Annales Mathématiques
Blaise Pascal
no. 2

A Note on Free Quantum Groups
[Une Note sur les Groupes Quantiques Libres]
Teodor Banica
Annales Mathématiques Blaise Pascal, Tome 15 (2008) no. 2, pp. 135-146.

On étudie l’opération de complexification libre pour les groupes quantiques compacts, G→G c . On montre qu’avec des définitions convenables, cette opération induit une bijection entre groupes quantiques orthogonaux libres de niveau infini, et groupes quantiques unitaires libres satisfaisant G=G c .

We study the free complexification operation for compact quantum groups, G→G c . We prove that, with suitable definitions, this induces a one-to-one correspondence between free orthogonal quantum groups of infinite level, and free unitary quantum groups satisfying G=G c .

DOI : https://doi.org/10.5802/ambp.243
Classification : 16W30
Mots clés : Groupe quantique libre 
@article{AMBP_2008__15_2_135_0,
     author = {Teodor Banica},
     title = {A {Note} on {Free} {Quantum} {Groups}},
     journal = {Annales Math\'ematiques Blaise Pascal},
     pages = {135--146},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {15},
     number = {2},
     year = {2008},
     doi = {10.5802/ambp.243},
     mrnumber = {2468039},
     zbl = {1188.46043},
     language = {en},
     url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.243/}
}
Teodor Banica. A Note on Free Quantum Groups. Annales Mathématiques Blaise Pascal, Tome 15 (2008) no. 2, pp. 135-146. doi : 10.5802/ambp.243. https://ambp.centre-mersenne.org/articles/10.5802/ambp.243/

[1] T. Banica Le groupe quantique compact libre U(n), Comm. Math. Phys., Volume 190 (1997), pp. 143-172 | Article | MR 1484551 | Zbl 0906.17009

[2] T. Banica Representations of compact quantum groups and subfactors, J. Reine Angew. Math., Volume 509 (1999), pp. 167-198 | Article | MR 1679171 | Zbl 0957.46038

[3] T. Banica; J. Bichon; B. Collins The hyperoctahedral quantum group, J. Ramanujan Math. Soc., Volume 22 (2007), pp. 345-384 | MR 2376808 | Zbl pre05252140

[4] T. Banica; B. Collins Integration over compact quantum groups, Publ. Res. Inst. Math. Sci., Volume 43 (2007), p. 377-302 | Article | MR 2341011 | Zbl 1129.46058

[5] A. Nica; R. Speicher Lectures on the combinatorics of free probability, Cambridge University Press, Cambridge, 2006 | MR 2266879 | Zbl 1133.60003

[6] D.V. Voiculescu Circular and semicircular systems and free product factors, Progress in Math., Volume 92 (1990), pp. 45-60 | MR 1103585 | Zbl 0744.46055

[7] S. Wang Free products of compact quantum groups, Comm. Math. Phys., Volume 167 (1995), pp. 671-692 | Article | MR 1316765 | Zbl 0838.46057

[8] S. Wang Quantum symmetry groups of finite spaces, Comm. Math. Phys., Volume 195 (1998), pp. 195-211 | Article | MR 1637425 | Zbl 1013.17008

[9] S.L. Woronowicz Compact matrix pseudogroups, Comm. Math. Phys., Volume 111 (1987), pp. 613-665 | Article | MR 901157 | Zbl 0627.58034

[10] S.L. Woronowicz Tannaka-Krein duality for compact matrix pseudogroups. Twisted SU(N) groups, Invent. Math., Volume 93 (1988), pp. 35-76 | Article | MR 943923 | Zbl 0664.58044