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Introduction
In their article [10], the authors described two semisimple Yetter-Drinfel'd Hopf algebras of dimension 8 over the group ring of the elementary abelian group of order 4. The purpose of the construction of these Yetter-Drinfel'd Hopf algebras was to show that the core of a group-like element is not always completely trivial. It was not even mentioned in this article that, via the Radford biproduct construction (cf. [23]), these Yetter-Drinfel'd Hopf algebras give rise to ordinary Hopf algebras of dimension 32. The purpose of the present article is to discuss how these Hopf algebras fit into the general theory of semisimple Hopf algebras of prime power dimension, as found for example in [6], [7], [8], [13], and [14]. As we will see, these ordinary Hopf algebras also behave differently from other known examples.
Every semisimple Hopf algebra of prime power dimension can in principle be constructed as an iterated crossed product (cf. [20], Theorem 3.5). However, for the dimensions where the description is more concrete, or even so concrete that all semisimple Hopf algebras of that given dimension can be classified, like in the articles just cited, the Hopf algebras often contain a commutative Hopf subalgebra of prime index, a situation sometimes also encountered for semisimple Hopf algebras that are not of prime power dimension, like in [15] and [21]. The examples constructed here do not contain such a Hopf subalgebra, as we show in Paragraph 4.4. More generally, we show that they cannot be constructed as central or cocentral abelian extensions.
Let us state the main results of the article more precisely, while we simultaneously explain its structure. Section 1 contains preliminaries on Yetter-Drinfel'd Hopf algebras, Radford biproducts, and extensions of Hopf algebras. In Section 2, we consider the eight-dimensional Yetter-Drinfel'd Hopf algebra A over the elementary abelian group of order 4 that appears in [10], Section 2. We describe the arising Radford biproduct B, which has dimension 32, give a presentation of B in terms of generators and relations, and compute its center. In Section 3, we show that the groups of group-like elements of both B and B* are elementary abelian of order 8, which means in particular that they have eight one-dimensional representations. In Theorem 3.2, we show that, up to isomorphism, both B and B* have in addition two irreducible representations of dimension 2 and one irreducible representation of dimension 4.
In Section 4, we show that B contains a unique sixteen-dimensional Hopf subalgebra N. It is normal and isomorphic to the Hopf algebra Hd:1,1 from the classification appearing in [6], Table 1. Our Hopf algebra B therefore fits into exactly one extension of the type
N ↪ B ↠ Z
with dim N = 16 and dim Z = 2. This extension is not abelian.
In Section 5, we show that the Hopf subalgebra N also arises as the unique sixteen-dimensional quotient of B. By construction, N is also a Radford biproduct, and the quotient map πN: B → N is induced from a map between the corresponding Yetter-Drinfel'd Hopf algebras. Although πN does not restrict to the identity on N, it is conormal, and therefore B fits into exactly one extension of the type
U ↪ B ↠ N
with dim U = 2 and dim N = 16. This extension is not abelian. We use these facts to describe the structure of the Grothendieck ring of B in Paragraph 5.3.
In Section 6, we begin by determining the eight-dimensional Hopf subalgebras of B. There are three, denoted by M1, M2, and M3, all of which are contained in N. The algebra M1 is just the span of the group-like elements, while the other two are dual to the group ring of the dihedral group. We show in Paragraph 6.2 that, of the three Hopf subalgebras, only M2 is normal, so that B fits into exactly one extension of the type
M ↪ B ↠ Q
with dim M = 8 and dim Q = 4. In this case, M = M2 ≅ KD8,
while Q ≅ K[Z2 × Z2]. This extension is abelian, but neither central nor cocentral.
In Paragraph 6.3, we determine the four-dimensional Hopf subalgebras of B. There are seven, but only one of them is normal. Consequently, the Hopf algebra B fits into exactly one extension of the type
P ↪ B ↠ F
with dim P = 4 and dim F = 8. In this case, P ≅ K[Z2 × Z2] and F ≅ K[D8]. This extension is abelian, but neither central nor cocentral.
Besides the Yetter-Drinfel'd Hopf algebra A, the authors considered in [10], Section 3 a second eight-dimensional Yetter-Drinfel'd Hopf algebra over K[Z2 × Z2], denoted here by A'. These two algebras are certainly not isomorphic, because A is commutative, while A' is not. However, the corresponding Radford biproducts B and B' are isomorphic, as we show in Section 7, so that it is not necessary to carry out the same analysis for B'.
Let us now state the conventions that are used throughout this article. Our base field K will be algebraically closed of characteristic zero. The multiplicative group of K will be denoted by K× := K∖{0}.
All vector spaces will be defined over K. The dual space of a vector space V will be denoted by V* = Hom(V,K), and the dual map of a linear map f, also called its transpose, will be denoted by f*.
The group algebra of a group G, which we will also call its group ring, will be denoted by K[G]. The corresponding dual space K[G]*, the dual group ring, will be denoted by KG. The character group of a group G will be denoted by Ĝ := Hom(G,K×). Its elements will be interchangeably called multiplicative characters, one-dimensional characters, or one-dimensional representations, a terminology that we will use not only for groups, but also for algebras. Clearly, the multiplicative characters of the group algebra correspond bijectively to the multiplicative characters of the group via restriction.
All algebras are assumed to have a unit element, and algebra homomorphisms are assumed to preserve these unit elements. The center of an algebra A will be denoted by Z(A), which should not be confused with a certain quotient Hopf algebra Z already mentioned above. The set of group-like elements in a coalgebra A will be denoted by G(A). The subalgebra generated by elements a1,…,an of an algebra A will be denoted by
K<a1,…,an>, whereas the subgroup of a group G generated by elements g1,…,gn will be denoted by <g1,…,gn>. The augmentation ideal of a Hopf algebra H will be denoted by H+ := ker(εH). The image of an element a in a quotient space will be denoted by ā. Finally, the symbol ⊂ denotes non-strict inclusion.
As already discussed, the article is divided into sections, which are divided further into relatively small paragraphs. A reference to Proposition 2.2 refers to the unique proposition in Paragraph 2.2, and definitions, theorems, lemmas, and corollaries are referenced in the same way.
The material discussed here and in our previous article [10] was presented at the AMS Spring Southeastern Sectional Meeting in Auburn in March 2019, in two consecutive talks. It was also presented at the International Workshop on Hopf Algebras and Tensor Categories in Nanjing in September 2019. The travel of the second author to these conferences as well as his work on this article were supported by NSERC grant RGPIN-2017-06543. The work of the first author on this article was supported by a Faculty Summer Research Grant from the College of Science and Health at DePaul University.
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