Solution group representations as quantum symmetries of graphs

Roberson, David E.; Schmidt, Simon

doi:10.1112/jlms.12664

Cited by 5 publications

(2 citation statements)

References 14 publications

(35 reference statements)

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“…In the subsequent years, there has been much activity, with the quantum automorphisms of some small graphs computed, and much interest in the question of which graphs have a non-commutative (that is, not arising from a classical group) quantum automorphism group. In this direction, let us just mention one recent paper [37] which constructs a finite graph which has a non-commutative, yet finite-dimensional, quantum automorphism group. More recently quantum automorphisms of quantum graphs have been defined and studied [34, Section V, Part C] and, more completely, [8, Section 3], compare [27,Section 4], [33,Section 2.3].…”

Section: Quantum Automorphisms Of Quantum Graphsmentioning

confidence: 99%

Untitled

2024

Comm. Amer. Math. Soc.

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We undertake a study of the notion of a quantum graph over arbitrary finite-dimensional 𝐶 * -algebras 𝐵 equipped with arbitrary faithful states. Quantum graphs are realised principally as either certain operators on 𝐿 2 (𝐵), the quantum adjacency matrices, or as certain operator bimodules over 𝐵 ′ . We present a simple, purely algebraic approach to proving equivalence between these settings, thus recovering existing results in the tracial state setting. For non-tracial states, our approach naturally suggests a generalisation of the operator bimodule definition, which takes account of (some aspect of) the modular automorphism group of the state. Furthermore, we show that each such "non-tracial" quantum graph corresponds to a "tracial" quantum graph which satisfies an extra symmetry condition. We study homomorphisms (or CPmorphisms) of quantum graphs arising from unital completely positive (UCP) maps, and the closely related examples of quantum graphs constructed from UCP maps. We show that these constructions satisfy automatic bimodule properties. We study quantum automorphisms of quantum graphs, give a definition of what it means for a compact quantum group to act on an operator bimodule, and prove an equivalence between this definition, and the usual notion defined using a quantum adjacency matrix. We strive to give a relatively self-contained, elementary, account, in the hope this will be of use to other researchers in the field.

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Section: Quantum Automorphisms Of Quantum Graphsmentioning

confidence: 99%

Untitled

2024

Comm. Amer. Math. Soc.

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“…In [1], the graphs where constructed from quantum solutions of binary constraint systems. Roberson and the author ( [12]) used colored versions of the graphs constructed in [1] and a decoloring procedure to obtain new quantum isomorphic, non-isomorphic graphs, but those still come from quantum solutions of binary constraint systems. A more general approach was presented by Musto, Reutter and Verdon in [11].…”

Section: Introductionmentioning

confidence: 99%

Quantum isomorphic strongly regular graphs from the E 8 root system

Schmidt

2024

Algebraic Combinatorics

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In this article, we give a first example of a pair of quantum isomorphic, nonisomorphic strongly regular graphs, that is, non-isomorphic strongly regular graphs having the same homomorphism counts from all planar graphs. The pair consists of the orthogonality graph of the 120 lines spanned by the E 8 root system and a rank 4 graph whose complement was first discovered by Brouwer, Ivanov and Klin. Both graphs are strongly regular with parameters (120, 63, 30, 36). Using Godsil-McKay switching, we obtain more quantum isomorphic, non-isomorphic strongly regular graphs with the same parameters.

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Quantum graphs: Different perspectives, homomorphisms and quantum automorphisms

Daws

2024

Comm. Amer. Math. Soc.

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We undertake a study of the notion of a quantum graph over arbitrary finite-dimensional C ∗ C^* -algebras B B equipped with arbitrary faithful states. Quantum graphs are realised principally as either certain operators on L 2 ( B ) L^2(B) , the quantum adjacency matrices, or as certain operator bimodules over B ′ B’ . We present a simple, purely algebraic approach to proving equivalence between these settings, thus recovering existing results in the tracial state setting. For non-tracial states, our approach naturally suggests a generalisation of the operator bimodule definition, which takes account of (some aspect of) the modular automorphism group of the state. Furthermore, we show that each such “non-tracial” quantum graph corresponds to a “tracial” quantum graph which satisfies an extra symmetry condition. We study homomorphisms (or CP-morphisms) of quantum graphs arising from unital completely positive (UCP) maps, and the closely related examples of quantum graphs constructed from UCP maps. We show that these constructions satisfy automatic bimodule properties. We study quantum automorphisms of quantum graphs, give a definition of what it means for a compact quantum group to act on an operator bimodule, and prove an equivalence between this definition, and the usual notion defined using a quantum adjacency matrix. We strive to give a relatively self-contained, elementary, account, in the hope this will be of use to other researchers in the field.

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Solution group representations as quantum symmetries of graphs

Cited by 5 publications

References 14 publications

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Quantum isomorphic strongly regular graphs from the E 8 root system

Quantum graphs: Different perspectives, homomorphisms and quantum automorphisms

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