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Further results on semisimple Hopf algebras of dimension p 2q 2

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Abstract

Let p, q be distinct prime numbers, and k an algebraically closed field of characteristic 0. Under certain restrictions on p, q, we discuss the structure of semisimple Hopf algebras of dimension p 2q 2. As an application, we obtain the structure theorems for semisimple Hopf algebras of dimension 9q 2 over k. As a byproduct, we also prove that odd-dimensional semisimple Hopf algebras of dimension less than 600 are of Frobenius type.

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... Their work has motivated great interest in this field which has produced many nice results. For example, using a similar method, Burciu [4], Dong and Dai [9], and Kashina et al [26] independently proved that if an odd-dimensional semisimple Hopf algebra has a 3-dimensional simple module then 3 divides the dimension of the Hopf algebra. ...
... In [42], Natale completed the classification of H by assuming that H and H * are both of Frobenius type. Some other applications of Kaplansky's sixth conjecture may be found in the authors' recent work [7], [8], [9], [10]. ...
... Motivated by the work of Nichols and Richmond, many algebraists intend to consider semisimple Hopf algebras with a simple module of dimension 3. Unfortunately, only partial answers have been obtained. For example, Burciu [4], Dong and Dai [9] and Kashina et al [26] independently proved: Theorem 2.5. If a semisimple Hopf algebra is of odd dimension and has a simple module of dimension 3, then the dimension of the Hopf algebra is divisible by 3. ...
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About 39 years ago, Kaplansky conjectured that the dimension of a semisimple Hopf algebra over an algebraically closed field of characteristic zero is divisible by the dimensions of its simple modules. Although it still remains open, some partial answers to this conjecture play an important role in classifying semisimple Hopf algebras. This paper focuses on the recent development of Kaplansky's sixth conjecture and its applications in classifying semisimple Hopf algebras.
... The lemma below was initially obtained in the Hopf setting, see [10, Lemma 2.2 and Lemma 2.5]. In fact, the proof in [10] only uses the properties of the Grothendieck ring of a semisimple Hopf algebra. Therefore, the proof of [10] also works in the integral fusion category setting. ...
... In fact, the proof in [10] only uses the properties of the Grothendieck ring of a semisimple Hopf algebra. Therefore, the proof of [10] also works in the integral fusion category setting. Lemma 3.3. ...
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We prove that a non-pointed maximally non-self-dual (MNSD) modular category of Frobenius-Perron (FP) dimension less than 2025 has at most two possible types, and all these types can be realized except those of FP dimension 675, 729 and 1125. We also prove that all these modular categories are group-theoretical except the modular categories of dimension 675. Our result shows that a non-group-theoretical MNSD modular category of smallest FP dimension may be the category of FP dimension 675, and non-pointed MNSD modular category of smallest FP dimension is the category of FP dimension 243.
... In particular, if n 0 is a prime number and d = 1, then d = p. On the other hand, if C is of type (1, n 0 ; d, n), then d divides n 0 (see [2,Lemma 5.2], [12, Theorem 5.1 (b)]), hence in this case C is of Frobenius type. ...
... This contradicts equality (2). ...
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Let k be an algebraically closed field of characteristic zero. In this paper we prove that fusion categories of Frobenius-Perron dimensions 84 and 90 are of Frobenius type. Combining this with previous results in the literature, we obtain that every weakly integral fusion category of Frobenius-Perron dimension less than 120 is of Frobenius type.
... In [10], the authors proved that if H has a simple module of dimension 2 then 2 divides the dimension of H. In [2,3,7], the authors proved that if dimH is odd and H has a simple module of dimension 3 then 3 divides the dimension of H. However, it is not known whether 3 divides dimH when dimH is even and H has a simple module of dimension 3. Our present work gives a partial answer to this question. ...
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