arXiv subjects
Publications and source records attributed to Junichiro Matsuda.
We study approximate representations of locally compact quantum groups and prove stability results in the sense of Ulam in this context. Our main result is that compact and amenable discrete quantum groups are representation stable. We also show that an analogous stability result holds for unitary compressions of general amenable locally compact quantum groups without the assumption of compactness or discreteness.
We establish a quantum version of Frucht's Theorem, proving that every finite quantum group is the quantum automorphism group of an undirected finite quantum graph. The construction is based on first considering several quantum Cayley graphs of the quantum group in question, and then providing a method to systematically combine them into a single quantum graph with the right symmetry properties. We also show that the dual $\widehat \Gamma$ of any non-abelian finite group $\Gamma$ is ``quantum rigid''. That is, $\widehat \Gamma$ always admits a quantum Cayley graph whose quantum automorphism group is exactly $\widehat \Gamma$.
Connectedness and bipartiteness are basic properties of classical graphs, and the purpose of this paper is to investigate the case of quantum graphs. We introduce the notion of connectedness and bipartiteness of quantum graphs in terms of graph homomorphisms. This paper shows that regular tracial quantum graphs have the same algebraic characterization of connectedness and bipartiteness as classical graphs. We also prove the equivalence between bipartiteness and two-colorability of quantum graphs by comparing two notions of graph homomorphisms respecting adjacency matrices or edge spaces. In particular, all kinds of quantum two-colorability are mutually equivalent for regular connected tracial quantum graphs.
Motivated by string diagrammatic approach to undirected tracial quantum graphs by Musto, Reutter, Verdon (2018), in the former part of this paper we diagrammatically formulate directed nontracial quantum graphs by Brannan, Chirvasitu, Eifler, Harris, Paulsen, Su, Wasilewski (2019). In the latter part, we supply a concrete classification of undirected reflexive quantum graphs on $M_2$ and their quantum automorphism groups in both tracial and nontracial settings. We also obtain quantum isomorphisms between tracial quantum graphs on $M_2$ and certain classical graphs, which reproves the monoidal equivalences between $SO(3)$ and $S_4^+$, and $O(2)$ and $H_2^+$.