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A refinement of the toral rank conjecture for 2-step nilpotent Lie algebras


Author: Paulo Tirao
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
Published electronically: April 28, 2000
MathSciNet review: 1664387
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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 $2^{\vert\mathfrak{z}\vert}$, where $\mathfrak{z}$is the center of $\mathfrak{g}$. We improve this result by showing that a better lower bound is $2^t$, where $t={\vert\mathfrak{z}\vert+\left[\frac{\vert v\vert+1}2\right]}$ 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 [Enhancements On Off] (What's this?)

  • 1. Armstrong G., Cairns G. and Jessup B., Explicit Betti numbers for a family of nilpotent Lie algebras, Proc. Amer. Math. Soc. 125 (1997), 381-385. MR 97d:17013
  • 2. Cairns G. and Jessup B., New bounds on the Betti numbers of nilpotent Lie algebras, Comm. in Algebra 25 (1997), 415-430. MR 98a:17032
  • 3. Cairns G., Jessup B., and Pitkethly J., On the Betti numbers of nilpotent Lie algebras of small dimension, Prog. in Math.: Integrable systems and foliations (Albert C. et al., eds.), vol. 145, Birkhauser, 1997, pp. 19-31.MR 98c:17018
  • 4. Deninger Ch. and Singhof W., On the cohomology of nilpotent Lie algebras, Bull. Soc. Math. France 116 (1988), 3-14. MR 90c:17023
  • 5. Halperin S., Le complexe de Koszul en algèbre et topologie, Ann. Inst. Fourier 37 (1987), 77-97. MR 89d:55040
  • 6. Seeley C., 7-dimensional nilpotent Lie algebras, Trans. Amer. Math. Soc. 335 (1993), 479-496. MR 93d:17015
  • 7. Sigg S., Laplacian and homology of free 2-step nilpotent Lie algebras, J. Algebra 185 (1996), 144-161. MR 97i:17014

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Additional 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

DOI: https://doi.org/10.1090/S0002-9939-00-05366-1
Keywords: Homology of Lie algebras, 2-step nilpotent Lie algebras, toral rank conjecture
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
Article copyright: © Copyright 2000 American Mathematical Society

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