We compute the second (and the first) cohomology groups of -algebras associated with the universal quantum unitary groups of not necessarily Kac type, extending our earlier results for the free unitary group . The extended setup forces us to use infinite-dimensional representations to construct the cocycles.
Nous calculons les deuxièmes (et premiers) groupes de cohomologi des algèbres involutives associées aux groupes quantiques unitaires universels pas nécessairement du type Kac, étendant ainsi nos résultats précédents pour le groupe unitaire libre . Dans ce cadre nous sommes obligés d’utiliser des représentations de dimension infinie pour la construction des cocycles.
Revised:
Accepted:
Published online:
Keywords: Hopf $^*$-algebra, cohomology group, compact quantum group.
Mot clés : Algèbre de Hopf involutive, groupe de cohomologie, groupe quantique compact.
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@article{AIF_2023__73_2_479_0,
author = {Das, Biswarup and Franz, Uwe and Kula, Anna and Skalski, Adam},
title = {Second cohomology groups of the {Hopf}$^*$-algebras associated to universal unitary quantum groups},
journal = {Annales de l'Institut Fourier},
pages = {479--509},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {73},
number = {2},
year = {2023},
doi = {10.5802/aif.3527},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3527/}
}
TY - JOUR AU - Das, Biswarup AU - Franz, Uwe AU - Kula, Anna AU - Skalski, Adam TI - Second cohomology groups of the Hopf$^*$-algebras associated to universal unitary quantum groups JO - Annales de l'Institut Fourier PY - 2023 SP - 479 EP - 509 VL - 73 IS - 2 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3527/ DO - 10.5802/aif.3527 LA - en ID - AIF_2023__73_2_479_0 ER -
%0 Journal Article %A Das, Biswarup %A Franz, Uwe %A Kula, Anna %A Skalski, Adam %T Second cohomology groups of the Hopf$^*$-algebras associated to universal unitary quantum groups %J Annales de l'Institut Fourier %D 2023 %P 479-509 %V 73 %N 2 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3527/ %R 10.5802/aif.3527 %G en %F AIF_2023__73_2_479_0
Das, Biswarup; Franz, Uwe; Kula, Anna; Skalski, Adam. Second cohomology groups of the Hopf$^*$-algebras associated to universal unitary quantum groups. Annales de l'Institut Fourier, Volume 73 (2023) no. 2, pp. 479-509. doi : 10.5802/aif.3527. https://aif.centre-mersenne.org/articles/10.5802/aif.3527/
[1] Le groupe quantique compact libre , Comm. Math. Phys., Volume 190 (1997) no. 1, pp. 143-172 | DOI | MR | Zbl
[2] Hochschild homology of Hopf algebras and free Yetter-Drinfeld resolutions of the counit, Compos. Math., Volume 149 (2013) no. 4, pp. 658-678 | DOI | MR | Zbl
[3] Homological invariants of discrete quantum groups, Lecture notes for a mini-course given at Seoul (2017) (available at http://math.univ-bpclermont.fr/~bichon/Seoul.pdf)
[4] Cohomological dimensions of universal cosovereign Hopf algebras, Publ. Mat., Volume 62 (2018) no. 2, pp. 301-330 | DOI | MR | Zbl
[5] Homological properties of quantum permutation algebras, New York J. Math., Volume 23 (2017), pp. 1671-1695 http://nyjm.albany.edu:8000/j/2017/23_1671.html | MR | Zbl
[6] Orthogonal free quantum group factors are strongly 1-bounded, Adv. Math., Volume 329 (2018), pp. 133-156 | DOI | MR | Zbl
[7] Homology of free quantum groups, C. R. Math. Acad. Sci. Paris, Volume 347 (2009) no. 5-6, pp. 271-276 | DOI | MR | Zbl
[8] Lévy-Khintchine decompositions for generating functionals on algebras associated to universal compact quantum groups, Infin. Dimens. Anal. Quantum Probab. Relat. Top., Volume 21 (2018) no. 3, 1850017, 36 pages | DOI | MR | Zbl
[9] The RFD and Kac quotients of the Hopf*-algebras of universal orthogonal quantum groups, Ann. Math. Blaise Pascal, Volume 28 (2021) no. 2, pp. 141-155 | DOI | MR | Zbl
[10] On the Lévy–Khinchin decomposition of generating functionals, Commun. Stoch. Anal., Volume 9 (2015) no. 4, pp. 529-544 | DOI | MR
[11] Quantum Bohr compactification, Illinois J. Math., Volume 49 (2005) no. 4, pp. 1245-1270 http://projecteuclid.org/euclid.ijm/1258138137 | DOI | MR | Zbl
[12] An invitation to quantum groups and duality. From Hopf algebras to multiplicative unitaries and beyond, EMS Textbooks in Mathematics, European Mathematical Society (EMS), Zürich, 2008, xx+407 pages | DOI | MR | Zbl
[13] Universal quantum groups, Internat. J. Math., Volume 7 (1996) no. 2, pp. 255-263 | DOI | MR | Zbl
[14] Compact matrix pseudogroups, Comm. Math. Phys., Volume 111 (1987) no. 4, pp. 613-665 http://projecteuclid.org/euclid.cmp/1104159726 | DOI | MR | Zbl
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