Riemannian Geometry on Associative Varieties

TL;DR

This paper develops a framework for defining Riemannian geometry on associative varieties by extending algebraic and differential geometric concepts to associative algebras over real and arbitrary fields.

Contribution

It introduces an associative generalization of algebraic varieties and develops differential geometric tools like connections and geodesics within this framework.

Findings

Defined associative varieties using local representations for simple modules.

Extended differential geometry concepts to associative algebras, including connections and geodesics.

Established a foundation for real geometry on associative algebras.

Abstract

We prove that the classical algebraic varieties over algebraically closed fields can be defined over arbitrary fields $k.$ Then we prove that for associative algebras $A$, there exist local representing objects $A_M$ for simple modules $M.$ Replacing the localization in maximal ideals in the commutative situation with the local representations in simple modules in the associative, we define an associative generalization of varieties. Now we realize that replacing $\mathbb R[x_1,\dots,x_n]=\mathbb R[n]$ with $C^\infty(\mathbb R^n),$ we can do differential geometry for associative $\mathbb R[n]$-algebras. This says that we can define a Riemannian geometry on associative varieties. This gives us the definition of connections and algebraic geodesic curves, introducing real geometry into associative algebras.