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On the classification of finite quasi-quantum groups by:1 (Source: Crossref)

    We give an overview of the classification results obtained so far for finite quasi-quantum groups over an algebraically closed field of characteristic zero. The main classification results on basic quasi-Hopf algebras are obtained by Etingof, Gelaki, Nikshych, and Ostrik, and on dual quasi-Hopf algebras by Huang, Liu and Ye. The objective of this survey is to help in understanding the tools and methods used for the classification.

    AMSC: 16T15, 57T05, 16G20


    • 1. V. G. Drinfel’d, Quasi-Hopf algebras, Leningrad Math. J. 1 (1990) 1419–1457. Google Scholar
    • 2. S. Majid, Quasi-quantum groups as internal symmetries of topological quantum field theories, Lett. Math. Phys. 22 (1991) 83–90. Crossref, ISI, ADSGoogle Scholar
    • 3. D. Altschuler and A. Coste, Quasi-quantum groups, knots, three-manifolds, and topological field theory, Comm. Math. Phys. 150 (1992) 83–107. Crossref, ISI, ADSGoogle Scholar
    • 4. S. Majid, Tannaka–Krein theorem for quasi-Hopf algebras, in Deformation Theory and Quantum Groups with Applications to Mathematical Physics Amherst, MA, 1990, Contemp. Math., Vol. 134 (Amer. Math. Soc., 1992), pp. 219–232. CrossrefGoogle Scholar
    • 5. S. Shnider and S. Sternberg, Quantum Groups. From Coalgebras to Drinfeld Algebras. A Guided Tour, Graduate Texts in Mathematical Physics, II (International Press, Cambridge, MA, 1993). Google Scholar
    • 6. P. Etingof and V. Ostrik, Finite tensor categories, Mosc. Math. J. 4(3) (2004) 627–654. Crossref, ISIGoogle Scholar
    • 7. P. Etingof and S. Gelaki, Finite-dimensional quasi-Hopf algebras with radical of codimension 2, Math. Res. Lett. 11 (2004) 685–696. Crossref, ISIGoogle Scholar
    • 8. P. Etingof and S. Gelaki, On radically graded finite dimensional quasi-Hopf algebras, Mosc. Math. J. 5 (2005) 371–378. Crossref, ISIGoogle Scholar
    • 9. P. Etingof and S. Gelaki, Liftings of graded quasi-Hopf algebras with radical of prime codimension, J. Pure Appl. Algebra 205 (2006) 310–322. Crossref, ISIGoogle Scholar
    • 10. P. Etingof, S. Gelaki, D. Nikshych and V. Ostrik, Tensor Categories, Mathematical Surveys and Monographs, Vol. 205 (American Mathematical Society, Providence, RI, 2015). CrossrefGoogle Scholar
    • 11. P. Etingof, D. Nikshych and V. Ostrik, On fusion categories, Ann. of Math. 162 (2005) 581–642. Crossref, ISIGoogle Scholar
    • 12. S. Gelaki, Basic quasi-Hopf algebras of dimension n3, J. Pure Appl. Algebra 198 (2005) 165–174. Crossref, ISIGoogle Scholar
    • 13. H.-L. Huang, Quiver approaches to quasi-Hopf algebras, J. Math. Phys. 50(4) (2009) 043501. Crossref, ISIGoogle Scholar
    • 14. M. E. Sweedler, Hopf Algebras (Benjamin, New York, 1969). Google Scholar
    • 15. C. Cibils and M. Rosso, Hopf quivers, J. Algebra, 254(2) (2002) 241–251. Crossref, ISIGoogle Scholar
    • 16. N. Andruskiewitsch and H.-J. Schneider, On the classification of finite-dimensional pointed Hopf algebras, Ann. of Math. 171 (2010) 375–417. Crossref, ISIGoogle Scholar
    • 17. E. Taft, The order of the antipode of finite-dimensional Hopf algebras, Proc. Natl. Acad. Sci. USA 68 (1971) 2631–2633. Crossref, ISI, ADSGoogle Scholar
    • 18. N. Andruskiewitsch and H.-J. Schneider, Finite quantum groups and Cartan matrices, Adv. Math. 154(1) (2000) 1–45. Crossref, ISIGoogle Scholar
    • 19. I. E. Angiono, Basic quasi-Hopf algebras over cyclic groups, Adv. Math. 225(6) (2010) 3545–3575. Crossref, ISIGoogle Scholar
    • 20. H.-L. Huang, From projective representations to quasi-quantum groups, Sci. China Math. 55(10) (2012) 2067–2080. Crossref, ISIGoogle Scholar
    • 21. F. Van Oystaeyen and P. Zhang, Quiver Hopf algebras, J. Algebra 280 (2004) 577–589. Crossref, ISIGoogle Scholar
    • 22. S. Shnider and S. Sternberg, The cobar resolution and a restricted deformation theory for Drinfeld algebras, J. Algebra 169 (1994) 343–366. Crossref, ISIGoogle Scholar
    • 23. A. Yu. Drozd, Tame and wild matrix problems, in Representation Theory II, Lecture Notes in Math., Vol. 832 (Springer, 1980), pp. 242–258. CrossrefGoogle Scholar
    • 24. H.-L. Huang, G. Liu and Yu Ye, Quivers, quasi-quantum groups and finite tensor categories, Comm. Math. Phys. 303(3) (2011) 595–612. Crossref, ISI, ADSGoogle Scholar
    • 25. H.-L. Huang and G. Liu, On coquasitriangular pointed Majid algebras, Comm. Algebra 40 (2012) 3609–3621. Crossref, ISIGoogle Scholar
    • 26. H.-L. Huang, G. Liu and Y. Ye, Graded elementary quasi-Hopf algebras of tame representation type, Israel Math. J. 209 (2015) 157–186. Crossref, ISIGoogle Scholar
    • 27. M. Balodi, H.-L. Huang and S. D. Kumar, Finite Majid algebras over Klein group, Comm. Algebra 42 (2014) 4962–4983. Crossref, ISIGoogle Scholar
    • 28. H.-L. Huang and Y. Yang, Quasi-quantum planes and quasi-quantum groups of dimension p3 and p4, Proc. Amer. Math. Soc. 143(10) (2015) 4245–4260. Crossref, ISIGoogle Scholar
    • 29. S. E. Akrami and S. Majid, Braided cyclic cocycles and nonassociative geometry, J. Math. Phys. 45(10) (2004) 3883–3911. Crossref, ISI, ADSGoogle Scholar
    • 30. S. Majid, Gauge theory on nonassociative spaces, J. Math. Phys. 46 (2005) 103519. Crossref, ISI, ADSGoogle Scholar