Hostname: page-component-7f64f4797f-vhqbp Total loading time: 0 Render date: 2025-11-04T09:19:39.499Z Has data issue: false hasContentIssue false
  • English
  • Français

Homotopical aspects of Lie algebras

Part of: Homological algebra

Published online by Cambridge University Press:  09 April 2009

Graham J. Ellis
Graham J. Ellis
Affiliation:
University College Galway, National University of Ireland, Galway, Ireland
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The Hurewicz theorem, Mayer-Vietoris sequence, and Whitehead's certain exact sequence are proved for simplicial Lie algebras. These results are applied, using crossed module techniques, to obtain information on the low dimensional homology of a Lie algebra, and information on aspherical presentations of Lie algebras.

MSC classification

Secondary: 18G55: Homotopical algebra

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 54 , Issue 3 , June 1993 , pp. 393 - 419
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Baues, H. J., Combinatorial homotopy and 4-dimensional complexes (Walter de Gruyter, Berlin, 1991).CrossRefGoogle Scholar
[2]Baues, H. J. and Conduché, D., ‘The central series for Peiffer commutators in groups with operators’, J. Algebra, 133 (1990), 1–34.CrossRefGoogle Scholar
[3]Brown, K. S., Cohomology of groups, Graduate Texts in Math. 87 (Springer, New York, 1982).CrossRefGoogle Scholar
[4]Brown, R., ‘Coproducts of crossed P-modules: applications to second homotopy groups and to the homology of groups’, Topology 23 (3) (1984), 337–345.CrossRefGoogle Scholar
[5]Conduché, D., ‘Modules croisés généralisés de Longueur 2’, J. Pure Appl. Algebra 34 (1984), 155–178.CrossRefGoogle Scholar
[6]Curtis, E. B., ‘Simplicial theory’, Advances in Math. 6 (1971), 107–209.CrossRefGoogle Scholar
[7]Dold, A. and Puppe, D., ‘Homologie nicht-additiver Funktoren, Anwendungen’, Ann. Inst. Fourier (Grenoble) 11 (1961) 201–312.CrossRefGoogle Scholar
[8]Duskin, J., Simplicial methods and the interpretation of triple cohomology, Mem. Amer. Math. Soc. 163 (1975).Google Scholar
[9]Ellis, G. J., ‘Nonablelian exterior product of Lie algebras and an exact sequence in the homology of Lie algebras’, J. Pure Applied Algebra 46 (1987), 111–115.CrossRefGoogle Scholar
[10]Ellis, G. J., ‘A nonabelian tensor product of Lie algebras’, Glasgow. Math. J. 33 (1991), 101–120.CrossRefGoogle Scholar
[11]Ellis, G. J., ‘Homology of 2-types’, J. London Math. Soc. (to appear)Google Scholar
[12]Eilenberg, S. and MacLane, S., ‘On the groups H(π, n), II’, Ann. of Math. (2) 60 (1954), 49–139.CrossRefGoogle Scholar
[13]Hilton, P. and Stammbach, U., A course in homological algebra, Graduate Texts in Math. 4 (Springer, New York, 1971).CrossRefGoogle Scholar
[14]Kassel, C. and Loday, J.-L., ‘Extensions centrales d'algèbres de Lie’, Ann. Inst. Fourier (Grenoble) 33 (1982), 119–142.CrossRefGoogle Scholar
[15]Lyndon, R. C., ‘Cohomology theory of groups with a single defining relation’, Ann. of Math. 52 (1950), 650–665.CrossRefGoogle Scholar
[16]Quillen, D. G., Homotopical algebra, Lecture Notes in Math. 43 (Springer, New York, 1967).CrossRefGoogle Scholar
[17]Simson, D. and Tyc, A., ‘Connected sequences of stable derived functors and their applications’, Dissertationes Math. (Rozprawy Mat.) 111 (1974).Google Scholar
[18]Witt, E., ‘Die Unterringe der freien Lieschen Ringe’, Math. Zeitschrift 64 (1956), 195–216.CrossRefGoogle Scholar
[19]Whitehead, J. H. C., ‘A certain exact sequence’, Ann. of Math. 52 (1950), 51–110.CrossRefGoogle Scholar
[20]Guttiérrez, M. A. and Ratcliffe, J. G., ‘On the second homotopy group’, Quart. J. Math. Oxford Ser. (2) 32 (1981), 45–55.CrossRefGoogle Scholar