M\"obius Strip Diagram Algebras

TL;DR

This paper introduces M"obius strip diagram algebras, extending diagram calculus with nonorientable features, and classifies their simple modules through cell theory, connecting to nonorientable cobordism categories.

Contribution

It defines M"obius strip diagram algebras and links them to nonorientable cobordism categories, providing a framework for module classification and dimension computation.

Findings

Defined M"obius strip diagram algebras and their categorical versions.

Connected diagram categories to nonorientable cobordism categories.

Classified simple modules and computed their dimensions.

Abstract

We introduce M\"obius strip diagram algebras (and their monoid and categorical versions) as subalgebras of a partition-style diagram calculus in which strands may carry handles and M\"obius strip features. We identify the resulting diagram category with a linear quotient of a nonorientable two-dimensional cobordism category. Finally, we develop the associated cell theory and use it to classify the simple modules and compute dimensions in a range of cases.