## Quantum automorphism groups of homogeneous graphs

Venue: | J. Funct. Anal |

Citations: | 23 - 11 self |

### BibTeX

@ARTICLE{Banica_quantumautomorphism,

author = {Teodor Banica},

title = {Quantum automorphism groups of homogeneous graphs},

journal = {J. Funct. Anal},

year = {},

pages = {243--280}

}

### OpenURL

### Abstract

Let X be a finite graph, with edges colored and possibly oriented, such that an oriented edge and a non-oriented one cannot have same color. The universal Hopf algebra H(X) coacting on X is in general non commutative, infinite dimensional, bigger than the algebra of functions on the usual symmetry group G(X). For a graph with no edges Tannakian duality makes H(X) correspond to a Temperley-Lieb algebra. We study some versions of this correspondence.

### Citations

264 | Gravity coupled with matter and the foundation of non-commutative geometry
- Connes
- 1996
(Show Context)
Citation Context ... says that H is the universal Hopf C ∗ -algebra coacting on (A, H, D). It is not clear where (A, H, D) comes from, the main problem here being to find a formulation involving Connes’ spectral triples =-=[14]-=-. Anyway, it is quite reasonable to consider that the “simplest” triple is (C n , C n , 0). The universal Hopf C ∗ -algebra coacting on it is Hn. 12 TEODOR BANICA There is a topological formulation o... |

255 |
Compact matrix pseudogroups
- Woronowicz
- 1987
(Show Context)
Citation Context ...s discovered a few years ago by Wang [30], by a construction with generators and relations. The C ∗ assumption shows that Hn has a Haar functional and is cosemisimple, thanks to results of Woronowicz =-=[33]-=-. The algebra of functions on the usual symmetric group Sn is a quotient of Hn. For n = 1, 2, 3 the quotient map is an isomorphism. For n ≥ 4 it is not. This is because of a free product example, wher... |

107 |
Addition of certain noncommuting random variables
- Voiculescu
- 1986
(Show Context)
Citation Context ... around the Temperley-Lieb algebra, SO(3) or Hn, which are “free” in some sense. For instance their common Poincaré series f satisfies the equation R(f(z 2 )) = z, where R is Voiculescu’s R-transform =-=[29]-=-. It is tempting to include all speculations in a single diagram. An A∞ subfactor can be added, but it is not clear how. In fact this is one of the motivating problems. {:::} −→ (C n , C n , 0) −→ Hn ... |

104 |
Roberts: A new duality theory for compact groups
- Doplicher, E
- 1989
(Show Context)
Citation Context ...licher-Roberts problem. Let C be a tensor C ∗ -category with duals. If C is symmetric and objects have integer dimensions the Tannakian functor exists and is unique, as shown by Doplicher and Roberts =-=[16]-=-. In the general case the problem is harder than the Yang-Baxter equation. However, there is a reasonable question, namely to construct invariants which vanish when the functor exists. So far there is... |

85 |
Tannaka-Krein duality for compact matrix pseudogroups
- Woronowicz
- 1988
(Show Context)
Citation Context ...sifying its irreducible corepresentations, together with their fusion rules. In this section we present a categorical and topological approach to this problem, by using Woronowicz’s Tannakian duality =-=[34]-=- and the spin planar algebra P(X) constructed by Jones in [18], [19]. The main result will be a Tannaka-Galois type correspondence between pairs (H, v) and subalgebras P ⊂ P(X).18 TEODOR BANICA Recal... |

70 | Quantum symmetry groups of finite spaces
- Wang
- 1998
(Show Context)
Citation Context ...erley-Lieb algebra. We study some versions of this correspondence. Introduction There is a universal Hopf C ∗ -algebra Hn coacting on the set {1, . . .,n}. This was discovered a few years ago by Wang =-=[30]-=-, by a construction with generators and relations. The C ∗ assumption shows that Hn has a Haar functional and is cosemisimple, thanks to results of Woronowicz [33]. The algebra of functions on the usu... |

65 | Operator algebras and conformal field theory III, preprint DPMMS
- Wassermann
(Show Context)
Citation Context ...hat in general β must be replaced with a quite complicated invariant, a spherical C ∗ -planar algebra. The analogue of G is a kind of “quantum group” of very general type. See for instance Wassermann =-=[32]-=-. Much simpler constructions, including Goldman’s, use groups or Hopf C ∗ -algebras. There are several interesting questions about these subfactors, related to generalisation of Goldman’s theorem, and... |

61 |
A theory of dimension, K-theory 11
- Longo, Roberts
- 1997
(Show Context)
Citation Context ...r equation. However, there is a reasonable question, namely to construct invariants which vanish when the functor exists. So far there is one invariant, coming from amenability. See Longo and Roberts =-=[24]-=-. In the projective non-braided case the fundamental symmetries come from the action of P, and it is natural to take the “quotient” of the problem by this action. For instance for Hopf C ∗ -algebras g... |

56 | Representations of symmetric groups and free probability
- Biane
- 1998
(Show Context)
Citation Context ...opf C ∗ -algebras of type Hn. These considerations belong somehow to representation theory, and a fundamental problem here is to find how to apply Voiculescu’s free probability techniques, like Biane =-=[7]-=- and Sniady [27] do. It is expected that something should happen around the Temperley-Lieb algebra, SO(3) or Hn, which are “free” in some sense. For instance their common Poincaré series f satisfies t... |

51 | Planar algebras
- Jones
(Show Context)
Citation Context ...ras of depth 1 planar algebras. See Kodiyalam, Landau and Sunder [21] and [2], [4] and section 5 below. The simplest depth 1 planar algebra is probably the spin planar algebra constructed by Jones in =-=[18]-=-, and its simplest subalgebra is the Temperley-Lieb algebra. These data produce Hn. In the subfactor point of view the inspiring result is Goldman’s theorem [17]. This says that subfactors of index β ... |

47 |
Algebras associated to intermediate subfactors
- Bisch, Jones
- 1997
(Show Context)
Citation Context ...t that a product of 2 simplexes gives a Fuss-Catalan algebra on 2 colors is known from [5]. This is a consequence of a straightforward computation in [3], using the presentation result of Bisch-Jones =-=[10]-=-. We give here a simpler proof, which works for s colors. This uses theorem 6.1 and the presentation result of Landau [23]. The Fuss-Catalan algebra on s colors is presented by a sequence of 2-boxes p... |

36 |
Compact quantum groups, Symétries quantiques (Les Houches
- Woronowicz
- 1995
(Show Context)
Citation Context ...tipode. Here ⊗ is any C ∗ -algebra tensor product and H op is the C ∗ -algebra H, but with opposite product. We assume that the square of the antipode is the identity, and that Woronowicz’s axioms in =-=[17]-=- are satisfied. Let X be a finite set. We denote by C(X) the algebra of complex functions on X. The linear form on C(X) which sums the values of the function is denoted Σ. Definition 1.1. A coaction o... |

34 |
Classification of amenable subfactors of type
- Popa
- 1994
(Show Context)
Citation Context ...17]. This says that subfactors of index β = 2 are all isomorphic, and appear as fixed point algebras under actions of G = Z2. Classification of subfactors, culminating with fundamental papers of Popa =-=[25]-=- and Jones [18] shows that in general β must be replaced with a quite complicated invariant, a spherical C ∗ -planar algebra. The analogue of G is a kind of “quantum group” of very general type. See f... |

32 | Representations of compact quantum groups and subfactors
- Banica
- 1999
(Show Context)
Citation Context ...mulation of the random walk approach. Pairs (H, v) are above are expected to be in Tannaka-Galois correspondence with subalgebras of depth 1 planar algebras. See Kodiyalam, Landau and Sunder [21] and =-=[2]-=-, [4] and section 5 below. The simplest depth 1 planar algebra is probably the spin planar algebra constructed by Jones in [18], and its simplest subalgebra is the Temperley-Lieb algebra. These data p... |

26 |
An axiomatization of the lattice of higher relative commutants of a subfactor
- Popa
- 1995
(Show Context)
Citation Context ...ctions, including Goldman’s, use groups or Hopf C ∗ -algebras. There are several interesting questions about these subfactors, related to generalisation of Goldman’s theorem, and to Popa’s problem in =-=[26]-=- about axiomatization of hyperfinite subfactors. From the complexity point of view of Bisch and Jones [12] it is natural to consider first problems involving Hopf C ∗ -algebras of type Hn. These consi... |

24 | Quantum automorphism groups of small metric spaces
- Banica
(Show Context)
Citation Context ...a simplex corresponds to a Temperley-Lieb algebra. This is the same as a Fuss-Catalan algebra on 1 color. The fact that a product of 2 simplexes gives a Fuss-Catalan algebra on 2 colors is known from =-=[5]-=-. This is a consequence of a straightforward computation in [3], using the presentation result of Bisch-Jones [10]. We give here a simpler proof, which works for s colors. This uses theorem 6.1 and th... |

23 | Quantum automorphism groups of finite graphs
- Bichon
(Show Context)
Citation Context ... n , 0) −→ Hn −→ TL(n) −→ 1 − √ 1 − 4z 2z −→ z Very little is known about Hn. In [31] Wang shows that Hn is simple in the compact quantum group sense. Certain quotients of Hn are studied by Bichon in =-=[8]-=-, [9]. A Kesten type criterion in [2] shows that Hn with n ≥ 5 is not amenable in the discrete quantum group sense. Some useful information about Hn might come from its “maximal” quotients. These are ... |

19 | Singly generated planar algebras of small dimension
- Bisch, Jones
(Show Context)
Citation Context ...ut these subfactors, related to generalisation of Goldman’s theorem, and to Popa’s problem in [26] about axiomatization of hyperfinite subfactors. From the complexity point of view of Bisch and Jones =-=[12]-=- it is natural to consider first problems involving Hopf C ∗ -algebras of type Hn. These considerations belong somehow to representation theory, and a fundamental problem here is to find how to apply ... |

14 |
Hopf algebras and subfactors associated to vertex models
- Banica
- 1998
(Show Context)
Citation Context ..., in the sense that both u and the blockwise transpose u t are unitaries, the vertex model is described in some sense by the corepresentation theory of a certain Hopf C ∗ -algebra H(u) constructed in =-=[1]-=-. The magic biunitarity condition appears naturally in this setting, as an intermediate step between spin models and vertex models. Here the vertex model is described by a coaction of H(u) on X. The s... |

13 | Free wreath product by the quantum permutation group
- Bichon
- 2004
(Show Context)
Citation Context ...0) −→ Hn −→ TL(n) −→ 1 − √ 1 − 4z 2z −→ z Very little is known about Hn. In [31] Wang shows that Hn is simple in the compact quantum group sense. Certain quotients of Hn are studied by Bichon in [8], =-=[9]-=-. A Kesten type criterion in [2] shows that Hn with n ≥ 5 is not amenable in the discrete quantum group sense. Some useful information about Hn might come from its “maximal” quotients. These are const... |

10 | Quantum groups and Fuss-Catalan algebras
- Banica
(Show Context)
Citation Context ...same as a Fuss-Catalan algebra on 1 color. The fact that a product of 2 simplexes gives a Fuss-Catalan algebra on 2 colors is known from [5]. This is a consequence of a straightforward computation in =-=[3]-=-, using the presentation result of Bisch-Jones [10]. We give here a simpler proof, which works for s colors. This uses theorem 6.1 and the presentation result of Landau [23]. The Fuss-Catalan algebra ... |

10 | The planar algebra of a bipartite graph - Jones - 2000 |

10 | Strictly outer actions of groups and quantum groups
- Vaes
(Show Context)
Citation Context ...k 5.3. Subfactor problem. Hopf C ∗ -algebras can probably help. Among missing pieces is a theory of outer actions on the hyperfinite factor R. Some recent advances on this subject are made by Vaes in =-=[28]-=-. 6. Exchange relations and multi-simplexes Let X be a finite set and d ∈ MX(C) be a self-adjoint matrix with indices in X. This matrix can be viewed as a 2-box in the spin planar algebra P(X), by usi... |

8 |
The annular structure of subfactors, in Essays on geometry and related topics
- Jones
- 2001
(Show Context)
Citation Context ...| ||| → ⎜ ⎟ Lnm Tn+m | | | an | | ⎜ ⎝ an||| ⎟ ⋒| ⎠ | This problem with two cases n ≤ m and n > m can be avoided by using an uniform approach, with discs with marked points instead of boxes. See Jones =-=[20]-=-. Consider the linear spaces formed by such maps. Qnm = {Lnm(Tn+m) : Pn(X) → Pm(X) | Tn+m ∈ Qn+m} Pictures show that these spaces form a tensor C ∗ -subcategory of the tensor C ∗ -category of linear m... |

8 |
The planar algebra associated to a Kac algebra
- Kodiyalam, Landau, et al.
(Show Context)
Citation Context ...gical formulation of the random walk approach. Pairs (H, v) are above are expected to be in Tannaka-Galois correspondence with subalgebras of depth 1 planar algebras. See Kodiyalam, Landau and Sunder =-=[21]-=- and [2], [4] and section 5 below. The simplest depth 1 planar algebra is probably the spin planar algebra constructed by Jones in [18], and its simplest subalgebra is the Temperley-Lieb algebra. Thes... |

7 |
On biunitary permutations matrices and some subfactors of index 9
- Krishnan, Sunder
- 1996
(Show Context)
Citation Context ...any other models can be used. The 2 rectangles and 2 oriented triangles are probably related to free compositions of Bisch and Jones [11]. The torus graph reminds vertex models of Krishnan and Sunder =-=[22]-=- and Bhattacharyya [6]. A statistical mechanical approach to computation of X → f is developed by Curtin in [15]. The underlying planar algebra is generated by a self-adjoint 2-box, and the whole subj... |

4 |
On subfactors of factors of type II1
- Goldman
- 1960
(Show Context)
Citation Context ...planar algebra constructed by Jones in [18], and its simplest subalgebra is the Temperley-Lieb algebra. These data produce Hn. In the subfactor point of view the inspiring result is Goldman’s theorem =-=[17]-=-. This says that subfactors of index β = 2 are all isomorphic, and appear as fixed point algebras under actions of G = Z2. Classification of subfactors, culminating with fundamental papers of Popa [25... |

4 |
Exchange relation planar algebras, Geometriae Dedicata 95
- Landau
- 2002
(Show Context)
Citation Context ...gs happening at n = 8. Interesting graphs here are the product of 3 segments, the 2 squares, the 8-spoke wheel, the cube, the 2 rectangles. A product of s simplexes gives exchange relations of Landau =-=[23]-=-, and corresponds to a Fuss-Catalan algebra on s colors. This applies to the product of 3 segments, where P = FC(2, 2, 2). A simple modification shows that the same happens for 2 squares. We extend th... |

3 | The planar algebra of a coaction
- Banica
(Show Context)
Citation Context ...ion of the random walk approach. Pairs (H, v) are above are expected to be in Tannaka-Galois correspondence with subalgebras of depth 1 planar algebras. See Kodiyalam, Landau and Sunder [21] and [2], =-=[4]-=- and section 5 below. The simplest depth 1 planar algebra is probably the spin planar algebra constructed by Jones in [18], and its simplest subalgebra is the Temperley-Lieb algebra. These data produc... |

3 |
A note on free composition of subfactors
- Bisch, Jones
- 1997
(Show Context)
Citation Context ...btained by applying planar diagrams to X, then counting pictures. Many other models can be used. The 2 rectangles and 2 oriented triangles are probably related to free compositions of Bisch and Jones =-=[11]-=-. The torus graph reminds vertex models of Krishnan and Sunder [22] and Bhattacharyya [6]. A statistical mechanical approach to computation of X → f is developed by Curtin in [15]. The underlying plan... |

3 | Some planar algebras related to graphs
- Curtin
(Show Context)
Citation Context ...s of Bisch and Jones [11]. The torus graph reminds vertex models of Krishnan and Sunder [22] and Bhattacharyya [6]. A statistical mechanical approach to computation of X → f is developed by Curtin in =-=[15]-=-. The underlying planar algebra is generated by a self-adjoint 2-box, and the whole subject is related to the classification program of Bisch and Jones [12], [13]. For arbitrary n we have the followin... |

2 |
Group actions on graphs related to Krishnan-Sunder subfactors
- Bhattacharyya
- 2003
(Show Context)
Citation Context ... used. The 2 rectangles and 2 oriented triangles are probably related to free compositions of Bisch and Jones [11]. The torus graph reminds vertex models of Krishnan and Sunder [22] and Bhattacharyya =-=[6]-=-. A statistical mechanical approach to computation of X → f is developed by Curtin in [15]. The underlying planar algebra is generated by a self-adjoint 2-box, and the whole subject is related to the ... |

2 | Free probability and representations of large symmetric groups
- Sniady
(Show Context)
Citation Context ...as of type Hn. These considerations belong somehow to representation theory, and a fundamental problem here is to find how to apply Voiculescu’s free probability techniques, like Biane [7] and Sniady =-=[27]-=- do. It is expected that something should happen around the Temperley-Lieb algebra, SO(3) or Hn, which are “free” in some sense. For instance their common Poincaré series f satisfies the equation R(f(... |

2 | Simple compact quantum groups I, preprint - Wang |