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Introduction
In a previous article [4], the authors introduced two families of Yetter-Drinfel'd Hopf algebras over the Klein four-group, both of which depended on a not necessarily primitive fourth root of unity. Because the algebras in the first family were commutative, while the algebras in the second family were not commutative, no algebra in the first family can be isomorphic to an algebra in the second family. However, what was not addressed in the article was the question whether the Yetter-Drinfel'd Hopf algebras within one of these families are isomorphic, i.e., whether the choice of a different fourth root of unity actually leads to a different Yetter-Drinfel'd Hopf algebra. In this article, we show that this is actually the case; in other words, Yetter-Drinfel'd Hopf algebras arising from different fourth roots of unity are not isomorphic.
Every Yetter-Drinfel'd Hopf algebra gives rise to an ordinary Hopf algebra via the Radford biproduct construction. The arising biproducts were already described in our article [6], where, in particular, we gave a presentation in terms of generators and relations and showed that the biproducts arising from the first family are isomorphic to the biproducts arising from the second family if in both cases the same root of unity is chosen. It is not easy to see whether or not biproducts corresponding to different roots of unity are isomorphic. In view of the result just mentioned, it suffices to discuss this question for the first family of Yetter-Drinfel'd Hopf algebras. As we show in Section 2, biproducts corresponding to two primitive fourth roots of unity are isomorphic, biproducts corresponding to two non-primitive fourth roots of unity are isomorphic, but two biproducts are not isomorphic if one of them corresponds to a primitive fourth root of unity, while the other corresponds to a non-primitive fourth root of unity. This is the main result of this article, which is stated in Theorem 2.6.
In the sequel, we assume that the reader is familiar with our previous work on the topic contained in the two references [4] and [6] already mentioned. The assumptions and the notation follow these references. We assume that our base field K contains a primitive eighth root of unity, whose square is a primitive fourth root of unity denoted by ι. Note that this assumption forces that the characteristic of K is different from 2. All Yetter-Drinfel'd modules, and therefore all Yetter-Drinfel'd Hopf algebras, will be left-left Yetter-Drinfel'd modules; i.e., they are left modules and left comodules. The module action is denoted by a dot. The set of group-like elements of a Hopf algebra H is denoted by G(H). Its augmentation ideal, i.e., the kernel of the counit, is denoted by H+. The transpose of a linear map f is denoted by f*. Residue classes in a quotient space are denoted by a bar. The symbol ≡ is used to express that the residue classes of two vectors in a quotient space are equal. 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.
Concerning the organization of the article, Section 1 covers the isomorphism problem for Yetter-Drinfel'd Hopf algebras. In Paragraph 1.1, we briefly review how the first family of Yetter-Drinfel'd Hopf algebras, which is commutative, is defined. As stated, this definition depends on a fourth root of unity. Paragraph 1.2 compares two such algebras that correspond to a pair of fourth roots of unity that differ by a sign, leading up to the result in Paragraph 1.3 that Yetter-Drinfel'd Hopf algebras in the first family cannot be isomorphic if they correspond to two different roots of unity. In the case where the fourth root of unity under consideration is not primitive, we also derive there a certain equation of degree 6 for one of the generators. Paragraph 1.4 briefly reviews the definition of the second family of Yetter-Drinfel'd Hopf algebras, which is noncommutative. As for the first family, two Yetter-Drinfel'd Hopf algebras in the second family cannot be isomorphic if they correspond to two different roots of unity, as shown in Paragraph 1.5. We also derive there an analogous equation of degree 6 for one of the generators, but now in the primitive case.
Section 2 treats the isomorphism problem for the corresponding Radford biproducts. Paragraph 2.1 describes Radford biproducts in general and lists some specific properties that are required for the proof of the main result. Paragraph 2.2 reviews the particular biproducts that arise from the Yetter-Drinfel'd Hopf algebras under consideration, which were first considered in [6]. Paragraph 2.3 and Paragraph 2.4 treat properties of these particular biproducts that will be needed in the sequel. Isomorphisms between different biproducts appear for the first time in Paragraph 2.5, where we construct an explicit isomorphism between the biproducts that arise from Yetter-Drinfel'd Hopf algebras corresponding to two fourth roots of unity that differ by a sign. Therefore, two Radford biproducts are isomorphic if the corresponding roots of unity are both primitive or both not primitive. The main result, namely that they are not isomorphic if one fourth root of unity is primitive and the other is not, is stated in Theorem 2.6. The rest of the article is concerned with the proof of that result. The fundamental technique of the proof is to analyze a certain group of group-like elements in the dual that has order 4. Under the transpose of a hypothetical isomorphism, it is mapped to another group of order 4, for which there are seven possibilities. These are treated on a case-by-case basis.
Both authors have presented results covered in this article, together with other results about the Yetter-Drinfel'd Hopf algebras considered here, at various conferences, either separately or jointly. The first author presented partial results at the AMS Spring Eastern Sectional Meeting in March 2021 and complete results at the AMS Spring Central Sectional Meeting in March 2022 as well as at the fourth 'International Workshop on Groups, Rings, Lie and Hopf Algebras' in St. John's in June 2022. The second author presented partial results at the Mathematical Congress of the Americas in July 2021 and complete results at the conference 'Hopf Algebras, Monoidal Categories and Related Topics' in Bucharest in July 2022. Both authors presented together at the Joint Mathematics Meeting in April 2022 in two consecutive talks.
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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