Second cohomology groups of the Hopf * -algebras associated to universal unitary quantum groups
Das, Biswarup1; Franz, Uwe2; Kula, Anna1; Skalski, Adam3
1 Instytut Matematyczny, Uniwersytet Wrocławski, pl.Grunwaldzki 2, 50-384 Wrocław (Poland)
2 Laboratoire de mathématiques de Besançon, Université de Bourgogne Franche-Comté, 16, route de Gray, 25030 Besançon cedex (France)
3 Institute of Mathematics of the Polish Academy of Sciences, ul. Śniadeckich 8, 00–656 Warszawa (Poland)
Annales de l'Institut Fourier, Online first, 31 p.

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 U d + . 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 U d + . Dans ce cadre nous sommes obligés d’utiliser des représentations de dimension infinie pour la construction des cocycles.

Online First:
DOI: 10.5802/aif.3527
Classification: 16E40,  16T20
Keywords: Hopf * -algebra, cohomology group, compact quantum group.
Das, Biswarup 1; Franz, Uwe 2; Kula, Anna 1; Skalski, Adam 3

1 Instytut Matematyczny, Uniwersytet Wrocławski, pl.Grunwaldzki 2, 50-384 Wrocław (Poland)
2 Laboratoire de mathématiques de Besançon, Université de Bourgogne Franche-Comté, 16, route de Gray, 25030 Besançon cedex (France)
3 Institute of Mathematics of the Polish Academy of Sciences, ul. Śniadeckich 8, 00–656 Warszawa (Poland) 
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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, Online first, 31 p.

[1] Banica, Teodor Le groupe quantique compact libre U(n), Comm. Math. Phys., Volume 190 (1997) no. 1, pp. 143-172 | DOI | MR | Zbl

[2] Bichon, Julien 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] Bichon, Julien 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] Bichon, Julien Cohomological dimensions of universal cosovereign Hopf algebras, Publ. Mat., Volume 62 (2018) no. 2, pp. 301-330 | DOI | MR | Zbl

[5] Bichon, Julien; Franz, Uwe; Gerhold, Malte 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] Brannan, Michael; Vergnioux, Roland Orthogonal free quantum group factors are strongly 1-bounded, Adv. Math., Volume 329 (2018), pp. 133-156 | DOI | MR | Zbl

[7] Collins, Benoît; Härtel, Johannes; Thom, Andreas Homology of free quantum groups, C. R. Math. Acad. Sci. Paris, Volume 347 (2009) no. 5-6, pp. 271-276 | DOI | MR | Zbl

[8] Das, Biswarup; Franz, Uwe; Kula, Anna; Skalski, Adam 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] Das, Biswarup; Franz, Uwe; Skalski, Adam 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] Franz, Uwe; Gerhold, Malte; Thom, Andreas On the Lévy–Khinchin decomposition of generating functionals, Commun. Stoch. Anal., Volume 9 (2015) no. 4, pp. 529-544 | DOI | MR

[11] Sołtan, Piotr M. Quantum Bohr compactification, Illinois J. Math., Volume 49 (2005) no. 4, pp. 1245-1270 http://projecteuclid.org/euclid.ijm/1258138137 | DOI | MR | Zbl

[12] Timmermann, Thomas 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] Van Daele, Alfons; Wang, Shuzhou Universal quantum groups, Internat. J. Math., Volume 7 (1996) no. 2, pp. 255-263 | DOI | MR | Zbl

[14] Woronowicz, S. L. Compact matrix pseudogroups, Comm. Math. Phys., Volume 111 (1987) no. 4, pp. 613-665 http://projecteuclid.org/euclid.cmp/1104159726 | MR | Zbl

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