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POLYNOMIAL COHOMOLOGY AND POLYNOMIAL MAPS ON NILPOTENT GROUPS

Part of: Low-dimensional topology Lie groups

Published online by Cambridge University Press:  02 October 2019

DAVID KYED Open the ORCID record for DAVID KYED [Opens in a new window]  and
HENRIK DENSING PETERSEN
DAVID KYED
Affiliation:
Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, DK-5230, Odense M, Denmark email: dkyed@imada.sdu.dk
HENRIK DENSING PETERSEN
Affiliation:
Stenhus Gymnasium, Stenhusvej 20, DK-4300, Holbæk, Denmark email: he@stenhus-gym.dk
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Abstract

We introduce a refined version of group cohomology and relate it to the space of polynomials on the group in question. We show that the polynomial cohomology with trivial coefficients admits a description in terms of ordinary cohomology with polynomial coefficients, and that the degree one polynomial cohomology with trivial coefficients admits a description directly in terms of polynomials. Lastly, we give a complete description of the polynomials on a connected, simply connected nilpotent Lie group by showing that these are exactly the maps that pull back to classical polynomials via the exponential map.

MSC classification

Primary: 22E41: Continuous cohomology
Secondary: 57M07: Topological methods in group theory 22E25: Nilpotent and solvable Lie groups
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 62 , Issue 3 , September 2020 , pp. 706 - 736
Copyright
© Glasgow Mathematical Journal Trust 2019

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References

Babakhanian, A., Cohomological methods in group theory (M. Dekker, New York, 1972).Google Scholar
Bader, U., Rosendal, C. and Sauer, R., On the cohomology of weakly almost periodic group representations, J. Topology and Analysis 6 (2014), 153–165.10.1142/S1793525314500125CrossRefGoogle Scholar
Baumslag, G., Lecture notes on nilpotent groups (American Mathematical Society, Providence, RI, 1971).Google Scholar
Bekka, B., de la Harpe, P. and Valette, A., Kazhdan’s property (T), New Mathematical Monographs, vol. 11 (Cambridge University Press, Cambridge, 2008).10.1017/CBO9780511542749CrossRefGoogle Scholar
Blanc, P., Sur la cohomologie continue des groupes localement compacts, Ann. Sci. École Norm. Sup. 12 (1979), 137–168.10.24033/asens.1364CrossRefGoogle Scholar
Borel, A., Linear algebraic groups, Graduate Texts in Mathematics, vol. 126, 2nd edition, (Springer-Verlag, New York, 1991).10.1007/978-1-4612-0941-6CrossRefGoogle Scholar
Buckley, J. T., Polynomial functions and wreath products, Illinois J. Math . 14 (1970), 274–282.10.1215/ijm/1256053186CrossRefGoogle Scholar
Corwin, L. J. and Greenleaf, F. P., Representations of nilpotent Lie groups and their applications, Part : Basic theory and examples (Cambridge University Press, Cambridge, 1990).Google Scholar
Delorme, P., 1-Cohomologie des représentations unitaires des groupes de Lie semi-simples et résolubles. Produits tensoriels continus de représentations, Bull. Soc. Math. France 105 (1977), 281–336.10.24033/bsmf.1853CrossRefGoogle Scholar
Guichardet, A., Cohomologie des groupes topologiques et des algèbres de Lie (CEDIC, Paris, 1980).Google Scholar
Hochschild, G., The structure of Lie groups (Holden-Day, Inc., San Francisco, London, Amsterdam), 1965), ix+230.Google Scholar
Jennings, S. A., The group ring of a class of infinite nilpotent groups, Canad. J. Math. 7 (1955), 169–187.10.4153/CJM-1955-022-5CrossRefGoogle Scholar
Kirillov, A., An introduction to Lie groups and lie algebras, Cambridge Studies in Advanced Mathematics, vol. 113, (Cambridge University Press, Cambridge, 2008).Google Scholar
Leibman, A., Polynomial mappings of groups, Israel Journal of Mathematics 129 (2002), 29–60. Erratum available.10.1007/BF02773152CrossRefGoogle Scholar
Mal’cev, A., On a class of homogeneous spaces, AMS Translation 39 (1951), 33.Google Scholar
Raghunathan, M. S., Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, (Springer-Verlag, New York, Heidelberg, 1972), ix+227.Google Scholar
Passi, I. B. S., Polynomial maps on groups, J. Algebra 9 (1968), 121–151.10.1016/0021-8693(68)90017-3CrossRefGoogle Scholar
Passi, I. B. S., Polynomial functors, Proc. Cambridge Philos. Soc. 66 (1969), 505–512.10.1017/S0305004100045254CrossRefGoogle Scholar
Passi, I. B. S., Polynomial maps on groups. II, Math. Z. 135 (1973/74), 137–141.10.1007/BF01189350CrossRefGoogle Scholar
Shalom, Y., Harmonic analysis, cohomology, and large-scale geometry of amenable groups, Acta Math . 192 (2004), 119–185.10.1007/BF02392739CrossRefGoogle Scholar
Székelyhidi, L., Note on exponential polynomials, Pacific J. Math . 103(2) (1982), 583–587.10.2140/pjm.1982.103.583CrossRefGoogle Scholar
Wang, H.-C., Discrete subgroups of solvable Lie groups,, Ann. Math. 64 (1956), 1–19.10.2307/1969948CrossRefGoogle Scholar