Geodesic
POLYNOMIAL COHOMOLOGY AND POLYNOMIAL MAPS ON NILPOTENT GROUPS
Glasgow mathematical journal, Tome 62 (2020) no. 3, pp. 706-736

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
KYED, DAVID; PETERSEN, HENRIK DENSING. POLYNOMIAL COHOMOLOGY AND POLYNOMIAL MAPS ON NILPOTENT GROUPS. Glasgow mathematical journal, Tome 62 (2020) no. 3, pp. 706-736. doi: 10.1017/S0017089519000429
@article{10_1017_S0017089519000429,
     author = {KYED, DAVID and PETERSEN, HENRIK DENSING},
     title = {POLYNOMIAL {COHOMOLOGY} {AND} {POLYNOMIAL} {MAPS} {ON} {NILPOTENT} {GROUPS}},
     journal = {Glasgow mathematical journal},
     pages = {706--736},
     year = {2020},
     volume = {62},
     number = {3},
     doi = {10.1017/S0017089519000429},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089519000429/}
}
TY  - JOUR
AU  - KYED, DAVID
AU  - PETERSEN, HENRIK DENSING
TI  - POLYNOMIAL COHOMOLOGY AND POLYNOMIAL MAPS ON NILPOTENT GROUPS
JO  - Glasgow mathematical journal
PY  - 2020
SP  - 706
EP  - 736
VL  - 62
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089519000429/
DO  - 10.1017/S0017089519000429
ID  - 10_1017_S0017089519000429
ER  - 
%0 Journal Article
%A KYED, DAVID
%A PETERSEN, HENRIK DENSING
%T POLYNOMIAL COHOMOLOGY AND POLYNOMIAL MAPS ON NILPOTENT GROUPS
%J Glasgow mathematical journal
%D 2020
%P 706-736
%V 62
%N 3
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089519000429/
%R 10.1017/S0017089519000429
%F 10_1017_S0017089519000429

[1] Babakhanian, A., Cohomological methods in group theory (M. Dekker, New York, 1972). Google Scholar

[2] 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/S1793525314500125 Google Scholar | DOI

[3] Baumslag, G., Lecture notes on nilpotent groups (American Mathematical Society, Providence, RI, 1971). Google Scholar

[4] 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/CBO9780511542749 Google Scholar | DOI

[5] Blanc, P., Sur la cohomologie continue des groupes localement compacts, Ann. Sci. École Norm. Sup. 12 (1979), 137–168.10.24033/asens.1364 Google Scholar | DOI

[6] Borel, A., Linear algebraic groups, Graduate Texts in Mathematics, vol. 126, 2nd edition, (Springer-Verlag, New York, 1991).10.1007/978-1-4612-0941-6 Google Scholar | DOI

[7] Buckley, J. T., Polynomial functions and wreath products, Illinois J. Math . 14 (1970), 274–282.10.1215/ijm/1256053186 Google Scholar | DOI

[8] 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

[9] 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.1853 Google Scholar | DOI

[10] Guichardet, A., Cohomologie des groupes topologiques et des algèbres de Lie (CEDIC, Paris, 1980). Google Scholar

[11] Hochschild, G., The structure of Lie groups (Holden-Day, Inc., San Francisco, London, Amsterdam), 1965), ix+230. Google Scholar

[12] Jennings, S. A., The group ring of a class of infinite nilpotent groups, Canad. J. Math. 7 (1955), 169–187.10.4153/CJM-1955-022-5 Google Scholar | DOI

[13] Kirillov, A., An introduction to Lie groups and lie algebras, Cambridge Studies in Advanced Mathematics, vol. 113, (Cambridge University Press, Cambridge, 2008). Google Scholar

[14] Leibman, A., Polynomial mappings of groups, Israel Journal of Mathematics 129 (2002), 29–60. Erratum available.10.1007/BF02773152 Google Scholar | DOI

[15] Mal’Cev, A., On a class of homogeneous spaces, AMS Translation 39 (1951), 33. Google Scholar

[16] 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

[17] Passi, I. B. S., Polynomial maps on groups, J. Algebra 9 (1968), 121–151.10.1016/0021-8693(68)90017-3 Google Scholar | DOI

[18] Passi, I. B. S., Polynomial functors, Proc. Cambridge Philos. Soc. 66 (1969), 505–512.10.1017/S0305004100045254 Google Scholar | DOI

[19] Passi, I. B. S., Polynomial maps on groups. II, Math. Z. 135 (1973/74), 137–141.10.1007/BF01189350 Google Scholar | DOI

[20] Shalom, Y., Harmonic analysis, cohomology, and large-scale geometry of amenable groups, Acta Math . 192 (2004), 119–185.10.1007/BF02392739 Google Scholar | DOI

[21] Székelyhidi, L., Note on exponential polynomials, Pacific J. Math . 103(2) (1982), 583–587.10.2140/pjm.1982.103.583 Google Scholar | DOI

[22] Wang, H.-C., Discrete subgroups of solvable Lie groups,, Ann. Math. 64 (1956), 1–19.10.2307/1969948 Google Scholar | DOI

Cité par Sources :