Weak algebra bundles and associator varieties - Canlubo, Clarisson Rizzie (2019)
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 by illustrating that the classifying object of algebra bundles and, consequently, of weak algebra bundles is a weak 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. (Author's abstract)