IMPORTANT NOTICE

The AMS website will be down for maintenance on May 23 between 6:00am - 8:00am EDT. For questions please contact AMS Customer Service at cust-serv@ams.org or (800) 321-4267 (U.S. & Canada), (401) 455-4000 (Worldwide).

 

Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-97-02005-9
MathSciNet review: 1443190
Full-text PDF

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, Rend. Sem. Mat. Univ. Politec. Torino 14 (1954/5), 117-139.MR 18:52a
  • [Bla] N. Blackburn, On a special class of p-groups, Acta Math. 100 (1958), 45-92.MR 21:1349
  • [Blo] Richard E. Block, Determination of the differentiably simple rings with a minimal ideal, Ann. of Math. 90 (1969), 433-459.MR 40:4319
  • [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 maths.anu.edu.au 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, Springer, Berlin, 1967.MR 37:302
  • [Jac] Nathan Jacobson, Lie algebras, Dover, New York, 1979.MR 80k:17001
  • [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), 271-276.MR 93c:20041
  • [M2] S. McKay, On the structure of a special class of p-groups II, Quart. J. Math. Oxford Ser. (2) 41 (1990), 431-448.MR 91k:20027
  • [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: constructive proof of the coclass conjectures, Invent. Math. 115 (1994), 315-345.MR 95j:20022b
  • [Sh2] Aner Shalev, Simple Lie algebras and Lie algebras of maximal class, Arch. Math. (Basel) 63 (1994), 297-301. MR 95j:17025
  • [Sh3] Aner Shalev, Finite p-groups, Proceedings of the NATO Advanced Study Institute on Finite and Locally Finite Groups, Istanbul, Turkey, 1994, NATO ASI Series, vol. 471, Kluwer Academic Publishers, Dordrecht/Boston/London, 1995, pp. 401-450.MR 96f:20002
  • [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
Email: caranti@science.unitn.it

S. Mattarei
Affiliation: Dipartimento di Matematica ed Applicazioni, Università degli Studi di Padova, via Belzoni 7, I-35131 Padova, Italy
Email: mattarei@pdmat1.math.unipd.it

M. F. Newman
Affiliation: School of Mathematical Sciences, Australian National University, Canberra, ACT 0200, Australia
Email: newman@maths.anu.edu.au

DOI: https://doi.org/10.1090/S0002-9947-97-02005-9
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

American Mathematical Society