## The Classification of Semisimple Hopf Algebras of dimension 16

Venue: | J. of Algebra |

Citations: | 13 - 1 self |

### BibTeX

@ARTICLE{Kashina_theclassification,

author = {Yevgenia Kashina},

title = {The Classification of Semisimple Hopf Algebras of dimension 16},

journal = {J. of Algebra},

year = {}

}

### OpenURL

### Abstract

Abstract. In this paper we completely classify nontrivial semisimple Hopf algebras of dimension 16. We also compute all the possible structures of the Grothendieck ring of semisimple non-commutative Hopf algebras of dimension 16. Moreover, we prove that non-commutative semisimple Hopf algebras of dimension p n, p is prime, cannot have a cyclic group of grouplikes. 1. Introduction. Recently various classification results were obtained for finite-dimensional semisimple Hopf algebras over an algebraically closed field of characteristic 0. The smallest dimension, for which the question was still open, was 16. In this paper we completely classify all nontrivial (i.e. noncommutative and non-cocommutative) Hopf algebras of dimension 16. Moreover, we consider all

### Citations

127 |
Theory of Groups of Finite Order
- Burnside
- 1911
(Show Context)
Citation Context ... and the Grothendieck rings K0 (H) (defined in Section 2). Here we consider twistings of group algebras kG, where G is a nonabelian group of order 16. There are exactly nine such groups, described in =-=[2]-=- (see Section 4). The twistings appearing here are explained in Section 7. The coproduct # α is explained in Section 8. H8 denotes the unique nontrivial semisimple Hopf algebra of dimension 8 (see [7]... |

50 |
A Hopf algebras freeness theorem
- Nichels, Zoeller
- 1989
(Show Context)
Citation Context ...ations of H of degree 1 are exactly the grouplike elements of H ∗ . Let G (H ∗ ) denote the group of grouplikes of H ∗ , then kG (H ∗ ) is a subHopfalgebra of H ∗ and thus, by Nichols-Zoeller Theorem =-=[23]-=-, |G (H ∗ )| = dimkG (H ∗ ) divides dim H ∗ = dimH = 16. Therefore by the Artin-Wedderburn Theorem, as an algebra H is isomorphic to either (1.1) (1.2) k (8) ⊕ M2 (k) ⊕ M2 (k) or k (4) ⊕ M2 (k) ⊕ M2 (... |

50 |
Matched pairs of groups and bismash products of Hopf algebras
- Takeuchi
- 1981
(Show Context)
Citation Context ...mological data was done in [18] and [1]. K is commutative and F is cocommutative and thus (F, K) form an abelian matched pair of Hopf algebras and (G, 〈t〉) form an abelian matched pair of groups (see =-=[30]-=-, [5], [29], [12, Section 1]). Therefore H becomes a bicrossed product K# θ σF with an action ⇀: F ⊗ K → K, a coaction ρ : F → F ⊗ K, a cocycle σ : F ⊗ F → K and a dual cocycle θ : F → K ⊗ K. G is a n... |

44 | Hopf Algebras and their - Montgomery - 1993 |

40 |
Normal basis and transitivity of crossed products for Hopf algebras
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- 1992
(Show Context)
Citation Context ...x 2, by [10, Proposition 2] or [20, Theorem 2.1.1] K is normal in H and thus we have an exact sequence of Hopf algebras (3.1) K i ֒→ H π ։ F where F = k 〈t〉 ∼ = kC2 and K = (kG) ∗ , which is cleft by =-=[26]-=- or [17]. Such a sequence is called an extension of F by K and was first studied by Kac in [6]. The construction of extensions from cohomological data was done in [18] and [1]. K is commutative and F ... |

35 | Gelaki: Some properties of finite-dimensional semisimple Hopf algebras
- Etingof, S
- 1998
(Show Context)
Citation Context ....4], (kG) J is non-cocommutative if and only if J −1 (τJ) does not lie in the centralizer of ∆(kG) in kG ⊗ kG. Moreover, by [24, Theorem 4.1] K0 ((kG) J ) ∼ = K0 (kG). Since J is a 2-cocycle, then by =-=[4]-=- (kG) J is triangular.CLASSIFICATION OF SEMISIMPLE HOPF ALGEBRAS OF DIMENSION 16. 29 We now discuss Examples 2, 12 and 13 in the tables. We used GAP to compute G (H) in Examples 12 and 13. Example 2.... |

33 | Hopf algebras of prime dimension - Zhu - 1994 |

27 |
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Citation Context ...equence of Hopf algebras (3.1) K i ֒→ H π ։ F where F = k 〈t〉 ∼ = kC2 and K = (kG) ∗ , which is cleft by [26] or [17]. Such a sequence is called an extension of F by K and was first studied by Kac in =-=[6]-=-. The construction of extensions from cohomological data was done in [18] and [1]. K is commutative and F is cocommutative and thus (F, K) form an abelian matched pair of Hopf algebras and (G, 〈t〉) fo... |

24 |
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Citation Context ...lasses of representations of H with the addition given by a direct sum. Then its enveloping group K0 (H) has the structure of an ordered ring with involution ∗ and is called the Grothendieck ring. In =-=[22]-=- the structure of K0 (H) was described for comodules; it was then translated into the language of modules in [25]. The multiplication in this ring is defined as follows: let [π1] and [π2] denote the c... |

21 |
Quasi-Hopf algebras, (Russian) Algebra i Analiz 1
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Citation Context ..., therefore each of them is spanned by two grouplikes. Thus L is spanned by 8 grouplikes and L is a group algebra.4 YEVGENIA KASHINA We will also need the notion of a twisting of a Hopf algebra (see =-=[3]-=-, [31], [24]): Definition 2.3. The twisting HΩ of a Hopf algebra H is a Hopf algebra with the same algebra structure and counit and with comultiplication and antipode given by ∆Ω (h) = Ω∆(h)Ω −1 SΩ (h... |

21 |
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- 1968
(Show Context)
Citation Context ..., 3 described in [8] and [9]), Hd:1,1 ∼ = k (D8 × C2) J and HC:σ1 (since there is no other choice for the dual). 2. Preliminaries. First we will need the following definition, which was introduced in =-=[28]-=-. Definition 2.1. Let K0 (H) + denote the abelian semigroup of all equivalence classes of representations of H with the addition given by a direct sum. Then its enveloping group K0 (H) has the structu... |

19 | The pn theorem for semisimple Hopf algebras - Masuoka - 1996 |

18 |
Finite ring groups
- Kac, Paljutkin
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Citation Context ...[2] (see Section 4). The twistings appearing here are explained in Section 7. The coproduct # α is explained in Section 8. H8 denotes the unique nontrivial semisimple Hopf algebra of dimension 8 (see =-=[7]-=- and [11]). Table 1. No. Example G (H) G (H ∗ ) K0 (H) Notes 1 Hd:−1,1 ∼ = H8 ⊗ kC2 (C2) 3 (C2) 3 2 Hd:1,1 ∼ = k (D8 × C2) J (C2) 3 (C2) 3 3 (Hc:σ1) ∗ (C2) 3 4 (Hb:1) ∗ (C2) 3 5 Hc:σ1 C2 × C4 (C2) 3 6... |

17 | Some remarks on exact sequences of quantum groups - Schneider - 1993 |

15 |
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- Nikshych
- 1998
(Show Context)
Citation Context ... each of them is spanned by two grouplikes. Thus L is spanned by 8 grouplikes and L is a group algebra.4 YEVGENIA KASHINA We will also need the notion of a twisting of a Hopf algebra (see [3], [31], =-=[24]-=-): Definition 2.3. The twisting HΩ of a Hopf algebra H is a Hopf algebra with the same algebra structure and counit and with comultiplication and antipode given by ∆Ω (h) = Ω∆(h)Ω −1 SΩ (h) = uS (h)u ... |

13 |
Self-dual Hopf algebras of dimension p 3 obtained by extension
- Masuoka
(Show Context)
Citation Context ...rdered rings with marked elements, and thus G (H ∗ ) ∼ = G ( (HΩ) ∗) . 3. Hopf algebras of dimension 16 with a commutative subHopfalgebra of dimension 8. We apply the methods used by Masuoka in [11], =-=[12]-=- and [14]. Let H be a nontrivial semisimple Hopf algebra of dimension 16 with a subHopfalgebra K = (kG) ∗ of dimension 8. Since K is a subHopfalgebra of index 2, by [10, Proposition 2] or [20, Theorem... |

13 | On semisimple Hopf algebras of dimension pq2 - Natale |

12 |
Extensions of Hopf algebras, Algebra i Analiz 7
- Andruskiewitsch, Devoto
- 1995
(Show Context)
Citation Context ...) ∗ , which is cleft by [26] or [17]. Such a sequence is called an extension of F by K and was first studied by Kac in [6]. The construction of extensions from cohomological data was done in [18] and =-=[1]-=-. K is commutative and F is cocommutative and thus (F, K) form an abelian matched pair of Hopf algebras and (G, 〈t〉) form an abelian matched pair of groups (see [30], [5], [29], [12, Section 1]). Ther... |

11 |
Extension theory for connected Hopf algebras
- Singer
- 1970
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Citation Context ...ata was done in [18] and [1]. K is commutative and F is cocommutative and thus (F, K) form an abelian matched pair of Hopf algebras and (G, 〈t〉) form an abelian matched pair of groups (see [30], [5], =-=[29]-=-, [12, Section 1]). Therefore H becomes a bicrossed product K# θ σF with an action ⇀: F ⊗ K → K, a coaction ρ : F → F ⊗ K, a cocycle σ : F ⊗ F → K and a dual cocycle θ : F → K ⊗ K. G is a normal subgr... |

11 |
2-Cocycles and Twisting of Kac algebras
- Vainerman
(Show Context)
Citation Context ...refore each of them is spanned by two grouplikes. Thus L is spanned by 8 grouplikes and L is a group algebra.4 YEVGENIA KASHINA We will also need the notion of a twisting of a Hopf algebra (see [3], =-=[31]-=-, [24]): Definition 2.3. The twisting HΩ of a Hopf algebra H is a Hopf algebra with the same algebra structure and counit and with comultiplication and antipode given by ∆Ω (h) = Ω∆(h)Ω −1 SΩ (h) = uS... |

8 |
Extensions of Hopf Algebras and Their Cohomological Description
- Hofstetter
- 1994
(Show Context)
Citation Context ...cal data was done in [18] and [1]. K is commutative and F is cocommutative and thus (F, K) form an abelian matched pair of Hopf algebras and (G, 〈t〉) form an abelian matched pair of groups (see [30], =-=[5]-=-, [29], [12, Section 1]). Therefore H becomes a bicrossed product K# θ σF with an action ⇀: F ⊗ K → K, a coaction ρ : F → F ⊗ K, a cocycle σ : F ⊗ F → K and a dual cocycle θ : F → K ⊗ K. G is a normal... |

8 | Calculations of Some Groups of Hopf algebra Extensions - Masuoka - 1997 |

7 |
Semisimple Hopf algebras of dimension 6
- Masuoka
- 1995
(Show Context)
Citation Context ... Section 4). The twistings appearing here are explained in Section 7. The coproduct # α is explained in Section 8. H8 denotes the unique nontrivial semisimple Hopf algebra of dimension 8 (see [7] and =-=[11]-=-). Table 1. No. Example G (H) G (H ∗ ) K0 (H) Notes 1 Hd:−1,1 ∼ = H8 ⊗ kC2 (C2) 3 (C2) 3 2 Hd:1,1 ∼ = k (D8 × C2) J (C2) 3 (C2) 3 3 (Hc:σ1) ∗ (C2) 3 4 (Hb:1) ∗ (C2) 3 5 Hc:σ1 C2 × C4 (C2) 3 6 Hb:1 C2 ... |

7 | Cocycle deformations and Galois objects for some co-semisimple Hopf algebras of finite dimension - Masuoka - 2000 |

6 |
Some further classification results on semisimple Hopf algebras
- Masuoka
- 1996
(Show Context)
Citation Context ...ngs with marked elements, and thus G (H ∗ ) ∼ = G ( (HΩ) ∗) . 3. Hopf algebras of dimension 16 with a commutative subHopfalgebra of dimension 8. We apply the methods used by Masuoka in [11], [12] and =-=[14]-=-. Let H be a nontrivial semisimple Hopf algebra of dimension 16 with a subHopfalgebra K = (kG) ∗ of dimension 8. Since K is a subHopfalgebra of index 2, by [10, Proposition 2] or [20, Theorem 2.1.1] K... |

6 |
Y.: “Generalization of Cleft Comodule Algebras
- Masuoka, Doi
- 1992
(Show Context)
Citation Context ...[10, Proposition 2] or [20, Theorem 2.1.1] K is normal in H and thus we have an exact sequence of Hopf algebras (3.1) K i ֒→ H π ։ F where F = k 〈t〉 ∼ = kC2 and K = (kG) ∗ , which is cleft by [26] or =-=[17]-=-. Such a sequence is called an extension of F by K and was first studied by Kac in [6]. The construction of extensions from cohomological data was done in [18] and [1]. K is commutative and F is cocom... |

5 | A result extended from groups to Hopf algebras, Commun. Algebra 25 - Kobayashi, Masuoka - 1997 |

1 |
Studies of semisimple finite-dimensional Hopf algebras
- Kashina
- 1999
(Show Context)
Citation Context ...ing of kD16. Remark 1.4. The following Hopf algebras are selfdual: Hd:−1,1 ∼ = H8 ⊗ kC2 (since H8 is selfdual), Hc:σ0 (since comparing K0-rings we see that ∼= Hc:σ0 A + 3 ∼ = ( A +) ∗, 3 described in =-=[8]-=- and [9]), Hd:1,1 ∼ = k (D8 × C2) J and HC:σ1 (since there is no other choice for the dual). 2. Preliminaries. First we will need the following definition, which was introduced in [28]. Definition 2.1... |

1 |
Examples of Hopf algebras of dimension 2 m , submitted
- Kashina
(Show Context)
Citation Context ...D16. Remark 1.4. The following Hopf algebras are selfdual: Hd:−1,1 ∼ = H8 ⊗ kC2 (since H8 is selfdual), Hc:σ0 (since comparing K0-rings we see that ∼= Hc:σ0 A + 3 ∼ = ( A +) ∗, 3 described in [8] and =-=[9]-=-), Hd:1,1 ∼ = k (D8 × C2) J and HC:σ1 (since there is no other choice for the dual). 2. Preliminaries. First we will need the following definition, which was introduced in [28]. Definition 2.1. Let K0... |

1 |
More examples of bicrossedproduct and double crossedproduct Hopf algebras
- Majid
- 1990
(Show Context)
Citation Context ...d K = (kG) ∗ , which is cleft by [26] or [17]. Such a sequence is called an extension of F by K and was first studied by Kac in [6]. The construction of extensions from cohomological data was done in =-=[18]-=- and [1]. K is commutative and F is cocommutative and thus (F, K) form an abelian matched pair of Hopf algebras and (G, 〈t〉) form an abelian matched pair of groups (see [30], [5], [29], [12, Section 1... |

1 |
Private communications
- Natale
- 1999
(Show Context)
Citation Context ...is selfdual. Remark 1.1. A part of Theorem 1.2, saying that if H has a nonabelian group of grouplikes then G (H) = D8 and G (H ∗ ) = C2 × C2 can also be obtained as a corollary to a theorem of Natale =-=[21]-=-, and Proposition 3.1. This theorem states that if G (H) is nonabelian then H ∗ has 4 central grouplikes. The author was supported in part by NSF grants DMS-9701755 and DMS 98-02086. 12 YEVGENIA KASH... |

1 |
On finite dimensional simple Hopf algebras
- Nikshych
(Show Context)
Citation Context ...structure of an ordered ring with involution ∗ and is called the Grothendieck ring. In [22] the structure of K0 (H) was described for comodules; it was then translated into the language of modules in =-=[25]-=-. The multiplication in this ring is defined as follows: let [π1] and [π2] denote the classes of representations equivalent to π1 and π2, then [π1] • [π2] is the class of the representation (π1 ⊗ π2) ... |