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.

中文翻译：

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幂零群上的多项式上同调和多项式映射

我们引入了一个改进的群上同调版本，并将其与所讨论的群上的多项式空间相关联。我们证明了具有平凡系数的多项式上同调可以根据具有多项式系数的普通上同调来进行描述，并且具有平凡系数的一阶多项式上同调可以直接根据多项式来进行描述。最后，我们通过证明这些正是通过指数映射回拉到经典多项式的映射，对连通的、简单连通的幂零李群上的多项式进行了完整的描述。