Weak Algebra Bundles and Associator Varieties
Clarisson Rizzie Canlubo
Institute of Mathematics, and National Science Research Institute
College of Science, University of the Philippines Diliman
Quezon City 1101 Philippines
*Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.
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.
INTRODUCTION
Weak algebra bundles are generalizations of (strict) algebra bundles. They are monoid objects in the category of vector bundles. Algebra bundles appear more frequently in the literature. The exterior bundle and the Clifford bundle are examples of (strict) algebra bundles. In Section 3, we look at the varieties of associative algebras of a fixed dimension – the so-called associator varieties. In Section 4, we will show that weak algebra bundles are more natural than algebra bundles by constructing the so-called classifying weak algebra bundle. In Section 5, we will give necessary and sufficient conditions for a weak algebra bundle to be locally trivial and, hence, strictness. We will introduce the notion of a differential connection. Existence of a differential connection together with a technical condition guarantee local triviality. . . . . read more
REFERENCES
CANLUBO CRP. 2017. Non-commutative covering spaces and their symmetries [Ph.D. Thesis] Copenhagen (Denmark): University of Copenhagen.
CHERN SS, CHEN WH, LAM KS. 1999. Lectures on Differential Geometry. In: Series on University Mathematics Vol 1. Hackensack, NJ: World Scientific Publishing Co.
CHIDAMBARA C, KIRANAGI BS. 1994. On cohomology of associative algebra bundles. Journal of the Ramanujan Mathematical Society 9(1): 1–12.
DADARLAT M. 2006. Continuous fields of C∗-algebras over finite dimensional spaces. arXiv:math/0611405.
DOUADY A, LAZARD M. 1966. Espaces fibres en algebre de Lie et en groups. Invent Math 1: 133–151.
FELL JMG. 1961. The structure of algebras of operator fields. Acta Math 106(3–4): 233–280.
HUSEMÖLLER D. 1994. Fiber Bundles. In: Graduate Texts in Mathematics Vol. 20. Berlin (Germany): Springer-Verlag.
KIRANAGI BS, RAJENDRA R. 2008. Revisiting Hochschild cohomology for algebra bundles. Journal of Algebra and its Applications 7(6): 685–715.
MAY PJ. 1999. A Concise Course in Algebraic Topology. Chicago, IL: The University of Chicago Press.
SHAFAREVICH IR. 1994. Basic Algebraic Geometry 1, 2nd rev. and expanded version, Springer Study Edition. Berlin (Germany): Springer-Verlag.
