arXiv subjects

Lukas Rollier

Publications and source records attributed to Lukas Rollier.

3 recordsLinked to original sources

arXiv2025

Equivariant representation theory for proper actions on discrete spaces

Starting from any proper action of any locally compact quantum group on any discrete quantum space, we show that its equivariant representation theory yields a concrete unitary 2-category of finite type Hilbert bimodules over the discrete quantum space, from which the quantum group and its action may be completely reconstructed as in a previous article by the author. In particular, this shows that any locally compact quantum group acting properly on a discrete quantum space must be an algebraic quantum group.

math.OA↗

arXiv2024

Equivariant Tannaka-Krein reconstruction and quantum automorphism groups of discrete structures

We define quantum automorphism groups of a wide range of discrete structures. The central tool for their construction is a generalisation of the Tannaka-Krein reconstruction theorem. For any direct sum of matrix algebras $M$, and any concrete unitary 2-category of finite type Hilbert-$M$-bimodules $\mathcal{C}$, under reasonable conditions, we construct an algebraic quantum group $\mathbb{G}$ which acts on $M$ by $α$, such that the category of $α$-equivariant corepresentations of $\mathbb{G}$ on finite type Hilbert-$M$-bimodules is equivalent to $\mathcal{C}$. Moreover, we explicitly describe how to get such categories from connected locally finite discrete structures. As an example, we define the quantum automorphism group of a quantum Cayley graph.

math.OA↗

arXiv2023

Quantum automorphism groups of connected locally finite graphs and quantizations of discrete groups

We construct for every connected locally finite graph $Π$ the quantum automorphism group $\text{QAut}\ Π$ as a locally compact quantum group. When $Π$ is vertex transitive, we associate to $Π$ a new unitary tensor category $\mathcal{C}(Π)$ and this is our main tool to construct the Haar functionals on $\text{QAut}\ Π$. When $Π$ is the Cayley graph of a finitely generated group, this unitary tensor category is the representation category of a compact quantum group whose discrete dual can be viewed as a canonical quantization of the underlying discrete group. We introduce several equivalent definitions of quantum isomorphism of connected locally finite graphs $Π$, $Π'$ and prove that this implies monoidal equivalence of $\text{QAut}\ Π$ and $\text{QAut}\ Π'$.

math.QA↗