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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: desi@ime.unicamp.br
C. MARTÍNEZ–PÉREZ
Affiliation:
Conchita Martínez-Pérez, Departamento de Matemáticas, Universidad de Zaragoza, 50009 Zaragoza, Spain. e-mail: conmar@unizar.es
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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 

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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