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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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