## Triviality Theorems for Yetter-Drinfel'd Hopf Algebras

### Yorck Sommerhäuser

- Preprint: XXX preprint archive: math.RA/1503.09181
- Journal: J. Algebra 454 (2016), 475-519

#### Abstract

Under suitable assumptions on the base field, we prove that a commutative semisimple Yetter-Drinfel'd Hopf algebra over a finite abelian group is trivial, i.e., is an ordinary Hopf algebra, if its dimension is relatively prime to the order of the finite abelian group. Furthermore, we prove that a finite-dimensional cocommutative cosemisimple Yetter-Drinfel'd Hopf algebra contains a trivial Yetter-Drinfel'd Hopf subalgebra of dimension greater than one, at least if the Yetter-Drinfel'd Hopf algebra itself has dimension greater than one.

#### Introduction

Usually, a Yetter-Drinfel'd Hopf algebra is not an ordinary Hopf algebra. The difference between the two notions is caused by the fact that the coproduct of a Yetter-Drinfel'd Hopf algebra is not an algebra homomorphism with respect to the canonical algebra structure on the second tensor power of the Yetter-Drinfel'd Hopf algebra. Rather, it is an algebra homomorphism with respect to a modified algebra structure on the second tensor power constructed via a special quasisymmetry that is characteristic for Yetter-Drinfel'd modules. However, it may happen that, for the specific Yetter-Drinfel'd Hopf algebra under consideration, this quasisymmetry coincides on the second tensor power of the algebra with the ordinary flip of tensor factors. Then the algebra structure is in fact not modified, and the Yetter-Drinfel'd Hopf algebra is an ordinary Hopf algebra.

In this case, the Yetter-Drinfel'd Hopf algebra is called trivial (cf. [30], Def. 1.1, p. 8). According to a result of P. Schauenburg, the behavior of the quasisymmetry just described characterizes this property: If a Yetter-Drinfel'd Hopf algebra is an ordinary Hopf algebra, then the quasisymmetry for Yetter-Drinfel'd modules coincides on the second tensor power of the algebra with the ordinary flip of tensor factors (cf. [25], Cor. 2, p. 262; see also [30], Prop. 1.1, p. 8).

The first main result of this article is that, in a certain situation, every Yetter-Drinfel'd Hopf algebra is trivial. In this situation, we consider a semisimple commutative Yetter-Drinfel'd Hopf algebra A over the group ring K[G] of a finite abelian group G, where K is a field of characteristic zero, and prove in Paragraph 3.9 the following triviality theorem:

#### Theorem

If dim(A) and |G| are relatively prime, then A is trivial.

This result was known in the case where |G| is prime (cf. [30], Cor. 6.7, p. 100).

The second main result of this article is concerned with a finite-dimensional cocommutative cosemisimple Yetter-Drinfel'd Hopf algebra A over the group ring K[G] of a finite abelian group G, where K is an algebraically closed field of characteristic zero. We also prove in Paragraph 3.9 that, although A itself need not be trivial, it contains at least a trivial part:

#### Proposition

If dim(A)>1, then A contains a trivial Yetter-Drinfel'd Hopf subalgebra B with dim(B)>1.

We develop the rather involved general theory of such Yetter-Drinfel'd Hopf algebras here only to the extent that is necessary to prove these two results. Section 1 contains a brief, but nonetheless very important, summary of basic facts about Yetter-Drinfel'd Hopf algebras. More detailed treatments of this material can be found in [26], [27], [28], [29], or [30]. Section 1 ends with Theorem 1.5, a version of the Nichols-Zoeller theorem for two Yetter-Drinfel'd Hopf algebras that are not necessarily defined over the same Hopf algebra. In Section 2, we discuss Yetter-Drinfel'd Hopf algebras over group rings of finite abelian groups. In particular, we introduce there a certain alternative description of the coaction that will be used in the entire article. This brief section also contains an important discussion of a way to modify the finite abelian group over which the Yetter-Drinfel'd Hopf algebra is defined. The main part of the article is Section 3, which deals with commutative semisimple Yetter-Drinfel'd Hopf algebras over group rings of finite abelian groups. As we will indicate during the discussion, most of the material presented there generalizes facts that were established in Chapter 6 of [30] for the case in which the finite abelian group has prime order. This final section of the article ends with the proof of the two main results stated above.

While the precise assumptions that are used throughout each section are listed in its first paragraph, there are some conventions that are common to all sections. The base field is always denoted by K, and while it is arbitrary in Section 1, we assume that it is algebraically closed of characteristic zero in Section 2 and Section 3 unless this is explicitly stated otherwise, which happens only in a minor comment in Paragraph 3.9. The multiplicative group of invertible elements in the field K is denoted by K^{×} := K∖{0}. All vector spaces are defined over K, and all unadorned tensor products are taken over K. The dual of a vector space V is denoted by V^{*}:=Hom_{K}(V,K), and the transpose of a linear map f, i.e., the induced map between the dual spaces, is denoted by f^{*}. The symbol ⊥ will be used in four different meanings, two for vector spaces and two for groups. 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. All rings, and therefore especially all algebras, are assumed to have a unit element, and ring and algebra homomorphisms are assumed to preserve unit elements. Unless stated otherwise, a module is a left module. The cardinality of a set X is 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 δ_{ij}, 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 first main result of this article was presented at the AMS Fall Eastern Sectional Meeting in Halifax in October 2014, while the second main result was presented at the Joint Mathematics Meeting in San Antonio in January 2015. The author thanks the organizers of these conferences for the invitation. He also thanks the Department of Mathematics at SUNY Buffalo for a visiting appointment during which most of this article was written.

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- Last modified: April 2, 2016