## Integration over the Pauli quantum group

Citations: | 9 - 8 self |

### BibTeX

@MISC{Banica_integrationover,

author = {Teodor Banica and Benoît Collins},

title = {Integration over the Pauli quantum group},

year = {}

}

### OpenURL

### Abstract

Abstract. We prove that the Pauli representation of the quantum permutation algebra A(S4) is faithful. This provides the second known model for a free quantum algebra. We use this model for performing some computations, with the result that at level of laws of diagonal coordinates, the Lebesgue measure appears between the Dirac mass and the free Poisson law.

### Citations

305 |
Free Random Variables
- Voiculescu, Dykema, et al.
- 1992
(Show Context)
Citation Context ...ements: Definition 3.1. Associated to s = 1,2,3,4 is the average Ms = 1 s (u11 + u22 + ... + uss) where u is the fundamental corepresentation of A(S4). We use notions from free probability from [15], =-=[16]-=-. Recall first that the free Poisson law is the following probability measure on [0,4]: ν = 1 √ 4x−1 − 1dx 2π In this paper we use probability measures supported on [0,1]. So, consider the correspondi... |

261 |
Compact matrix pseudogroups
- WORONOWICZ
- 1987
(Show Context)
Citation Context ...uij) = ∑ uik ⊗ ukj ε(uij) = δij S(uij) = uji are called the comultiplication, counit and antipode of A(Sn). This algebra, discovered by Wang in [18], fits into Woronowicz’s quantum group formalism in =-=[20]-=-. In fact, the quantum group Qn defined by the heuristic formula A(Sn) = C(Qn) is a free analogue of the symmetric group Sn. This quantum group doesn’t exist of course: the idea is just that various p... |

72 | Quantum symmetry groups of finite spaces
- Wang
- 1998
(Show Context)
Citation Context ...t at level of laws of diagonal coordinates, the Lebesgue measure appears between the Dirac mass and the free Poisson law. Introduction The notion of free quantum group appeared in Wang’s papers [17], =-=[18]-=-. The idea is that given a group G ⊂ U(n), the matrix coordinates uij ∈ C(G) commute with each other, and satisfy certain relations R. One can define then the universal algebra A(G) generated by abstr... |

58 |
Free products of compact quantum groups
- WANG
- 1995
(Show Context)
Citation Context ...lt that at level of laws of diagonal coordinates, the Lebesgue measure appears between the Dirac mass and the free Poisson law. Introduction The notion of free quantum group appeared in Wang’s papers =-=[17]-=-, [18]. The idea is that given a group G ⊂ U(n), the matrix coordinates uij ∈ C(G) commute with each other, and satisfy certain relations R. One can define then the universal algebra A(G) generated by... |

57 | Representations of symmetric groups and free probability
- Biane
- 1998
(Show Context)
Citation Context ...e faithful in some weak sense. At that time there was no technique for proving or disproving the fact that π is faithful. This problem can be solved now by using integration techniques, inspired from =-=[10]-=-, [11], [12], [19] and introduced in [7], [8], and our result is that π is faithful. In other words, we have a model for the abstract algebra A(S4). This is the second known model for a free quantum a... |

53 | Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral, and free
- Collins
- 2003
(Show Context)
Citation Context ...hful in some weak sense. At that time there was no technique for proving or disproving the fact that π is faithful. This problem can be solved now by using integration techniques, inspired from [10], =-=[11]-=-, [12], [19] and introduced in [7], [8], and our result is that π is faithful. In other words, we have a model for the abstract algebra A(S4). This is the second known model for a free quantum algebra... |

44 | Integration with respect to the Haar measure on unitary, orthogonal and symplectic group
- Collins
(Show Context)
Citation Context ...d only recently: the idea is that with n → ∞ the integral geometry of G is governed by probabilistic independence, while that of G, by freeness in the sense of Voiculescu. This follows from [7], [8], =-=[12]-=-. (2) A number of subgroups G ⊂ Sn have been considered recently, most of them being symmetry groups of graphs, or of other combinatorial structures. Some freeness appears here as well, for instance i... |

42 |
Free Probability Theory
- Voiculescu
- 1997
(Show Context)
Citation Context ...ing elements: Definition 3.1. Associated to s = 1,2,3,4 is the average Ms = 1 s (u11 + u22 + ... + uss) where u is the fundamental corepresentation of A(S4). We use notions from free probability from =-=[15]-=-, [16]. Recall first that the free Poisson law is the following probability measure on [0,4]: ν = 1 √ 4x−1 − 1dx 2π In this paper we use probability measures supported on [0,1]. So, consider the corre... |

33 |
Le groupe quantique compact libre
- BANICA
- 1997
(Show Context)
Citation Context ... In other words, we have a model for the abstract algebra A(S4). This is the second known model for a free quantum algebra, the first one being a certain embedding A(U(2)) → C(T) ∗ C(SU(2)), found in =-=[1]-=-. (2) The model can be used for working out in detail the integral geometry of the corresponding quantum group. As in our previous work [7], [8], the problem we consider is that of computing laws of s... |

25 | Symmetries of a generic coaction
- Banica
- 1999
(Show Context)
Citation Context ...group. As in our previous work [7], [8], the problem we consider is that of computing laws of sums of diagonal coordinates uii. Each such coordinate is a projection of trace 1/4, and it is known from =-=[2]-=- that the sum u11 + u22 + u33 + u44 is free Poisson on [0,4]. Our main result here is that the law of u11+u22 is an average between a Dirac mass at 0, and the Lebesgue measure on [0,2]. In other words... |

24 | Quantum automorphism groups of small metric spaces
- Banica
(Show Context)
Citation Context ...ight be the “simplest” Hopf algebra, and we believe indeed that it is so. This algebra corresponds for instance to the situation “index 4, graph A∞”, known from work of Jones to be the basic one. See =-=[3]-=- for more comments on this subject. We have two results about A(S4): (1) An explicit realization of the magic condition is found in [9], by using the Pauli matrices. This gives a representation π : A(... |

23 | Asymptotic behavior of group integrals in the limit of infinite rank - Weingarten - 1978 |

18 | Integration over compact quantum groups
- Banica, Collins
(Show Context)
Citation Context ... understood only recently: the idea is that with n → ∞ the integral geometry of G is governed by probabilistic independence, while that of G, by freeness in the sense of Voiculescu. This follows from =-=[7]-=-, [8], [12]. (2) A number of subgroups G ⊂ Sn have been considered recently, most of them being symmetry groups of graphs, or of other combinatorial structures. Some freeness appears here as well, for... |

18 |
The boundary of universal discrete quantum groups, exactness and factoriality
- Vaes, Vergnioux
(Show Context)
Citation Context ...he fact that the irreducible corepresentations have dimensions bigger than those of SO(3), and this makes this algebra non-amenable. In fact, much worse is expected to be true: there is evidence from =-=[14]-=- that the reduced version should be simple, and that the corresponding von Neumann algebra should be a prime II1-factor. In other words, this algebra has bad analytic properties. 2. The Pauli represen... |

15 | Free product formulae for quantum permutation groups
- Banica, Bichon
(Show Context)
Citation Context ...tion G → G transforms usual wreath products into free wreath products. However, the main result so far goes somehow in the opposite sense: for certain classes of groups, with n → ∞ we have G = G. See =-=[4]-=-, [5], [6]. Summarizing, the study of free quantum groups focuses on several ad-hoc constructions G → G. The following two questions are of particular interest in connection with the axiomatization pr... |

13 | Integration over quantum permutation groups
- Banica, Collins
(Show Context)
Citation Context ...rstood only recently: the idea is that with n → ∞ the integral geometry of G is governed by probabilistic independence, while that of G, by freeness in the sense of Voiculescu. This follows from [7], =-=[8]-=-, [12]. (2) A number of subgroups G ⊂ Sn have been considered recently, most of them being symmetry groups of graphs, or of other combinatorial structures. Some freeness appears here as well, for inst... |

7 | Quantum automorphism groups of vertex-transitive graphs of order ≤ 11
- Banica, Bichon
(Show Context)
Citation Context ...G → G transforms usual wreath products into free wreath products. However, the main result so far goes somehow in the opposite sense: for certain classes of groups, with n → ∞ we have G = G. See [4], =-=[5]-=-, [6]. Summarizing, the study of free quantum groups focuses on several ad-hoc constructions G → G. The following two questions are of particular interest in connection with the axiomatization problem... |

7 | On the structure of quantum permutation groups - Banica, Moroianu |

6 |
Graphs having no quantum symmetry, Ann
- Banica, Bichon, et al.
(Show Context)
Citation Context ... transforms usual wreath products into free wreath products. However, the main result so far goes somehow in the opposite sense: for certain classes of groups, with n → ∞ we have G = G. See [4], [5], =-=[6]-=-. Summarizing, the study of free quantum groups focuses on several ad-hoc constructions G → G. The following two questions are of particular interest in connection with the axiomatization problem: whe... |

6 | Graphs having no quantum symmetry
- Banica, Bichon, et al.
(Show Context)
Citation Context ...ransforms usual wreath products into free wreath products. However, the main result so far goes somehow in the opposite sense: for certain classes of groups, with n → ∞ we have G + = G. See [4], [5], =-=[6]-=-. Summarizing, the study of free quantum groups focuses on several ad-hoc constructions G → G + . The following two questions are of particular interest in connection with the axiomatization problem: ... |