Equivariant Eilenberg-Watts theorems for locally compact quantum groups

Abstract: Given actions of a locally compact quantum group $G$ on the von Neumann algebras $A$ and $B$, we can associate to it the category $\operatorname{Corr}^G(A,B)$ of G-A-B-correspondences. Special cases of this category include the category $\operatorname{Rep}(A)$ of unital, normal $*$-representations of $A$ on Hilbert spaces and the category $\operatorname{Rep}^G(A)$ of unital, normal, $G$-representations on Hilbert spaces. We construct actions $\operatorname{Rep}^G(A)\curvearrowleft \operatorname{Rep}(G)$ and $\operatorname{Rep}(A)\curvearrowleft \operatorname{Rep}(\hat{G})$, providing us with natural examples of module categories. We show that the categories of module functors $\operatorname{Rep}(B)\to \operatorname{Rep}(A)$ and $\operatorname{Rep}^G(B)\to \operatorname{Rep}^G(A)$ are both equivalent to the category of $G$-$A$-$B$-correspondences, providing equivariant versions of the von Neumann algebraic Eilenberg-Watts theorem.

category theoryoperator algebrasquantum algebra

Audience: researchers in the topic

Quantum Groups Seminar [QGS]

Series comments: This online seminar aims to bring together experts in the area of quantum groups. The seminar topics will cover the theory of quantum groups and related structures in a large sense: Hopf algebras, operator algebras, q-deformations, higher categories and related branches of noncommutative mathematics.

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