Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 082, 40 pages      arXiv:2305.12494      https://doi.org/10.3842/SIGMA.2024.082

First Cohomology with Trivial Coefficients of All Unitary Easy Quantum Group Duals

Alexander Mang
Hamburg University, Bundesstraße 55, 20146 Hamburg, Germany

Received September 24, 2023, in final form August 22, 2024; Published online September 12, 2024

Abstract
The first quantum group cohomology with trivial coefficients of the discrete dual of any unitary easy quantum group is computed. That includes those potential quantum groups whose associated categories of two-colored partitions have not yet been found.

Key words: discrete quantum group; quantum group cohomology; trivial coefficients; easy quantum group; category of partitions.

pdf (739 kb)   tex (67 kb)  

References

  1. Banica T., Bichon J., Complex analogues of the half-classical geometry, Münster J. Math. 10 (2017), 457-483, arXiv:1703.03970.
  2. Banica T., Bichon J., Matrix models for noncommutative algebraic manifolds, J. Lond. Math. Soc. 95 (2017), 519-540, arXiv:1606.01115.
  3. Banica T., Curran S., Speicher R., Classification results for easy quantum groups, Pacific J. Math. 247 (2010), 1-26, arXiv:0906.3890.
  4. Banica T., Speicher R., Liberation of orthogonal Lie groups, Adv. Math. 222 (2009), 1461-1501, arXiv:0808.2628.
  5. Bhowmick J., D'Andrea F., Das B., Dąbrowski L., Quantum gauge symmetries in noncommutative geometry, J. Noncommut. Geom. 8 (2014), 433-471, arXiv:1112.3622.
  6. Bhowmick J., D'Andrea F., Dąbrowski L., Quantum isometries of the finite noncommutative geometry of the standard model, Comm. Math. Phys. 307 (2011), 101-131, arXiv:1009.2850.
  7. Bichon J., Franz U., Gerhold M., Homological properties of quantum permutation algebras, New York J. Math. 23 (2017), 1671-1695, arXiv:1704.00589.
  8. Cébron G., Weber M., Quantum groups based on spatial partitions, Ann. Fac. Sci. Toulouse Math. 32 (2023), 727-768, arXiv:1609.02321.
  9. Das B., Franz U., Kula A., Skalski A., Lévy-Khintchine decompositions for generating functionals on algebras associated to universal compact quantum groups, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 21 (2018), 1850017, 36 pages, arXiv:1711.02755.
  10. Das B., Franz U., Kula A., Skalski A., Second cohomology groups of the Hopf*-algebras associated to universal unitary quantum groups, Ann. Inst. Fourier (Grenoble) 73 (2023), 479-509, arXiv:2104.07933.
  11. Dijkhuizen M.S., Koornwinder T.H., CQG algebras: a direct algebraic approach to compact quantum groups, Lett. Math. Phys. 32 (1994), 315-330, arXiv:hep-th/9406042.
  12. Franz U., Gerhold M., Thom A., On the Lévy-Khinchin decomposition of generating functionals, Commun. Stoch. Anal. 9 (2015), 529-544, arXiv:1510.03292.
  13. Freslon A., On the partition approach to Schur-Weyl duality and free quantum groups, Transform. Groups 22 (2017), 707-751, arXiv:1409.1346.
  14. Ginzburg V., Calabi-Yau algebras, arXiv:math.AG/0612139.
  15. Gromada D., Classification of globally colorized categories of partitions, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 21 (2018), 1850029, 25 pages, arXiv:1805.10800.
  16. Hochschild G., Relative homological algebra, Trans. Amer. Math. Soc. 82 (1956), 246-269.
  17. Kustermans J., Locally compact quantum groups in the universal setting, Internat. J. Math. 12 (2001), 289-338, arXiv:math.OA/9902015.
  18. Kustermans J., Locally compact quantum groups, in Quantum Independent Increment Processes. I, Lecture Notes in Math., Vol. 1865, Springer, Berlin, 2005, 99-180.
  19. Kustermans J., Vaes S., Locally compact quantum groups, Ann. Sci. École Norm. Sup. 33 (2000), 837-934.
  20. Kustermans J., Vaes S., Locally compact quantum groups in the von Neumann algebraic setting, Math. Scand. 92 (2003), 68-92, arXiv:math.OA/0005219.
  21. Kyed D., L2-invariants for quantum groups, Ph.D. Thesis, University of Copenhagen, 2008, available at https://www.imada.sdu.dk/u/dkyed/PhD-thesis-final-hyperref.pdf.
  22. Kyed D., Raum S., On the $\ell^2$-Betti numbers of universal quantum groups, Math. Ann. 369 (2017), 957-975, arXiv:1610.05474.
  23. Maaßen L., Representation categories of compact matrix quantum groups, Ph.D. Thesis, RWTH Aachen University, 2021, available at https://doi.org/10.18154/RWTH-2021-06610.
  24. Mančinska L., Roberson D.E., Quantum isomorphism is equivalent to equality of homomorphism counts from planar graphs, in 2020 IEEE 61st Annual Symposium on Foundations of Computer Science, IEEE Computer Soc., Los Alamitos, CA, 2020, 661-672, arXiv:1910.06958.
  25. Mang A., Classification and homological invariants of compact quantum groups of combinatorial type, Ph.D. Thesis, Universität des Saarlandes, 2022, available at https://doi.org/10.22028/D291-39286.
  26. Mang A., Weber M., Categories of two-colored pair partitions part I: categories indexed by cyclic groups, Ramanujan J. 53 (2020), 181-208, arXiv:1809.06948.
  27. Mang A., Weber M., Categories of two-colored pair partitions Part II: Categories indexed by semigroups, J. Combin. Theory Ser. A 180 (2021), 105409, 43 pages, arXiv:1901.03266.
  28. Mang A., Weber M., Non-hyperoctahedral categories of two-colored partitions part I: new categories, J. Algebraic Combin. 54 (2021), 475-513, arXiv:1907.11417.
  29. Mang A., Weber M., Non-hyperoctahedral categories of two-colored partitions Part II: All possible parameter values, Appl. Categ. Structures 29 (2021), 951-982, arXiv:2003.00569.
  30. Raum S., Weber M., The combinatorics of an algebraic class of easy quantum groups, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 17 (2014), 1450016, 17 pages, arXiv:1312.1497.
  31. Raum S., Weber M., Easy quantum groups and quantum subgroups of a semi-direct product quantum group, J. Noncommut. Geom. 9 (2015), 1261-1293, arXiv:1311.7630.
  32. Raum S., Weber M., The full classification of orthogonal easy quantum groups, Comm. Math. Phys. 341 (2016), 751-779, arXiv:1312.3857.
  33. Tarrago P., Weber M., Unitary easy quantum groups: the free case and the group case, Int. Math. Res. Not. 2017 (2017), 5710-5750, arXiv:1512.00195.
  34. Tarrago P., Weber M., The classification of tensor categories of two-colored noncrossing partitions, J. Combin. Theory Ser. A 154 (2018), 464-506.
  35. Van Daele A., Discrete quantum groups, J. Algebra 180 (1996), 431-444.
  36. Van Daele A., An algebraic framework for group duality, Adv. Math. 140 (1998), 323-366.
  37. Van Daele A., Wang S., Universal quantum groups, Internat. J. Math. 7 (1996), 255-263.
  38. Wang S., Free products of compact quantum groups, Comm. Math. Phys. 167 (1995), 671-692.
  39. Weber M., On the classification of easy quantum groups, Adv. Math. 245 (2013), 500-533, arXiv:1201.4723.
  40. Wendel A., Hochschild cohomology of free easy quantum groups, Bachelor's Thesis, Saarland University, 2020.
  41. Woronowicz S.L., Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), 613-665.
  42. Woronowicz S.L., Tannaka-Krein duality for compact matrix pseudogroups. Twisted ${\rm SU}(N)$ groups, Invent. Math. 93 (1988), 35-76.
  43. Woronowicz S.L., A remark on compact matrix quantum groups, Lett. Math. Phys. 21 (1991), 35-39.
  44. Woronowicz S.L., Compact quantum groups, in Symétries Quantiques (Les Houches, 1995), North-Holland, Amsterdam, 1998, 845-884.

Previous article  Next article  Contents of Volume 20 (2024)