A formula for the second cohomology of two-step nilpotent groups

TL;DR

This paper provides an explicit formula for the second cohomology of two-step nilpotent groups by connecting polynomial cocycles with Lie algebra structures, enabling concrete cocycle construction.

Contribution

It makes the abstract expression for second cohomology concrete for two-step nilpotent groups using Lie algebra methods, which was previously more theoretical.

Findings

Explicit cocycles for second cohomology constructed

Connection between polynomial cocycles and Lie algebra established

Provides a practical method for cohomology computation

Abstract

In [5], the notion of polynomial cocycles is used to give an expression for the second cohomology of T-groups with coefficients in a torsion-free nilpotent module. We make this expression concrete in the case of a T-group G of nilpotency class <=2 and coefficients in a trivial G-module, using a Lie algebra associated to the group. This approach allows us to construct explicit cocycles representing the elements of the second cohomology group.