Equivariant Eilenberg-Watts theorems for locally compact quantum groups
Joeri De Ro
TL;DR
This work generalizes the W*-algebraic Eilenberg–Watts theorem to the setting of locally compact quantum groups by formulating and proving equivariant equivalences between categories of correspondences and module functors. Using the framework of $\mathrm{Rep}(\mathbb{G})$-module categories, the authors establish canonical equivalences Corr$^{\mathbb{G}}(A,B) \simeq$ Fun$_{\mathrm{Rep}(\mathbb{G})}(\mathrm{Rep}^{\mathbb{G}}(B), \mathrm{Rep}^{\mathbb{G}}(A))$ and Corr$^{\mathbb{G}}(A,B) \simeq$ Fun$_{\mathrm{Rep}(\hat{\mathbb{G}})}(\mathrm{Rep}(B), \mathrm{Rep}(A))$, with Quillen-type Morita reductions allowing passage to the crossed-product pictures. A parallel TEXT-module version is developed, and the Drinfeld center result is extended to LCQGs by showing Rep$(D(\mathbb{G})) \cong \mathcal{Z}(\mathrm{Rep}(\mathbb{G}))$, thus unifying representation theory with categorical half-braidings in this analytic setting. The results provide a robust category-theoretic toolkit for equivariant Morita theory and representation theory of locally compact quantum groups, including integrable Morita equivalences and duality phenomena linked to crossed products and Drinfeld doubles.
Abstract
Given two von Neumann algebras $A$ and $B$, the $W^*$-algebraic Eilenberg-Watts theorem, due to M. Rieffel, asserts that there is a canonical equivalence $\operatorname{Corr}(A,B)\simeq \operatorname{Fun}(\operatorname{Rep}(B), \operatorname{Rep}(A))$ of categories, where $\operatorname{Corr}(A,B)$ denotes the category of all $A$-$B$-correspondences, $\operatorname{Rep}(A)$ is the category of all unital normal $*$-representations of $A$ on Hilbert spaces and $\operatorname{Fun}(\operatorname{Rep}(B), \operatorname{Rep}(A))$ denotes the category of all normal $*$-functors $\operatorname{Rep}(B)\to \operatorname{Rep}(A)$. In this paper, we upgrade the von Neumann algebras $A$ and $B$ with actions $A\curvearrowleft \mathbb{G}$ and $B\curvearrowleft \mathbb{G}$ of a locally compact quantum group $\mathbb{G}$, and we provide several equivariant versions of the $W^*$-algebraic Eilenberg-Watts theorem using the language of module categories. We also prove that for a locally compact quantum group $\mathbb{G}$ with Drinfeld double $D(\mathbb{G})$, the category of unitary $D(\mathbb{G})$-representations is isomorphic to the Drinfeld center of $\operatorname{Rep}(\mathbb{G})$, generalizing a result by Neshveyev-Yamashita from the compact to the locally compact setting.