Type $C$ Webs

TL;DR

This paper constructs a pivotal category called Web(𝔰𝔭_{2n}) over the field of rational functions in q, proving its equivalence to a subcategory of finite-dimensional quantum symplectic algebra representations, solving a longstanding problem.

Contribution

It defines a new web category for type C Lie algebras and establishes its equivalence to a subcategory of quantum group representations, addressing an open problem from 1996.

Findings

Established an equivalence between Web(𝔰𝔭_{2n}) and quantum symplectic representations

Provided a diagrammatic calculus for type C Lie algebra representations

Solved the main open problem from Kuperberg's 1996 paper.

Abstract

We define a $\mathbb{C}(q)$-linear pivotal category $\mathbf{Web}(\mathfrak{sp}_{2n})$ and prove that it is equivalent to the full subcategory of finite-dimensional representations of $U_q(\mathfrak{sp}_{2n})$ tensor-generated by the fundamental representations. This answers the type $C$ case of the main open problem from Kuperberg's 1996 paper "Spiders for rank 2 Lie algebras" (arXiv:q-alg/9712003).