Representations and cohomologies of Kleinian 4-rings

TL;DR

This paper introduces Kleinian 4-rings, a new class of algebras over discrete valuation rings, and describes their lattices and cohomologies, providing explicit cocycles for regular lattices.

Contribution

It defines Kleinian 4-rings and details their lattice structures and cohomology, including explicit cocycles for regular lattices.

Findings

Kleinian 4-rings generalize the group algebra of the Kleinian 4-group.

Explicit cocycles are obtained for regular lattices.

The structure of lattices and cohomologies of Kleinian 4-rings is characterized.

Abstract

We introduce a new class of algebras over discrete valuation rings, called Kleinian 4-rings, which generalize the group algebra of the Kleinian 4-group. For these algebras we describe the lattices and their cohomologies. In the case of regular lattices we obtain an explicit form of cocycles defining the cohomology classes.