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Introduction
Even if the base field is algebraically closed of characteristic zero, there is at present no complete structure theory for finite-dimensional cocommutative cosemisimple Yetter-Drinfel'd Hopf algebras over the group ring of a finite abelian group. Such a structure theory is only available if the finite abelian group has prime order (cf. [12]). In the prime order case, it turns out that the so-called core of a group-like element is always completely trivial in the sense that both the action and the coaction on the core are trivial. The notion of the core itself can be defined in the case of a general finite abelian group. This raises the question whether also in this case the core of a group-like element is always completely trivial. The purpose of this article is to construct examples that show that this is not the case.
The examples that we construct are eight-dimensional Yetter-Drinfel'd Hopf algebras over the group ring of the elementary abelian group Z2 × Z2. They have a basis consisting of group-like elements and are therefore cocommutative and cosemisimple. After a general discussion of Yetter-Drinfel'd Hopf algebras in Section 1, where we also discuss the notions of triviality and complete triviality, we construct in Section 2 a commutative example of such a Yetter-Drinfel'd Hopf algebra. The example depends on a parameter ζ, which is a not necessarily primitive fourth root of unity. In Section 3, we construct a second such example, which is very similar, but noncommutative. Also the second example depends on a not necessarily primitive fourth root of unity ζ.
In Section 4, we first review parts of the theory developed in [13] in a dualized form for cocommutative instead of commutative Yetter-Drinfel'd Hopf algebras. In particular, we review the notion of the core of a group-like element. We then discuss the cores that appear in the examples constructed before and in particular observe that some are not completely trivial, because the finite abelian group acts and coacts on them nontrivially. However, these cores are trivial in the sense that they are ordinary Hopf algebras, while the Yetter-Drinfel'd Hopf algebras that we constructed are nontrivial; i.e., they are not ordinary Hopf algebras. The article concludes with the conjecture that this is a general phenomenon: We conjecture that the core of a group-like element is always trivial, at least under the hypotheses that we have been making throughout the article.
The expert reader will realize immediately that the examples that we construct are crossed products over a group with two elements, and wonder why this fact is not mentioned in the text. The reason is that much more is true: Not only the algebra structure arises from a specific construction, but also the coalgebra structure, and it is possible to describe the necessary compatibility conditions between the data involved. We plan to address these topics in the forthcoming article [4]. However, as a consequence of the fact that the cores in our examples are not completely trivial, our examples are not diagonal realizations in the sense of [1], Sec. 3, a construction that can also yield crossed products over a group with two elements.
Concerning conventions, the precise assumptions that are used throughout each section are listed in its first paragraph. In particular, this holds for the assumptions on the base field, which is always denoted by K. It is arbitrary in Section 1, but in Section 2, we assume that it contains a primitive eighth root of unity, which forces that its characteristic is different from 2. In Section 3, we assume that it contains a primitive fourth root of unity. Finally, in Section 4, we assume that it is algebraically closed of characteristic zero. All vector spaces are defined over K, and all unadorned tensor products are taken over K; more general tensor products appear only once in Paragraph 3.6. The dual of a vector space V is denoted by V*:=HomK(V,K), and the transpose of a linear map f, i.e., the induced map between the dual spaces, is denoted by f*. The set of matrices with m rows, n columns, and entries in K is denoted by M(m × n, K), and the n × n-identity matrix is denoted by En ∈ M(n × n, K).
All algebras are assumed to have a unit element, and algebra homomorphisms are assumed to preserve unit elements. Unless stated otherwise, a module is a left module. The multiplicative group of invertible elements in the base field K is denoted by K× := K∖{0}. In this article, a character is a one-dimensional character, i.e., a group homomorphism to the multiplicative group K× in the case of a group character, or an algebra homomorphism to the base field K in the case of the character of an algebra. The character group of an abelian group G is the group of all such characters, i.e., the group Ĝ := Hom(G,K×). The symbol ⊥ will not be used for vector spaces, but only for abelian groups. It has two meanings: For a subset of the group G, it denotes the set of characters that take the value 1 on all elements of the set, and for a subset of the character group Ĝ, it denotes the set of all group elements on which all of the given characters take the value 1.
The cardinality of a set X will be denoted by |X|. The symbol ⊂ denotes non-strict inclusion, so that we have X ⊂ X for every set X. Also, we use the so-called Kronecker symbol δij, which is equal to 1 if i=j and equal to 0 otherwise. With respect to enumeration, we use the convention that propositions, definitions, and similar items are referenced by the paragraph in which they occur; i.e., a reference to Proposition 1.1 refers to the unique proposition in Paragraph 1.1.
The material discussed here was first presented at the AMS Spring Southeastern Sectional Meeting in Auburn in March 2019. The travel of the second author to this conference 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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