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Mixed vs Stable Anti-Yetter–Drinfeld Contramodules
Symmetry, integrability and geometry: methods and applications, Tome 17 (2021) Cet article a éte moissonné depuis la source Math-Net.Ru

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We examine the cyclic homology of the monoidal category of modules over a finite dimensional Hopf algebra, motivated by the need to demonstrate that there is a difference between the recently introduced mixed anti-Yetter–Drinfeld contramodules and the usual stable anti-Yetter–Drinfeld contramodules. Namely, we show that Sweedler's Hopf algebra provides an example where mixed complexes in the category of stable anti-Yetter–Drinfeld contramodules (previously studied) are not equivalent, as differential graded categories to the category of mixed anti-Yetter–Drinfeld contramodules (recently introduced).
Keywords: Hopf algebras, homological algebra, Taft algebras. 
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     author = {Ilya Shapiro},
     title = {Mixed vs {Stable} {Anti-Yetter{\textendash}Drinfeld} {Contramodules}},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2021_17_a25/}
}
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Ilya Shapiro. Mixed vs Stable Anti-Yetter–Drinfeld Contramodules. Symmetry, integrability and geometry: methods and applications, Tome 17 (2021). http://geodesic.mathdoc.fr/item/SIGMA_2021_17_a25/

[1] Andruskiewitsch N., Angiono I., García Iglesias A., Torrecillas B., Vay C., “From Hopf algebras to tensor categories”, Conformal Field Theories and Tensor Categories, Math. Lect. Peking Univ., Springer, Heidelberg, 2014, 1–31, arXiv: 1204.5807 | DOI | MR | Zbl

[2] Andruskiewitsch N., Radford D., Schneider H. J., “Complete reducibility theorems for modules over pointed Hopf algebras”, J. Algebra, 324 (2010), 2932–2970, arXiv: 1001.3977 | DOI | MR | Zbl

[3] Barrett J. W., Westbury B. W., “Spherical categories”, Adv. Math., 143 (1999), 357–375, arXiv: hep-th/9310164 | DOI | MR | Zbl

[4] Ben-Zvi D., Francis J., Nadler D., “Integral transforms and Drinfeld centers in derived algebraic geometry”, J. Amer. Math. Soc., 23 (2010), 909–966, arXiv: 0805.0157 | DOI | MR | Zbl

[5] Brzeziński T., “Hopf-cyclic homology with contramodule coefficients”, Quantum groups and noncommutative spaces, Aspects Math., E41, Vieweg + Teubner, Wiesbaden, 2011, 1–8, arXiv: 0806.0389 | DOI | MR | Zbl

[6] Connes A., Moscovici H., “Cyclic cohomology and Hopf algebras”, Lett. Math. Phys., 48 (1999), 97–108, arXiv: math.QA/9904154 | DOI | MR | Zbl

[7] Drinfeld V. G., “Quantum groups”, Proceedings of the International Congress of Mathematicians (Berkeley, Calif., 1986), v. 1, 2, Amer. Math. Soc., Providence, RI, 1987, 798–820 | MR

[8] Hajac P. M., Khalkhali M., Rangipour B., Sommerhäuser Y., “Hopf-cyclic homology and cohomology with coefficients”, C. R. Math. Acad. Sci. Paris, 338 (2004), 667–672, arXiv: math.KT/0306288 | DOI | MR | Zbl

[9] Hajac P. M., Khalkhali M., Rangipour B., Sommerhäuser Y., “Stable anti-Yetter–Drinfeld modules”, C. R. Math. Acad. Sci. Paris, 338 (2004), 587–590, arXiv: math.QA/0405005 | DOI | MR | Zbl

[10] Halbig S., Generalised Taft algebras and pairs in involution, arXiv: 1908.10750 | Zbl

[11] Jara P., Ştefan D., “Hopf-cyclic homology and relative cyclic homology of Hopf–Galois extensions”, Proc. London Math. Soc., 93 (2006), 138–174 | DOI | MR | Zbl

[12] Kassel C., “Cyclic homology, comodules, and mixed complexes”, J. Algebra, 107 (1987), 195–216 | DOI | MR | Zbl

[13] Kerler T., “Mapping class group actions on quantum doubles”, Comm. Math. Phys., 168 (1995), 353–388, arXiv: hep-th/9402017 | DOI | MR | Zbl

[14] Larson R. G., Radford D. E., “Finite-dimensional cosemisimple Hopf algebras in characteristic $0$ are semisimple”, J. Algebra, 117 (1988), 267–289 | DOI | MR | Zbl

[15] Larson R. G., Radford D. E., “Semisimple cosemisimple Hopf algebras”, Amer. J. Math., 110 (1988), 187–195 | DOI | MR | Zbl

[16] Nenciu A., “Quasitriangular pointed Hopf algebras constructed by Ore extensions”, Algebr. Represent. Theory, 7 (2004), 159–172 | DOI | MR | Zbl

[17] Radford D. E., “Minimal quasitriangular Hopf algebras”, J. Algebra, 157 (1993), 285–315 | DOI | MR | Zbl

[18] Radford D. E., Westreich S., “Trace-like functionals on the double of the Taft Hopf algebra”, J. Algebra, 301 (2006), 1–34 | DOI | MR | Zbl

[19] Shapiro I., “On the anti-Yetter–Drinfeld module-contramodule correspondence”, J. Noncommut. Geom., 13 (2019), 473–497, arXiv: 1704.06552 | DOI | MR | Zbl

[20] Shapiro I., Categorified Chern character and cyclic cohomology, arXiv: 1904.04230

[21] Taft E. J., “The order of the antipode of finite-dimensional Hopf algebra”, Proc. Nat. Acad. Sci. USA, 68 (1971), 2631–2633 | DOI | MR | Zbl

[22] Toën B., “The homotopy theory of $dg$-categories and derived Morita theory”, Invent. Math., 167 (2007), 615–667, arXiv: math.AG/0408337 | DOI | MR | Zbl