Type $B$ Webs

Abstract

We solve the type $B$ case of the main open problem from Kuperberg's 1996 paper "Spiders for rank 2 Lie algebras." That is, we define a $\mathbb{C}(q)$-linear pivotal category $\mathbf{Web}({\mathfrak{so}}_{2n+1})$ and prove that it is equivalent to the full subcategory of finite-dimensional representations of $U_q({\mathfrak{so}}_{2n+1})$ tensor-generated by the fundamental representations. A consequence of our main result is an explicit construction of braid group symmetries for nonclassical finite-dimensional representations of the $\iota$quantum group $U_{-q^2}^{\iota }({\mathfrak{so}}_m)$, which may be of independent interest.

65 pages, many diagrams, minimal use of color, comments welcome!