Projective representations from quantum enhanced graph symmetries

We define re–gaugings and enhanced symmetries for graphs with group labels on their edges. These give rise to interesting projective representations of subgroups of the automorphism groups of the graphs. We furthermore embed this construction into several higher levels of generalization using category theory and show that they are natural in that language. These include projective representations of the re–gauging groupoid and a novel generalization to all symmetries of the graph. Introduction In [4], we developed a method of re–gaugings and actions of enhanced graphs symmetries for labelled graphs. The upshot were projective representations that are of interest in condensed matter physics. In those applications there is an underlying geometry at work, but the method itself is more general and can be generalized or reduced to a combinatorial group theoretic framework, which we will present. Presently, we will label the edges of a graph by elements of a group G and we give a presentation of the actions that is precise and concise. The precision is needed, since there are several actions (both left and right) at work which need to be disentangled. We present a new result on the action of general symmetries. We will also recall the particular projective representations we found in [4] since their occurrence “in nature” as natural symmetries might be of interest to group theorists as well as physicists and give an example of the new result on the action of general symmetries. Lastly, we give a new presentation of our constructions in the language of categories and show that they become very natural. This paper is at the same time more general and more specific than [4]. In loc. cit. the labels were invertible elements of a C∗–algebra, here they live in a general group. The geometry of [4] is then recovered by specializing the group to invertible functions on tori, or more generally invertible elements in a not necessarily commutative C∗ algebra. We also give a more technically precise account of the actions. On the other hand, [4] deals with the possibility of an actual groupoid representation, that is invertible morphisms between different vector spaces, while here, we work in the situation where there is only one underlying vector space. The categorical interpretation is entirely new. 30th International Colloquium on Group Theoretical Methods in Physics (Group30) IOP Publishing Journal of Physics: Conference Series 597 (2015) 012048 doi:10.1088/1742-6596/597/1/012048 Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1 1. Combinatorics and graphs In this section, we give the details of the following construction of representations: Fix a connected graph Γ with k vertices and a labelling lab of its directed edges by group elements of a fixed group G such that edges in opposite orientation are labelled by inverse group elements. We define a group of quantum automorphisms Autq(Γ, lab) of such a Γ–labelled graph. This is a subgroup of the automorphisms of the graph, which preserves the labels up to re-gauging, defined below. We show that after fixing an ordered rooted spanning tree (ORST) τ of the graph, there is a natural way to attach a k × k matrix with coefficients in G to each quantum automorphism. In the case that G is Abelian these matrices give a projective representation of the group of quantum automorphisms of (Γ, lab), into the group of k×k matrix with coefficients in G. The co–cycle is explicitly given and defines a group extension Âutq(Γ, lab). Thus applying any representation to G, we get a projective representation of Autq(Γ, lab) and a representation of Âutq(Γ, lab). A different choice of ORST gives rise to a projective representation. 1.1. Graphs, paths and spanning trees In this paragraph, we fix the notations we will be using. This is necessary to be able to be precise later on. A graph is a collection of vertices V , flags or half–edges F , a boundary map ∂ : F → V which attaches to each half–edge its vertex and a fixed point free involution ı : F → F on F . An edge is then a pair of half–edges {f1, f2} constituting an orbit of ı: f1 = ı(f2). We denote the set of edges by E. Then ∂ associates to each edge the set of its endpoints. An orientation of an edge is the choice of ordering of its two flags. Each edge has two possible orientations, which we call opposite. For an oriented edge → e= (f1, f2) we set s(e) = ∂(f1) and t(e) = ∂(f2) and call them source and target. Notice that for small loops, whose flags are incident to the same vertex, both orientations have the same sources and targets. This is why we chose to use the more elaborate way to present graphs above. We can also identify an orientation of an edge by the choice of the first half edge. In this way the set F is naturally the set of all oriented edges. The bijection is given by f ↔ (f, ı(f)). Using this bijection, we define s(f) = f and t(f) = ı(f) and from now on think of flags as oriented edges. In this notation ı flips the orientation if f ↔ (f, ı(f)) =e then ı(f)↔ (ı(f), f) =: ı(e ) =:e . 1.1.1. Spanning trees, orders and action of the permutation groups A spanning tree of a graph is a subgraph that is a tree (i.e. its realization is contractible) whose vertices are all the vertices of the tree. A rooted spanning tree (RST) is such a spanning tree together with the choice of a root. The edges in a rooted tree have a natural orientation by directing the edges away from the root; see Figure 1. In any rooted tree τ , any vertex v has a unique shortest path to and from the root vertex vrt, which we will call γ τ vrtv and γ τ vvrt = (γ τ vrtv) −1. An order on a graph is a bijection ord : V → {1, . . . , |V |}. A compatible order for a rooted spanning tree has to have ord(vrt) = 1. Denote v = ord −1 and write v(i) = vi then ord(vi) = i, and for a rooted spanning tree with a compatible order vrt = v1 1. An ordered rooted spanning tree (ORST) is a spanning tree together with a compatible order. Notice that an order for a spanning tree gives an order for the graph. The permutation group Sk, k = |V |, naturally acts on orders and inverse orders via σ(v) = v ◦ σ−1. Setting v′ = σ(v) this means that v′ i = vσ−1(i). Furthermore, σ ∈ Sn induces a bijection σV : V → V on the set of vertices via σV ◦ v = v ◦ σ−1 (1.1) that is σV = v ◦ σ−1 ◦ ord. In other words, σV (vi) = v′ i = vσ−1(i). And vice–versa given ΣV : V → V a bijection, it determines an element σ of Sk via σ = ord ◦ σ−1 V ◦ v. 1 In [4] we used v0 for the root, which would mean that v0 = v1. 30th International Colloquium on Group Theoretical Methods in Physics (Group30) IOP Publishing Journal of Physics: Conference Series 597 (2015) 012048 doi:10.1088/1742-6596/597/1/012048