Equivariant Eilenberg-Watts theorems for locally compact quantum groups
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909887784550400 |
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| author | De Ro, Joeri |
| author_facet | De Ro, Joeri |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_06206 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Equivariant Eilenberg-Watts theorems for locally compact quantum groups De Ro, Joeri Operator Algebras Category Theory Quantum Algebra 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. |
| title | Equivariant Eilenberg-Watts theorems for locally compact quantum groups |
| topic | Operator Algebras Category Theory Quantum Algebra |
| url | https://arxiv.org/abs/2510.06206 |