Quantum Webs of Type Q

TL;DR

This paper develops a new web diagram framework for quantum type Q superalgebras, establishing a functorial correspondence with their module categories and exploring applications to link invariants.

Contribution

It introduces a monoidal supercategory of quantum type Q webs and demonstrates its equivalence to categories of quantum superalgebra modules, extending web calculus to this setting.

Findings

Established a full, essentially surjective functor from web category to quantum superalgebra modules.

Identified a ribbon subcategory within the web category.

Discussed applications to link invariants and representation theory.

Abstract

Webs are combinatorial diagrams used to encode homomorphisms between representations of Lie (super)algebras and related objects. This paper extends the theory of webs to the quantum group of type Q. We define a monoidal supercategory of quantum type Q webs and show it admits a full, essentially surjective functor onto the monoidal supercategory of $U_q(\mathfrak{q}_n)$-modules generated by the quantum symmetric powers of the natural representation and their duals. We also show that a certain subcategory of the web category is a ribbon category and discuss applications to the representation theory of $U_q(\mathfrak{q}_n)$ and to invariants of oriented, framed links.