Polynomial Cohomology and Polynomial Maps on Nilpotent Groups

TL;DR

This paper develops a refined theory connecting polynomial cohomology to polynomial maps on nilpotent groups, providing new descriptions and classifications of polynomials in this setting.

Contribution

It introduces a refined version of group cohomology related to polynomial maps and characterizes polynomials on connected, simply connected nilpotent Lie groups.

Findings

Polynomial cohomology with trivial coefficients relates to ordinary cohomology with polynomial coefficients.

Degree one polynomial cohomology with trivial coefficients can be described directly via polynomials.

Polynomials on connected, simply connected nilpotent Lie groups are exactly those maps pulling back to classical polynomials via the exponential map.

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.