On Finite-Dimensional Semisimple Hopf Algebras of Dimension $n(n+1)$
Matematičeskie zametki, Tome 91 (2012) no. 2, pp. 253-269.

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We study finite-dimensional semisimple Hopf algebras over an algebraically closed field which have only one summand of dimension greater than $1$ in their semisimple decompositions and assume that the group of group elements in the dual Hopf algebra is cyclic and has minimal order. Under given constraints, we obtain a detailed description of the comultiplication and the antipode.
Keywords: semisimple Hopf algebra, coassociativity of comultiplication, cocommutative Hopf algebra.
Mots-clés : comultiplication, antipode, semisimple decomposition, homomorphism
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S. Yu. Spiridonova. On Finite-Dimensional Semisimple Hopf Algebras of Dimension $n(n+1)$. Matematičeskie zametki, Tome 91 (2012) no. 2, pp. 253-269. http://geodesic.mathdoc.fr/item/MZM_2012_91_2_a7/

[1] Y. Zhu, “Hopf algebras of prime dimension”, Internat. Math. Res. Notices, 1994, no. 1, 53–59 | DOI | MR | Zbl

[2] A. Masuoka, “The $p^n$ theorem for semisimple Hopf algebras”, Proc. Amer. Math. Soc., 124:3 (1996), 735–737 | DOI | MR | Zbl

[3] M. Hilgemann, S.-H. Ng, “Hopf algebras of dimension $2p^2$”, J. London Math. Soc. (2), 80:2 (2009), 295–310 | DOI | MR | Zbl

[4] S.-H. Ng, “Non-semisimple Hopf algebras of dimension $p^2$”, J. Algebra, 255:1 (2002), 182–197 | DOI | MR | Zbl

[5] A. Masuoka, “Some further classification results on semisimple Hopf algebras”, Comm. Algebra, 24:1 (1996), 307–329 | DOI | MR | Zbl

[6] P. Etingof, S. Gelaki, “Semisimple Hopf algebras of dimension $pq$ are trivial”, J. Algebra, 210:2 (1998), 664–669 | DOI | MR | Zbl

[7] S. Montgomery, Hopf Algebras and Their Actions on Rings, CBMS Regional Conf. Ser. in Math., 82, Amer. Math. Soc., Providence, RI, 1993 | MR | Zbl

[8] V. A. Artamonov, “O poluprostykh konechnomernykh algebrakh Khopfa”, Matem. sb., 198:9 (2007), 3–28 | MR | Zbl

[9] V. A. Artamonov, I. A. Chubarov, “Properties of some semisimple Hopf algebras”, Algebras, Representations and Applications, Contemp. Math., 483, Amer. Math. Soc., Providence, RI, 2009, 23–36 | MR | Zbl

[10] S. MacLane, Homology, Grundlehren Math. Wiss., 114, Springer-Verlag, Berlin, 1963 | MR | Zbl

[11] G. Seitz, “Finite groups having only one irreducible representation of degree greater than one”, Proc. Amer. Math. Soc., 19:2 (1968), 459–461 | DOI | MR | Zbl