Non-semisimple Hopf algebras of dimension p2

Ng, Siu Hung

doi:10.1016/s0021-8693(02)00139-4

Cited by 39 publications

(37 citation statements)

References 15 publications

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“…[29] Radford and Schneider proved Proposition 3.3 in the case where n = 1 using Lemma 3.1 (b). This result provides an alternative proof of the classification of Hopf algebras of dimension p 2 , which was recently finished by Ng [21]. Indeed, if H is semisimple, then by a result of Masuoka [16], it is isomorphic to a group algebra of order p 2 .…”

Section: On Hopf Algebras Of Dimension Pmentioning

confidence: 83%

On Hopf algebras of dimension $P^{3}$

García¹

2005

Tsukuba J. Math.

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Abstract. We discuss some general results on finite-dimensional Hopf algebras over an algebraically closed field k of characteristic zero and then apply them to Hopf algebras H of dimension p 3 over k. There are 10 cases according to the group-like elements of H and H * . We show that in 8 of the 10 cases, it is possible to determine the structure of the Hopf algebra. We also give a partial classification of the quasitriangular Hopf algebras of dimension p 3 over k, after studying extensions of a group algebra of order p by a Taft algebra of dimension p 2 . In particular, we prove that every ribbon Hopf algebra of dimension p 3 over k is either a group algebra or a Frobenius-Lusztig kernel. Finally, using some results from [1] and [4] on bounds for the dimension of the first term H 1 in the coradical filtration of H, we give the complete classification of the quasitriangular Hopf algebras of dimension 27. IntroductionWe work over an algebraically closed field k of characteristic zero. Let p be an odd prime number and let G p be the cyclic group of pth roots of unity. We denote by T (q), the Taft algebra of parameter q ∈ G p {1},

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Section: On Hopf Algebras Of Dimension Pmentioning

confidence: 83%

On Hopf algebras of dimension $P^{3}$

García¹

2005

Tsukuba J. Math.

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“…A Hopf algebra of dimension pq where (p, q) = (3, 11), (3,13), (3,19), (5,17), (5,19), (5,23), (5,29), (7,17), (7,19), (7,23), (7,29), (11, 29), (13, 29) is semisimple and isomorphic to a group algebra or the dual of a group algebra.…”

Section: Corollary 12mentioning

confidence: 99%

STRUCTURE OF CORADICAL FILTRATION AND ITS APPLICATION TO HOPF ALGEBRAS OF DIMENSIONpq

Fukuda

2008

Glasgow Math. J.

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“…The Kac-Zhu Theorem [Z], states that a Hopf algebra of prime dimension is isomorphic to a group algebra. S.-H. Ng [Ng1] proved that in dimension p 2 , the only Hopf algebras are the group algebras and the Taft algebras, using previous results in [AS1], [Mas3]. It is a common belief that a Hopf algebra of dimension pq, where p and q are distinct prime numbers, is semisimple.…”

Section: Introductionmentioning

confidence: 99%

“…Clearly, the order of S 2 H | R divides the order of S 2 H and hence is a power of 2. If Tr(S 2 H | R ) = 0, then by [Ng1,Lemma 1.4] dim R is even, a contradiction. Thus R is semisimple.…”

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confidence: 98%

Classifying Hopf algebras of a given dimension

Beattie

García²

2013

Contemporary Mathematics

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Abstract. Classifying all Hopf algebras of a given finite dimension over C is a challenging problem which remains open even for many small dimensions, not least because few general approaches to the problem are known. [ChNg]. In this paper, we add to the classification tools in [BG] and apply our results to Hopf algebras of dimension rpq and 8p where p, q, r are distinct primes. At the end of this paper we summarize in a table the status of the classification for dimensions up to 100 to date.

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Non-semisimple Hopf algebras of dimension p2

Cited by 39 publications

References 15 publications

On Hopf algebras of dimension $P^{3}$

On Hopf algebras of dimension $P^{3}$

STRUCTURE OF CORADICAL FILTRATION AND ITS APPLICATION TO HOPF ALGEBRAS OF DIMENSIONpq

Classifying Hopf algebras of a given dimension

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