Abstract
A complete description of conjugate classes of maximal tori and corresponding 1-sections is obtained for the exceptional simple Lie p-algebra of characteristic 3 of the Frank series. In particular, it is proved that all maximal tori are two-dimensional and they are Cartan subalgebras.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. E. Brown, “Properties of 29-dimensional simple Lie algebra of characteristic 3,” Math. Ann., 261, 487–492 (1982).
G. E. Brown, “A class of simple Lie algebras of characteristic 3,” Proc. Am. Math. Soc., 107, 901–905 (1989).
G. E. Brown, “On the structure of some Lie algebras of Kuznetsov,” Michigan Math. J., 39, No. 7, 85–90 (1992).
Chan Nam Zung, “On two classes of simple Lie algebras over a field of characteristic 3,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 2, 12–15 (1992).
S. P. Demushkin, “Cartan subalgebras of the simple Lie p-algebras W n and S n ,” Sib. Math. J., 11, 233–245 (1970).
S. P. Demushkin, “Cartan subalgebras of simple nonclassical Lie p-algebras,” Math. USSR Izv., 6, 905–924 (1972).
Ju. B. Ermolaev, “On a family of simple Lie algebras over a field of characteristic 3,” in: V All-Union Symp. on Rings, Algebras and Modules. Abstracts [in Russian], Novosibirsk (1982), pp. 52–53.
M. S. Frank, “A new simple Lie algebra of characteristic three,” Proc. Am. Math. Soc., 38, 43–46 (1973).
A. I. Kostrikin, “A parametric family of simple Lie algebras,” Math. USSR Izv., 4, 751–764 (1970).
M. I. Kuznetsov, “Simple modular Lie algebras with a solvable maximal subalgebra,” Math. USSR Sb., 30, 68–76 (1976).
M. I. Kuznetsov, “Classification of simple graded Lie algebras with non-semisimple component L 0,” Math. USSR Sb., 180, 147–158 (1990).
M. I. Kuznetsov, “Truncated induced modules over transitive Lie algebras of characteristic p,” Math. USSR Izv., 34, 575–608 (1990).
M. I. Kuznetsov and O. A. Mulyar, “Automorphisms of Frank algebras,” Vestn. NNGU. Ser. Mat., No. 1 (3), 64–75 (2005).
M. I. Kuznetsov, O. A. Mulyar, and D. V. Reshetnikov, “Tori of Frank algebra,” Vestn. NNGU. Ser. Mat., No. 1 (4), 49–58 (2006).
M. I. Kuznetsov and V. A. Yakovlev, “Elementary proof of Demushkin’s theorem on tori in special Lie p-algebras of Cartan type,” Commun. Algebra, 25, 3979–3983 (1997).
M. I. Kuznetsov and V. A. Yakovlev, “An elementary proof of Demushkin’s theorem on tori in Hamiltonian Lie p-algebras,” Commun. Algebra, 27, 2779–2784 (1999).
A. A. Ladilova, “Filtered deformations of Frank algebras,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 8, 53–56 (2009).
O. A. Mulyar, “Maximal subalgebras of Frank algebras,” Vestn. NNGU. Ser. Mat., No. 1 (3), 109–113 (2005).
S. M. Skryabin, “New series of simple Lie algebras of characteristic 3,” Russ. Acad. Sci. Sb. Math., 76, 389–406 (1993).
S. M. Skryabin, “On the structure of the graded Lie algebra associated with a noncontractible filtration,” J. Algebra, 197, 178–230 (1997).
S. M. Skryabin, “Tori in the Melikian algebra,” J. Algebra, 243, 69–95 (2001).
H. Strade, Simple Lie Algebras over Fields of Positive Characteristic. I. Structure Theory, De Gruyter Exp. Math., Vol. 38, Walter de Gruyter, Berlin (2004).
B. Ju. Weisfeiler, “On subalgebras of simple Lie algebras of characteristic p > 0,” Trans. Am. Math. Soc., 286, 471–503 (1984).
B. Ju. Weisfeiler and V. G. Kac, “Exponentials in Lie algebras of characteristic p,” Math. USSR Izv., 5, 777–803 (1971).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 17, No. 1, pp. 143–154, 2011/12.
Rights and permissions
About this article
Cite this article
Kuznetsov, M.I., Mulyar, O.A. Maximal tori of the Frank algebra. J Math Sci 185, 440–447 (2012). https://doi.org/10.1007/s10958-012-0926-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-012-0926-9