Deligne defined symmetric monoidal categories Rep(GL_t), Rep(O_t) and Rep(S_t) (t in C) which interpolate the usual finite dimensional representation categories of GL(n), O(n) and S_n to complex parameters t. Even earlier, Turaev had defined quantized versions of the GL_t and O_t categories. They are also known as the idempotent completions of the oriented skein category (the GL_t case) and Kauffmann category (the O_t case). I will explain why I am interested in these categories and what I would like to know about them. The main result is a classification of thick tensor ideals for Rep(U_q(o_t))$ if $q$ is generic (joint work with Hans Wenzl). If time permits and I feel ambitious, I might say something about the root of unity case.

A quiver quantum loop algebra is a quantum algebra U(Q,T) attached to a quiver Q and a collection T of non-zero complex parameters assigned to its edges. This construction generalizes several familiar families of algebras: quantum affine and quantum toroidal algebras associated with finite-dimensional simple Lie algebras, in Drinfeld’s loop presentation; more generally, the quantum affinizations studied by Hernandez; and the K-theoretic Hall algebras of nilpotent representations of preprojective algebras, as presented by Negut, Sala, and Schiffmann.

In this talk, I will define the algebras U(Q,T) and introduce a natural category O of representations, equipped with a rational tensor structure arising from the Drinfeld coproduct. I will then explain how an abelian q-difference equation emerges in this setting, and how its canonical solutions near zero and infinity give rise to two meromorphic braidings on O, related by unitarity.

This construction extends the analogous finite-type result of Gautam and Toledano Laredo. The resulting formulae for abelian R-matrices generalize those of Damiani in finite type and of Hernandez for quantum affinizations. This is joint work in progress with Sachin Gautam.