Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-15T15:57:52.059Z Has data issue: false hasContentIssue false
  • English
  • Français

A non-abelian tensor product of Lie algebras

Published online by Cambridge University Press:  18 May 2009

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

Extract

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.

A generalized tensor product of groups was introduced by R. Brown and J.-L. Loday [6], and has led to a substantial algebraic theory contained essentially in the following papers: [6, 7, 1, 5, 11, 12, 13, 14, 18, 19, 20, 23, 24] ([9, 27, 28] also contain results related to the theory, but are independent of Brown and Loday's work). It is clear that one should be able to develop an analogous theory of tensor products for other algebraic structures such as Lie algebras or commutative algebras. However to do so, many non-obvious algebraic identities need to be verified, and various topological proofs (which exist only in the group case) have to be replaced by purely algebraic ones. The work involved is sufficiently non-trivial to make it interesting.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 33 , Issue 1 , January 1991 , pp. 101 - 120
Copyright
Copyright © Glasgow Mathematical Journal Trust 1991

References

REFERENCES

1.Aboughazi, R., Produit tensoriel du groupe d' Heisenberg, Bull. Soc. Math. France 115 (1987), 1–12.Google Scholar
2.Barr, M. and Beck, J., Acyclic models and triples, in Eilenberg, S., Harrison, D. K., MacLane, S., Röhrl, H., ed., Proceedings of the conference on categorical algebra La Jolla 1965 (Springer, 1966), 344–354.CrossRefGoogle Scholar
3.Barr, M. and Beck, J., Homology and standard constructions, in Eckmann, B., ed., Seminar on triples and categorical homology theory, Lecture Notes in Mathematics 80 (Springer, 1969), 245–335.CrossRefGoogle Scholar
4.Brown, R. and Ellis, G. J., Hopf formulae for the higher homology of a group, Bull. London Math. Soc. 20 (1988), 124–128.CrossRefGoogle Scholar
5.Brown, R., Johnson, D. L. and Robertson, E. F., Some computations of nonabelian tensor products of groups, J. Algebra 111 (1987), 177–202.CrossRefGoogle Scholar
6.Brown, R. and Loday, J.-L., Van Kampen theorems for diagrams of spaces, Topology 26 (1987), 311–335.CrossRefGoogle Scholar
7.Brown, R. and Loday, J.-L., Homotopical excision, and Hurewicz theorems, for n-cubes of spaces, Proc. London Math. Soc. (3) 54 (1987), 176–192.CrossRefGoogle Scholar
8.Curtis, E. B., Simplicial homotopy theorey, Adv. Math. 6 (1971), 107–204.CrossRefGoogle Scholar
9.Dennis, K., In search of new homology functors having a close relationship to K-theory, preprint, 1976.Google Scholar
10.Ellis, G. J., Crossed modules and their higher dimensional analogues (Ph.D. thesis, University of Wales, 1984).Google Scholar
11.Ellis, G. J., Nonabelian exterior products of groups and an exact sequence in the homology of groups, Glasgow Math. J. 29 (1987), 13–19.CrossRefGoogle Scholar
12.Ellis, G. J., The nonabelian tensor product of finite groups is finite, J. Algebra 111 (1987), 203–205.CrossRefGoogle Scholar
13.Ellis, G. J., An algebraic derivation of a certain exact sequence, J. Algebra 117 (1989), 178–181.CrossRefGoogle Scholar
14.Ellis, G. J., On the higher universal quadratic functors and related computations (Galway Preprint, 1989).Google Scholar
15.Ellis, G. J., Nonabelian exterior products of Lie algebras and an exact sequence in the homology of Lie algebras, J. Pure Appl. Algebra 46 (1987), 111–115.CrossRefGoogle Scholar
16.Ellis, G. J., Relative derived functors and the homology of groups, Cahiers Topologie Géom. Différentielle Catégoriques, to appear.Google Scholar
17.Ellis, G. J. and Porter, T., Free and projective crossed modules and the second homology of a group, Pure Appl. Algebra 40 (1986), 27–31.CrossRefGoogle Scholar
18.Gilbert, N. D., The nonabelian tensor square of a free group, Arch. Math. (Basel), 48 (1987), 369–375.CrossRefGoogle Scholar
19.Gilbert, N. D. and Higgins, P. J., The nonabelian tensor product of groups and related constructions, Glasgow Math. J. 31 (1989), 17–29.CrossRefGoogle Scholar
20.Guin, D., “Cohomologie et homologie non abgliennes des groupes,” J. Pure Appl. Algebra 50 (1988), 109–137.CrossRefGoogle Scholar
21.Guin, D., Cohomologie non abélienne des algèbres de Lie (Strasbourg Preprint, 1987).Google Scholar
22.Hilton, P. J. and Stammbach, U., A course in homological algebra. Graduate Texts in Mathematics 4 (Springer, 1971).CrossRefGoogle Scholar
23.Johnson, D. L., The nonabelian tensor square of a finite split metacyclic group, Proc. Edinburgh Math. Soc. (2) 30 (1987), 91–96.CrossRefGoogle Scholar
24.Korkes, F. J., The cohomological and combinatorial group theory of profinite groups (Ph.D. thesis, University of Wales, 1986).Google Scholar
25.Keune, F., The relativization of K 2, J. Algebra 54 (1978), 159–177.CrossRefGoogle Scholar
26.Kassel, C. and Loday, J.-L., Extensions centrales d'algebres de Lie, Ann. Inst. Fourier (Grenoble) 33 (1982), 119–142.CrossRefGoogle Scholar
27.Lue, A. S.-T., The Ganea map for nilpotent groups, J. London Math. Soc. (2) 14 (1976), 309–312.CrossRefGoogle Scholar
28.Miller, C., The second homology group of a group, Proc. Amer. Math. Soc. 3 (1952), 588–595.CrossRefGoogle Scholar
29.Quillen, D. G., Homotopical Algebra, Lecture Notes in Mathematics 43 (Springer, 1967).CrossRefGoogle Scholar
30.Ratcliffe, J. G., Free and projective crossed modules, J. London Math. Soc. (2) 22 (1980), 66–74.CrossRefGoogle Scholar
31.Simson, D. and Tyc, A., Connected sequences of stable derived functors and their applications, Dissertationes Mathematicae (Rozprawy Mat.) 111 (1974).Google Scholar