associative algebras. Comm. Algebra 29 (2001), no. 9, 4095-4102.
Mathematical Reviews (MathSciNet): MR1857029
Zentralblatt MATH: 1054.17025
Digital Object Identifier: doi:10.1081/AGB-100105990
Bahturin, Y. A., Shestakov, I. P. and Zaicev, M. V.: Gradings on simple Jordan and Lie algebras. J. Algebra 283 (2005), no. 2, 849-868.
Mathematical Reviews (MathSciNet): MR2111225
Zentralblatt MATH: 1066.17018
Digital Object Identifier: doi:10.1016/j.jalgebra.2004.10.007
Bahturin, Y. A. and Tvalavadze, M.: Group gradings on $G_2$. Comm. Algebra 37 (2009), no. 3, 885-893.
Mathematical Reviews (MathSciNet): MR2503183
Zentralblatt MATH: 05542216
Digital Object Identifier: doi:10.1080/00927870802278529
Bahturin, Y. A. and Zaicev, M. V.: Group gradings on matrix algebras. Canad. Math. Bull. 45 (2002), no. 4, 499-508.
Mathematical Reviews (MathSciNet): MR1941224
Borel, A.: Linear algebraic Groups (second enlarged edition). Graduate Texts in Mathematics 126. Springer-Verlag, New-York, 1991.
Mathematical Reviews (MathSciNet): MR1102012
Conway, J. H. and Smith, D. A.: On quaternions and octonions: their geometry, arithmetic, and symmetry. A. K. Peters, Natick, MA, 2003.
Mathematical Reviews (MathSciNet): MR1957212
Zentralblatt MATH: 1098.17001
Draper, C.: Models of the Lie algebra $F_4$. Linear Algebra Appl. 428 (2008), no. 11-12, 2813-2839.
Mathematical Reviews (MathSciNet): MR2416591
Zentralblatt MATH: 1140.17006
Digital Object Identifier: doi:10.1016/j.laa.2008.01.009
Draper, C. and Martín, C.: Gradings on $\frakg_2$. Linear Algebra Appl. 418 (2006), no. 1, 85-111.
Draper, C., Martín, C. and Viruel, A.: Fine gradings on the exceptional Lie algebra $\frakd_4$. To appear in Forum Math.
Elduque, A.: Gradings on octonions. J. Algebra 207 (1998), no. 1, 342-354.
Mathematical Reviews (MathSciNet): MR1643126
Zentralblatt MATH: 0915.17022
Digital Object Identifier: doi:10.1006/jabr.1998.7474
Elduque, A.: A Lie grading which is not a semigroup grading. Linear Algebra Appl. 418 (2006), no. 1, 312-314.
Mathematical Reviews (MathSciNet): MR2257598
Zentralblatt MATH: 1146.17028
Digital Object Identifier: doi:10.1016/j.laa.2006.02.006
Fulton, W. and Harris, J.: Representation Theory. A first course. Graduate Texts in Mathematics 129. Springer-Verlag, New York, 1991.
Mathematical Reviews (MathSciNet): MR1153249
Carter, R. W.: Finite groups of Lie type. Conjugacy classes and complex characters. Pure and Applied Mathematics. A Wiley-Interscience Publication. John Wiley & Sons., New York, 1985.
Mathematical Reviews (MathSciNet): MR794307
Griess, R. L.: Elementary abelian $p$-subgroups of algebraic groups. Geom. Dedicata 39 (1991), no. 3, 253-305.
Mathematical Reviews (MathSciNet): MR1123145
Zentralblatt MATH: 0733.20023
Digital Object Identifier: doi:10.1007/BF00150757
Havlïček, M., Patera, J., Pelantová, E. and Tolar, J.: Automorphisms of fine gradings of $\hboxsl(n,\C)$ associated with the generalized Pauli matrices. J. Math. Phys. 43 (2002), no. 2, 1083-1094.
Mathematical Reviews (MathSciNet): MR1878988
Zentralblatt MATH: 1059.17018
Digital Object Identifier: doi:10.1063/1.1430046
Havlïček, M., Patera, J. and Pelantová, E.: On Lie gradings II. Linear Algebra Appl. 277 (1998), no. 1-3, 97-125.
Mathematical Reviews (MathSciNet): MR1624532
Zentralblatt MATH: 0939.17020
Digital Object Identifier: doi:10.1016/S0024-3795(97)10039-8
Havlïček, M., Patera, J. and Pelantová, E.: On Lie gradings III. Gradings of the real forms of classical Lie algebras. Linear Algebra Appl. 314 (2000), no. 1-3, 1-47.
Mathematical Reviews (MathSciNet): MR1769013
Zentralblatt MATH: 1017.17027
Digital Object Identifier: doi:10.1016/S0024-3795(00)00099-9
Havlïček, M., Patera, J., Pelantová, E. and Tolar, J.: On fine gradings and their symmetries. Czechoslovak J. Phys. 51 (2001), no. 4, 383-391.
Mathematical Reviews (MathSciNet): MR1848647
Digital Object Identifier: doi:10.1023/A:1017501925328
Hesselink, W. H.: Special and pure gradings of Lie algebras. Math. Z. 179 (1982), no. 1, 135-149.
Mathematical Reviews (MathSciNet): MR643052
Zentralblatt MATH: 0455.17007
Digital Object Identifier: doi:10.1007/BF01173920
Humphreys, J. E.: Introduction to Lie algebras and representation theory. Graduate Texts in Mathematics 9. Springer-Verlag, New York-Berlin, 1972.
Mathematical Reviews (MathSciNet): MR499562
Humphreys, J. E.: Linear Algebraic Groups. Graduate Texts in Mathematics 21. Springer-Verlag, New York-Heidelberg, 1975.
Mathematical Reviews (MathSciNet): MR396773
Jacobson, N.: Lie algebras. Dover Publications, New York, 1979.
Mathematical Reviews (MathSciNet): MR559927
Jacobson, N.: Structure and representations of Jordan algebras. Amer. Math. Soc. Colloq. Publ. 39. Amer. Math. Soc., Providence, RI, 1968.
Mathematical Reviews (MathSciNet): MR251099
Zentralblatt MATH: 0218.17010
Kac, V. G.: Infinite dimensional Lie algebras. Cambridge University Press, Cambridge, 1990.
Mathematical Reviews (MathSciNet): MR1104219
Knus, M. A., Merkurjev, A., Rost, M. and Tignol, J. P.: The book of involutions. Amer. Math. Soc. Colloq. Publ. 44. Amer. Math. Soc., Providence, RI, 1998.
Mathematical Reviews (MathSciNet): MR1632779
Zentralblatt MATH: 0955.16001
Milne, J. S.: Algebraic geometry. Taiaroa Publishing Erehwon, Version 5.00, 2005 (available in http://www.jmilne.org/math/).
De Montigny, M. and Patera, J.: Discrete and continuous graded contractions of Lie algebras and superalgebras. J. Phys. A 24 (1991), no. 3, 525-547.
Mathematical Reviews (MathSciNet): MR1098200
Zentralblatt MATH: 0731.17001
Digital Object Identifier: doi:10.1088/0305-4470/24/3/012
Neumann, F.: A theorem of Tits, normalizers of maximal tori and fibrewise Bousfield-Kan completions. Publ. Res. Inst. Math. Sci. 35 (1999), no. 5, 711-723.
Mathematical Reviews (MathSciNet): MR1739304
Digital Object Identifier: doi:10.2977/prims/1195143419
Onishchnik, A. L. and Vinberg, E. B.: Lie groups and Lie algebras III Encyclopaedia of Mathematical Sciences 41. Springer-Verlag, Berlin, 1991.
Mathematical Reviews (MathSciNet): MR1349140
Patera, J., Pelantová, E. and Svobodová, M.: The eight fine gradings of $\hboxsl(4,\C)$ and $\hboxo(6,\C)$. J. Math. Phys. 43 (2002), no. 12, 6353-6378.
Mathematical Reviews (MathSciNet): MR1940451
Zentralblatt MATH: 1060.17016
Digital Object Identifier: doi:10.1063/1.1508434
Patera, J., Pelantová, E. and Svobodová, M.: Fine gradings of $\hboxo(4,\C)$. J. Math. Phys. 45 (2004), no. 6, 2188-2198.
Mathematical Reviews (MathSciNet): MR2059687
Digital Object Identifier: doi:10.1063/1.1737054
Patera, J. and Zassenhaus, H.: On Lie gradings. I. Linear Algebra Appl. 112 (1989), 87-159.
Mathematical Reviews (MathSciNet): MR976333
Zentralblatt MATH: 0675.17001
Digital Object Identifier: doi:10.1016/0024-3795(89)90591-0
Platonov, V. P.: The theory of algebraic linear groups and periodic groups. American Society Translations (2) 69 (1968), 61-110.
Schafer, R. D.: An introduction to nonassociative algebras. Pure and Applied Mathematics 22. Academic Press, New York-London, 1966.
Mathematical Reviews (MathSciNet): MR210757
Zentralblatt MATH: 0145.25601
Vavpetič, A. and Viruel, A.: On the homotopy type of the classifying space of the exceptional Lie group $F_4$. Manuscripta Math. 107 (2002), no. 4, 521-540.
Mathematical Reviews (MathSciNet): MR1906774
Zentralblatt MATH: 1007.55011
Digital Object Identifier: doi:10.1007/s002290200250
Vinberg, È. B.: Classification of homogeneous nilpotent elements of a semisimple graded Lie algebra. Trudy Sem. Vektor. Tenzor. Anal. (1979), no. 19, 155-177.
Mathematical Reviews (MathSciNet): MR549013
Yoshii, Y.: Coordinate algebras of extended affine Lie algebras of type $A_1$. J. Algebra 234 (2000), no. 1, 128-168.
Mathematical Reviews (MathSciNet): MR1799481
Zentralblatt MATH: 0974.17026
Digital Object Identifier: doi:10.1006/jabr.2000.8450
2012 © Departamento de Matemáticas, Universidad Autónoma de Madrid
Revista Matemática Iberoamericana
