Quantum symmetries of graph -algebras
The study of graph -algebras has a long history in operator algebras. Surprisingly, their quantum symmetries have never been computed so far. We close this gap by proving that the quantum automorphism group of a finite, directed graph without multiple edges acts maximally on the corresponding graph -algebra. This shows that the quantum symmetry of a graph coincides with the quantum symmetry of the graph -algebra. In our result, we use the definition of quantum automorphism groups of graphs as given by Banica in 2005. Note that Bichon gave a different definition in 2003; our action is inspired from his work. We review and compare these two definitions and we give a complete table of quantum automorphism groups (with respect to either of the two definitions) for undirected graphs on four vertices.
Key words and phrases:finite graphs, graph automorphisms, automorphism groups, quantum automorphisms, graph -algebras, quantum groups, quantum symmetries
2010 Mathematics Subject Classification:46LXX (Primary); 20B25, 05CXX (Secondary)
Symmetry constitutes one of the most important properties of a graph. It is captured by its automorphism group
where is a finite graph with vertices and no multiple edges, is its adjacency matrix, and is the symmetric group. In modern mathematics, notably in operator algebras, symmetries are no longer described only by groups, but by quantum groups. In 2005, Banica [QBan] gave a definition of a quantum automorphism group of a finite graph within Woronowicz’s theory of compact matrix quantum groups [CMQG2]. In our notation, is based on the -algebra
where is Wang’s quantum symmetric group [WanSn] and are the relations
Earlier, in 2003, Bichon [QBic] defined a quantum automorphism group via
where are the relations
and and are range and source maps respectively. We immediately see that
holds, in the sense that there are surjective -homomorphisms:
Relatively little is known about these two quantum automorphism groups of graphs and we refer to Section LABEL:SectLit for an overview on all published articles in this area.
Graph -algebras in turn are well-established objects in operator algebras. They emerged from Cuntz and Krieger’s work [CK] in the 1980’s and they developed to be one of the most important classes of examples of -algebras, see for instance Raeburn’s book for an overview [Rae]. Given a finite graph the associated graph -algebra is defined as
A natural question is then: What is the quantum symmetry group of the graph -algebra and is it one of the above two quantum automorphism groups of the underlying graphs? The answer is: It is given by the one defined by Banica. Note however, that Bichon’s definition has its justification in other contexts such as in [GoBhSk, Bic2] or in the recent work by Speicher and the second author [SW16]. Moreover, Bichon’s work [QBic] inspired us how to formulate our main theorem, see also Remark LABEL:RemBicInsp.
1. Main result
Intuitively speaking, our main result is that the quantum symmetry of a finite graph without multiple edges coincides with the quantum symmetry of the associated graph -algebra. In other words, the following diagram is commutative: