Weak Algebra Bundles and Associator Varieties

TL;DR

This paper introduces the concept of weak algebra bundles, explores their local triviality conditions, and studies associator varieties as geometric objects analogous to Grassmannians for algebra bundles.

Contribution

It formalizes weak algebra bundles, establishes criteria for their local triviality, and analyzes associator varieties as key geometric structures in algebra bundle theory.

Findings

Weak algebra bundles are more natural than strict ones.

Necessary and sufficient conditions for local triviality are provided.

Associator varieties are characterized as geometric analogs of Grassmannians.

Abstract

Algebra bundles, in the strict sense, appear in many areas of geometry and physics. However, the structure of an algebra is flexible enough to vary non-trivially over a connected base, giving rise to a structure of a weak algebra bundle. We will show that the notion of a weak algebra bundle is more natural than that of a strict algebra bundle. We will give necessary and sufficient conditions for weak algebra bundles to be locally trivial. The collection of non-trivial associative algebras of a fixed dimension forms a projective variety, called associator varieties. We will show that these varieties play the role the Grassmannians play for principal $O(n)$-bundles.