A refinement of the toral rank conjecture for 2-step nilpotent Lie algebras
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- by Paulo Tirao
- Proc. Amer. Math. Soc. 128 (2000), 2875-2878
- DOI: https://doi.org/10.1090/S0002-9939-00-05366-1
- Published electronically: April 28, 2000
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Abstract:
It is known that the total (co)-homoloy of a 2-step nilpotent Lie algebra $\mathfrak {g}$ is at least $2^{|\mathfrak {z}|}$, where $\mathfrak {z}$ is the center of $\mathfrak {g}$. We improve this result by showing that a better lower bound is $2^t$, where $t={|\mathfrak {z}|+\left [\frac {|v|+1}2\right ]}$ and $v$ is a complement of $\mathfrak {z}$ in $\mathfrak {g}$. Furthermore, we provide evidence that this is the best possible bound of the form $2^t$.References
- Grant F. Armstrong, Grant Cairns, and Barry Jessup, Explicit Betti numbers for a family of nilpotent Lie algebras, Proc. Amer. Math. Soc. 125 (1997), no. 2, 381–385. MR 1353371, DOI 10.1090/S0002-9939-97-03607-1
- Grant Cairns and Barry Jessup, New bounds on the Betti numbers of nilpotent Lie algebras, Comm. Algebra 25 (1997), no. 2, 415–430. MR 1428787, DOI 10.1080/00927879708825863
- Grant Cairns, Barry Jessup, and Jane Pitkethly, On the Betti numbers of nilpotent Lie algebras of small dimension, Integrable systems and foliations/Feuilletages et systèmes intégrables (Montpellier, 1995) Progr. Math., vol. 145, Birkhäuser Boston, Boston, MA, 1997, pp. 19–31. MR 1432905, DOI 10.1007/978-1-4612-4134-8_{2}
- Ch. Deninger and W. Singhof, On the cohomology of nilpotent Lie algebras, Bull. Soc. Math. France 116 (1988), no. 1, 3–14 (English, with French summary). MR 946276
- Stephen Halperin, Le complexe de Koszul en algèbre et topologie, Ann. Inst. Fourier (Grenoble) 37 (1987), no. 4, 77–97 (French, with English summary). MR 927392
- Craig Seeley, $7$-dimensional nilpotent Lie algebras, Trans. Amer. Math. Soc. 335 (1993), no. 2, 479–496. MR 1068933, DOI 10.1090/S0002-9947-1993-1068933-4
- Stefan Sigg, Laplacian and homology of free two-step nilpotent Lie algebras, J. Algebra 185 (1996), no. 1, 144–161. MR 1409979, DOI 10.1006/jabr.1996.0317
Bibliographic Information
- Paulo Tirao
- Affiliation: International Centre for Theoretical Physics (ICTP), Trieste, Italy; Facultad de Matemática, Astronomía y Física, Córdoba, Argentina
- Address at time of publication: Heinrich-Heine-Universität, Mathematisches Institut, 40225 Düsseldorf, Germany
- Email: ptirao@bart.cs.uni-duesseldorf.de, Paulo.Tirao@FamaF.uncor.edu.ar
- Received by editor(s): August 24, 1998
- Received by editor(s) in revised form: November 22, 1998
- Published electronically: April 28, 2000
- Communicated by: Roe Goodman
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 2875-2878
- MSC (2000): Primary 17B56, 17B30
- DOI: https://doi.org/10.1090/S0002-9939-00-05366-1
- MathSciNet review: 1664387