On a noncommutative algebraic geometry

TL;DR

This paper explores quaternionic functions and their hyperholomorphic properties to define Hamilton 4-manifolds, extending complex algebraic geometry concepts into four dimensions using noncommutative algebra.

Contribution

It introduces a framework for quaternionic functions and Hamilton 4-manifolds, broadening algebraic geometry into noncommutative, four-dimensional settings.

Findings

Quaternionic functions with hyperholomorphic properties are characterized.

Hamilton 4-manifolds are constructed as analogs to Riemann surfaces.

A new class of four-dimensional manifolds is proposed.

Abstract

Several sets of quaternionic functions are described and studied with respect to hy-perholomorphy, addition and (non commutative) multiplication, on open sets of H, then Hamil-ton 4-manifolds analogous to Riemann surfaces, for H instead of C, are defined, and so begin to describe a class of four dimensional manifolds.