Quantum symmetries of graph C^{*}-algebras

Quantum symmetries of graph -algebras

Simon Schmidt  and  Moritz Weber Saarland University, Fachbereich Mathematik, 66041 Saarbrücken, Germany simon.schmidt@math.uni-sb.de, weber@math.uni-sb.de
July 15, 2019
Abstract.

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)
The second author was partially funded by the ERC Advanced Grant NCDFP, held by Roland Speicher. This work was part of the first author’s Master’s thesis. This work was also supported by the DFG project Quantenautomorphismen von Graphen

Introduction

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:

Comments 0
Request Comment
You are adding the first comment!
How to quickly get a good reply:
  • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
  • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
  • Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
""
The feedback must be of minimum 40 characters and the title a minimum of 5 characters
   
Add comment
Cancel
Loading ...
254417
This is a comment super asjknd jkasnjk adsnkj
Upvote
Downvote
""
The feedback must be of minumum 40 characters
The feedback must be of minumum 40 characters
Submit
Cancel

You are asking your first question!
How to quickly get a good answer:
  • Keep your question short and to the point
  • Check for grammar or spelling errors.
  • Phrase it like a question
Test
Test description