Philippine Journal of Science
148 (2): 315-322, June 2019
ISSN 0031 – 7683
Date Received: 13 Dec 2018

 

Weak Algebra Bundles and Associator Varieties

Clarisson Rizzie Canlubo

 

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 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.