×

Graded Lie algebras of maximal class. II. (English) Zbl 0971.17015

This paper refines results of an earlier paper [Trans. Am. Math. Soc. 349, 4021-4051 (1997; Zbl 0895.17037] by the authors and S. Mattarei. They describe the isomorphism classes of infinite-dimensional Lie algebras \(L = \bigoplus_{i=1}^\infty L_i\) over a field of characteristic \(p\) such that \(\dim L_1 = 2\) and \([L_i, L_1] = L_{i+1}\) for \(i \geqq 1\). The construction of such algebras makes use of Albert-Frank-Shalev algebras and inflation steps (concepts introduced in the earlier paper). A computer has been used in many of the calculations.
Reviewer: G.Brown (Boulder)

MSC:

17B70 Graded Lie (super)algebras
17B65 Infinite-dimensional Lie (super)algebras
17-04 Software, source code, etc. for problems pertaining to nonassociative rings and algebras
17B50 Modular Lie (super)algebras

Citations:

Zbl 0895.17037
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Albert, A.A.; Frank, M.S., Simple Lie algebras of characteristic p, Rend. sem. univ. politec. Torino, 14, 117-139, (1954-1955)
[2] Benkart, G.; Kostrikin, A.I.; Kuznetzov, M.I., Finite-dimensional simple Lie algebras with a nonsingular derivation, J. algebra, 171, 894-916, (1995) · Zbl 0815.17018
[3] Block, R.E., Determination of the differentiably simple rings with a minimal ideal, Ann. math. (2), 90, 433-459, (1969) · Zbl 0216.07303
[4] C. Carrara, (Finite) Presentations of Loop Algebras of Albert-Frank Lie Algebras, Ph.D. thesis, Trento, 1998. · Zbl 1177.17018
[5] C. Carrara, (Finite) presentations of loop algebras of Albert-Frank Lie algebras, to appear in, Ball. Un. Mat. Ital. A. · Zbl 1177.17018
[6] Caranti, A.; Jurman, G., Quotients of maximal class of thin Lie algebras: the odd characteristic case, Comm. algebra, 27, 5741-5748, (1999) · Zbl 0940.17013
[7] Caranti, A.; Mattarei, S.; Newman, M.F., Graded Lie algebras of maximal class, Trans. amer. math. soc., 349, 4021-4051, (1997) · Zbl 0895.17031
[8] Goze, M.; Khakimdjanov, Y., Nilpotent Lie algebras, (1996), Kluwer Academic Dordrecht
[9] G. Havas, M. F. Newman, and, E. A. O’Brien, ANU p-quotient program (version 1.4), written in C, available as a share library with GAP and as part of Magma, or from, http://wwwmaths.anu.edu.au/services/ftp.html, School of Mathematical Sciences, Australian National University, Canberra, 1997.
[10] G. Jurman, On Graded Lie Algebras in Characteristic Two, Ph.D. thesis, Trento, 1998. · Zbl 1053.17515
[11] G. Jurman, Graded Lie algebra of maximal class, III, in preparation. · Zbl 1058.17020
[12] Kostrikin, A.I.; Kuznetsov, M.I., Finite-dimensional Lie algebras with a nonsingular derivation, Algebra and analysis, kazan, 1994, (1996), de Gruyter Berlin, p. 81-90 · Zbl 0890.17024
[13] Knuth, D.E.; Wilf, H.S., The power of a prime that divides a generalized binomial coefficient, J. reine angew. math., 396, 212-219, (1989) · Zbl 0657.10008
[14] Leedham-Green, C.R., The structure of finite p-groups, J. London math. soc. (2), 50, 49-67, (1994) · Zbl 0822.20018
[15] Lucas, È., Sur LES congruences des nombres eulériens et des coefficients différentiels des fonctions trigonométriques, suivant un module premier, Bull. soc. math. France, 6, 49-54, (1878) · JFM 10.0139.04
[16] Shalev, A., Simple Lie algebras and Lie algebras of maximal class, Arch. math. (basel), 63, 297-301, (1994) · Zbl 0803.17006
[17] Shalev, A., The structure of finite p-groups: effective proof of the coclass conjectures, Invent. math., 115, 315-345, (1994) · Zbl 0795.20009
[18] Shalev, A.; Zelmanov, E.I., Pro-p groups of finite coclass, Math. proc. Cambridge philos. soc., 111, 417-421, (1992) · Zbl 0813.20030
[19] Shalev, A.; Zelmanov, E.I., Narrow Lie algebras: A coclass theory and a characterization of the Witt algebra, J. algebra, 189, 294-331, (1997) · Zbl 0886.17008
[20] Vergne, M., Réductibilité de la variété des algèbres de Lie nilpotentes, C. R. acad. sci. Paris Sér. A-B, 263, A4-A6, (1966) · Zbl 0145.25902
[21] Vergne, M., Cohomologie des algèbres de Lie nilpotentes. application à l’étude de la variété des algèbres de Lie nilpotentes, C. R. acad. sci. Paris Sér. A-B, 267, A867-A870, (1968) · Zbl 0244.17010
[22] Zassenhaus, H., Über liesche ringe mit primzahlcharakteristik, Abh. math. sem. univ. Hamburg, 13, 1-100, (1940) · JFM 65.0090.01
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.