Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Graded Lie Algebras of Maximal Class

Authors: A. Caranti, S. Mattarei and M. F. Newman
Journal: Trans. Amer. Math. Soc. 349 (1997), 4021-4051
MSC (1991): Primary 17B70, 17B65, 17B05, 17B30, 17B40, 20D15, 20F40
MathSciNet review: 1443190
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study graded Lie algebras of maximal class over a field $ \mathbf {F}$ of positive characteristic $p$. A. Shalev has constructed infinitely many pairwise non-isomorphic insoluble algebras of this kind, thus showing that these algebras are more complicated than might be suggested by considering only associated Lie algebras of p-groups of maximal class. Here we construct $| \mathbf {F}|^{\aleph _{0}}$ pairwise non-isomorphic such algebras, and $\max \{| \mathbf {F}|, \aleph _{0} \}$ soluble ones. Both numbers are shown to be best possible. We also exhibit classes of examples with a non-periodic structure. As in the case of groups, two-step centralizers play an important role.

References [Enhancements On Off] (What's this?)

  • [AF] A. A. Albert and M. S. Frank, Simple Lie algebras of characteristic 𝑝, Univ. e Politec. Torino. Rend. Sem. Mat. 14 (1954–55), 117–139. MR 0079222
  • [Bla] N. Blackburn, On a special class of 𝑝-groups, Acta Math. 100 (1958), 45–92. MR 0102558
  • [Blo] Richard E. Block, Determination of the differentiably simple rings with a minimal ideal., Ann. of Math. (2) 90 (1969), 433–459. MR 0251088
  • [CMNS] A. Caranti, S. Mattarei, M. F. Newman and C. M. Scoppola, Thin groups of prime-power order and thin Lie algebras, Quart. J. Math. Oxford Ser. (2) 47 (1996), 279-296. CMP 97:02
  • [CN$^{3}$] F. Celler, M. F. Newman, W. Nickel and A. C. Niemeyer, An algorithm for computing quotients of prime-power order for finitely presented groups and its implementation in GAP, Research Report 127, Australian National University, Canberra, 1993.
  • [HNO] George Havas, M.F. Newman and E.A. O'Brien, ANU p-Quotient Program (Version 1.2), written in C, available from by anonymous ftp in the directory pub/PQ, as a share library with GAP 3.4 and as part of Magma, 1995.
  • [Hu] B. Huppert, Endliche Gruppen. I, Die Grundlehren der Mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR 0224703
  • [Jac] Nathan Jacobson, Lie algebras, Dover Publications, Inc., New York, 1979. Republication of the 1962 original. MR 559927
  • [L] Édouard Lucas, Sur les congruences des nombres eulériens et des coefficients différentiels des fonctions trigonométriques, suivant un module premier, Bull. Soc. Math. France 6 (1878), 49-54.
  • [Ma] Avinoam Mann, Space groups and groups of prime power order. VII. Powerful 𝑝-groups and uncovered 𝑝-groups, Bull. London Math. Soc. 24 (1992), no. 3, 271–276. MR 1157263, 10.1112/blms/24.3.271
  • [M2] Susan McKay, On the structure of a special class of 𝑝-groups. II, Quart. J. Math. Oxford Ser. (2) 41 (1990), no. 164, 431–448. MR 1081105, 10.1093/qmath/41.4.431
  • [S+] Martin Schönert et al., GAP - Groups, Algorithms, and Programming, Release 3.4, Lehrstuhl D für Mathematik, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany, 1995.
  • [Sh1] Aner Shalev, The structure of finite 𝑝-groups: effective proof of the coclass conjectures, Invent. Math. 115 (1994), no. 2, 315–345. MR 1258908, 10.1007/BF01231763
  • [Sh2] Aner Shalev, Simple Lie algebras and Lie algebras of maximal class, Arch. Math. (Basel) 63 (1994), no. 4, 297–301. MR 1290602, 10.1007/BF01189564
  • [Sh3] B. Hartley, G. M. Seitz, A. V. Borovik, and R. M. Bryant (eds.), Finite and locally finite groups, NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 471, Kluwer Academic Publishers Group, Dordrecht, 1995. MR 1362803
  • [SZ] Aner Shalev and E. I. Zelmanov, Narrow Lie algebras I: a coclass theory and a characterization of the Witt algebra, J. Algebra 189 (1997), 294-330.
  • [Z] Hans Zassenhaus, Über Liesche Ringe mit Primzahlcharakteristik, Abh. Math. Sem. Univ. Hamburg 13 (1939), 1-100.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 17B70, 17B65, 17B05, 17B30, 17B40, 20D15, 20F40

Retrieve articles in all journals with MSC (1991): 17B70, 17B65, 17B05, 17B30, 17B40, 20D15, 20F40

Additional Information

A. Caranti
Affiliation: Dipartimento di Matematica, Università degli Studi di Trento, via Sommarive 14, I-38050 Povo (Trento), Italy

S. Mattarei
Affiliation: Dipartimento di Matematica ed Applicazioni, Università degli Studi di Padova, via Belzoni 7, I-35131 Padova, Italy

M. F. Newman
Affiliation: School of Mathematical Sciences, Australian National University, Canberra, ACT 0200, Australia

Keywords: Graded Lie algebras of maximal class, {\em p}\/-groups of maximal class, pro-{\em p}\/-groups of finite coclass, nilpotent Lie algebras
Received by editor(s): March 1, 1996
Additional Notes: The first two authors are members of CNR–GNSAGA, Italy, and acknowledge support of MURST, Italy. The third author acknowledges support from CNR-GNSAGA, Italy, and the University of Trento, Italy.
Article copyright: © Copyright 1997 American Mathematical Society