Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-26T07:39:25.223Z Has data issue: false hasContentIssue false
  • English
  • Français

Subdirect sums of Lie algebras

Published online by Cambridge University Press:  16 November 2017

D. H. KOCHLOUKOVA  and
C. MARTÍNEZ–PÉREZ
D. H. KOCHLOUKOVA
Affiliation:
Department of Mathematics, University of Campinas, 13083-859, Campinas, SP, Brazil. e-mail: [email protected]
C. MARTÍNEZ–PÉREZ
Affiliation:
Conchita Martínez-Pérez, Departamento de Matemáticas, Universidad de Zaragoza, 50009 Zaragoza, Spain. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We show Lie algebra versions of some results on homological finiteness properties of subdirect products of groups. These results include a version of the 1-2-3 Theorem.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 166 , Issue 1 , January 2019 , pp. 147 - 171
Copyright
Copyright © Cambridge Philosophical Society 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

Supported by “bolsa de produtividade em pesquisa” CNPq, 303350/2013-0, CNPq, Brazil and by FAPESP, 2016/05678-3, Brazil.

‡

Supported by Gobierno de Aragon, European Regional Development Funds and MTM2015-67781-P (MINECO/FEDER)

References

REFERENCES

[1] Bakhturin, Yu. A. Identities in Lie algebras. Nauka (Moscow, 1985).Google Scholar
[2] Baumslag, B. Free algebras and free groups. J. London math. Soc. (2) 4 (1971/72), 523–532.Google Scholar
[3] Baumslag, B. Residually free group. Proc. London Math. Soc. (3) 17 (1967) 402–418.Google Scholar
[4] Baumslag, G. On generalised free products. Math. Z. 78 (1962) 423–438.Google Scholar
[5] Baumslag, G., Bridson, M. R., Miller, C. F. III and Short, H. Fibre Products, non-positive curvature, and decision problems. Comment. Math. Helv. 75 (2000), 457–477.Google Scholar
[6] Baumslag, G. and Roseblade, J. Subgroups of direct products of free groups. J. London Math. Soc. (2) 30 (1984), no. 1, 44–52.Google Scholar
[7] Bichon, J. and Carnovale, G. Lazy cohomology: an analogue of the Schur multiplier for arbitrary Hopf algebras. J. Pure Appl. Algebra 204 (2006), no. 3, 627–665.Google Scholar
[8] Bieri, R. Homological dimension of discrete groups Queen Mary College Mathematical Notes. (Queen Mary College Department of Pure Mathematics, London, second edition, 1981).Google Scholar
[9] Bourbaki, N. Lie Groups and Lie Algebras, Part 1 (Addison-Wesley, Reading, MA, 1975) Chapters, 1–3.Google Scholar
[10] Bridson, M. R., Howie, J., Miller, C. F. III and Short, H. The subgroups of direct products of surface groups. Dedicated to John Stallings on the occasion of his 65th birthday. Geom. Dedicata 92 (2002), 95–103.Google Scholar
[11] Bridson, M. R., Howie, J., Miller, C. F. III and Short, H. Subgroups of direct products of limit groups. Ann. of Math. (2) 170 (2009), no. 3, 1447–1467.Google Scholar
[12] Bridson, M. R., Howie, J., Miller, C. F. III and Short, H. On the finite presentation of subdirect products and the nature of residually free groups. Amer. J. Math. 135 (2013), no. 4, 891–933.Google Scholar
[13] Brown, K. Cohomology of groups (Springer–Verlag 1982).Google Scholar
[14] Bryant, R. M. and Groves, J. R. J. Finitely presented Lie algebras. J. of Algebra 218 (1999), 1–25.Google Scholar
[15] Bryant, R. M. and Groves, J. R. J. Finite presentation of abelian-by-finite-dimensional Lie algebras. J. London Math. Soc. (2) 60 (1999), no. 1, 45–57.Google Scholar
[16] Groves, J. R. J. and Kochloukova, D. H. Homological finiteness properties of Lie algebras. J. Algebra 279 (2004), no. 2, 840–849.Google Scholar
[17] Jacobson, N. Lie algebras. Interscience Tracts in Pure and Applied Mathematics, No. 10 (Interscience Publishers (a division of John Wiley & Sons), New York-London, 1962).Google Scholar
[18] Kochloukova, D. H. On the homological finiteness properties of some modules over metabelian Lie algebras. Israel J. Math. 129 (2002), 221–239.Google Scholar
[19] Kochloukova, D. H. On subdirect products of type FPm of limit groups. J. Group Theory 13 (2010), no. 1, 1–19.Google Scholar
[20] Kochloukova, D. H. and Lima, F. F. Homological finiteness properties of fibre products, submittedGoogle Scholar
[21] Kuckuck, B. Subdirect products of groups and the n - (n + 1) - (n + 2) conjecture. Q. J. Math. 65 (2014), no. 4, 1293–1318.Google Scholar
[22] Lam, T. Y. Lectures on Modules and Rings. Graduate Texts in Mathematics, vol. 189 (Springer–Verlag, New York, 1999).Google Scholar
[23] Rotman, J. An Introduction to Homological Algebra (Academic Press, New York-London, 1979).Google Scholar
[24] Širsov, A. I. Subalgebras of free Lie algebras. Math Sb (2), 33 (1953), 441–452.Google Scholar
[25] Witt, E. Die Unterringe der freien Lieschen Ringe. Math. Z. 64 (1956), 195–216.Google Scholar
[26] Weibel, C. A. An Introduction to Homological Algebra (Cambridge University Press, Cambridge, 1994).Google Scholar