Available in electronic format
Available in print format		 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826 (e) ISSN 0002-9939 (p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

The restricted isomorphism problem for metacyclic restricted Lie algebras

Author(s): Hamid Usefi
Journal: Proc. Amer. Math. Soc. 136 (2008), 4125-4133.
MSC (2000): Primary 17B35, 17B50; Secondary 20C05
Posted: July 23, 2008
Retrieve article in: PDF DVI PostScript

Abstract | References | Similar articles | Additional information

Abstract: Let $ L$ be a restricted Lie algebra with the restricted enveloping algebra $ u(L)$ over a perfect field of positive characteristic $ p$. The restricted isomorphism problem asks what invariants of $ L$ are determined by $ u(L)$. This problem is the analogue of the modular isomorphism problem for finite $ p$-groups. Bagiński and Sandling have given a positive answer to the modular isomorphism problem for metacyclic $ p$-groups. In this paper, we provide a positive answer to the restricted isomorphism problem in case $ L$ is metacyclic and $ p$-nilpotent.

References:

1.
C. Bagiński, A. Konovalov, The modular isomorphism problem for finite $ p$-groups with a cyclic subgroup of index $ p\sp 2$, Groups St. Andrews 2005. Vol. 1, 186-193, London Math. Soc. Lecture Note Ser., 339 (Cambridge Univ. Press, 2007). MR 2327323 (2008f:20009)

2.
C. Bagiński, The isomorphism question for modular group algebras of metacyclic $ p$-groups, Proc. Amer. Math. Soc. 104 (1988), no. 1, 39-42. MR 958039 (89i:20016)

3.
N. Jacobson, Lie Algebras (Interscience, New York, 1962). MR 0143793 (26:1345)

4.
D.M. Riley, A. Shalev, Restricted Lie algebras and their envelopes, Canad. J. Math., 47 (1995), 146-164. MR 1319694 (96e:16031)

5.
D.M. Riley, A. Shalev, The Lie structure of enveloping algebras, J. Algebra, 162 (1993), no. 1, 46-61. MR 1250526 (94k:17019)

6.
D.M. Riley, H. Usefi, The isomorphism problem for universal enveloping algebras of Lie algebras, Algebras and Representation Theory, 10 (2007), no. 6, 517-532. MR 2350224

7.
R. Sandling, The modular group algebra problem for metacyclic $ p$-groups, Proc. Amer. Math. Soc., 124 (1996), no. 5, 1347-1350. MR 1343723 (96g:20003)

8.
R. Sandling, The isomorphism problem for group rings: a survey, Orders and Their Applications (Oberwolfach, 1984), Lecture Notes in Math., 1142 (Springer, Berlin, 1985), 256-288. MR 812504 (87b:20007)

9.
H. Strade, R. Farnsteiner, Modular Lie Algebras and Their Representations, Monographs and Textbooks in Pure and Applied Mathematics, 116 (Marcel Dekker, New York, 1988). MR 929682 (89h:17021)

10.
H. Usefi, Isomorphism invariants of restricted enveloping algebras, preprint is available on arXiv:0804.2281.

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 17B35, 17B50, 20C05

Retrieve articles in all Journals with MSC (2000): 17B35, 17B50, 20C05

Additional Information:

Hamid Usefi
Affiliation: Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, Canada, V6T 1Z2
Email: usefi@math.ubc.ca

DOI: 10.1090/S0002-9939-08-09632-9
PII: S 0002-9939(08)09632-9
Keywords: Restricted Lie algebras, metacylic, enveloping algebras, isomorphism problem, modular group algebras
Received by editor(s): September 28, 2007
Posted: July 23, 2008
Communicated by: Gail R. Letzter
Copyright of article: Copyright 2008, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google