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YetterDrinfel'd Hopf Algebras over

Abstract


IntroductionThroughout the whole discussion, we focus on YetterDrinfel'd Hopf algebras that are cosemisimple and cocommutative. The consideration of such a restricted situation is motivated by the structure theory of semisimple Hopf algebras, a field that recently has experienced considerable progress, mainly through the work of A. Masuoka (cf. [47]  [54]). In trying to understand the possible structures for a semisimple Hopf algebra of a given dimension, the analysis often proceeds in the following steps: First, the possible dimensions of the simple modules are determined. This information can sometimes be used to prove the existence of nontrivial grouplike elements. If this can be shown, the existence of analogous nontrivial grouplike elements in the dual may lead to a projection onto a group ring of a cyclic group of prime order. In this case, the Radford projection theorem (cf. [65]) leads to a decomposition of the given Hopf algebra into a tensor product of a YetterDrinfel'd Hopf algebra and the group ring of the cyclic group. The YetterDrinfel'd Hopf algebras arising in this way are semisimple and tend to be commutative, because the dimensions of their simple modules are often only a fraction the dimensions of the simple modules of the initial Hopf algebra, the denominator of the fraction being the order of the cyclic group. By applying the same reasoning to the dual Hopf algebra, it is sometimes even possible to conclude that the YetterDrinfel'd Hopf algebra is also cocommutative. In many cases, it is then possible to prove that this commutative and cocommutative YetterDrinfel'd Hopf algebra must be trivial, which leads to the conclusion that the initial Hopf algebra was a group ring or a dual group ring. It is therefore reasonable to conjecture that such a YetterDrinfel'd Hopf algebra over a cyclic group of prime order is always trivial, i. e., has trivial action or coaction. This conjecture was the starting point of the present considerations. However, the conjecture turns out to be false: Although rare objects, YetterDrinfel'd Hopf algebras of the form described above exist, although not in all dimensions. The main result of the present investigation is the precise description, under certain assumptions on the base field, of YetterDrinfel'd Hopf algebras of this type, i. e., nontrivial, cosemisimple, cocommutative YetterDrinfel'd Hopf algebras over groups of prime order: The structure theorem obtained in Section 7 asserts that these YetterDrinfel'd Hopf algebras can be decomposed into a tensor product of the dual group ring of the cyclic group under consideration and the group ring of another group. The coalgebra structure of this tensor product is the ordinary tensor product coalgebra structure, whereas the algebra structure is that of a crossed product. In particular, the dimension of such a YetterDrinfel'd Hopf algebra is divisible by the order of the cyclic group under consideration, a fact that in many interesting cases rules out the existence of a nontrivial YetterDrinfel'd Hopf algebra with these properties. The analysis that leads to this decomposition can be reversed in order to compose nontrivial YetterDrinfel'd Hopf algebras. In particular, we construct in this way YetterDrinfel'd Hopf algebras of dimension p^{2} that are nontrivial, commutative, cocommutative, semisimple, and cosemisimple. We prove that these examples exhaust all possibilities, and therefore see that there are, up to isomorphism, p(p1) YetterDrinfel'd Hopf algebras of this form. On the other hand, we show that the dimension p^{2} is, in a sense, the minimal case; more precisely, we prove that, if a nontrivial, cocommutative, and cosemisimple YetterDrinfel'd Hopf algebra is also commutative, its dimension must be divisible by p^{2}. We therefore reach the conclusion that in dimension n the existence of a nontrivial, commutative, cocommutative, semisimple, and cosemisimple YetterDrinfel'd Hopf algebra is possible if and only if p^{2} divides n. Finally, we apply these results to semisimple Hopf algebras along the lines indicated above. The presentation is organized as follows: In Section 1, we recall the basic facts on YetterDrinfel'd Hopf algebras, representation theory, and group cohomology that will be needed in the sequel. In Section 2, which is also preliminary, we recall the basic facts from Clifford theory. In our context, Clifford theory is used for the correspondence between the modules of the YetterDrinfel'd Hopf algebra and the modules of its Radford biproduct. The part of Clifford theory used in the proof of the main results is a theorem of W. Chin that sets up a correspondence between orbits of the centrally primitive idempotents of the YetterDrinfel'd Hopf algebra and orbits of the simple modules of the Radford biproduct under the action of the onedimensional characters. In the applications, Clifford theory is used to prove that the YetterDrinfel'd Hopf algebras occurring are indeed commutative or cocommutative. In Section 3, we construct examples of nontrivial YetterDrinfel'd Hopf algebras that are cocommutative and cosemisimple. We work in a framework created by N. Andruskiewitsch, or a very slight generalization thereof, that describes nicely the requirements that have to be satisfied in form of a compatibility condition. We then find various ways to satisfy this compatibility condition; in particular, we attach a YetterDrinfel'd Hopf algebra of this form to every group homomorphism from a finite group to the additive group of a finite ring. In Section 4, we discuss under which circumstances the YetterDrinfel'd Hopf algebras constructed in Section 3 are isomorphic. In particular, we determine the number of their isomorphism classes in dimension p^{2}. In Section 5, we investigate the Hopf algebras arising via the Radford biproduct construction and the second construction from [78], [79] from the YetterDrinfel'd Hopf algebras considered in Section 3, describe their integrals, and prove that these Hopf algebras are semisimple. In this way, we construct new classes of noncommutative, noncocommutative Hopf algebras with triangular decomposition that are semisimple and cosemisimple. In particular, Masuoka's examples of semisimple Hopf algebras of dimension p^{3} appear in these Hopf algebras as a kind of Borel subalgebra. However, these two constructions provide only one way to look at these Hopf algebras: They can also be understood from the point of view of extension theory. We show that these Hopf algebras are extensions of group rings by dual group rings, determine the corresponding groups, and find explicit normal bases for these extensions. In Section 6, we study nontrivial YetterDrinfel'd Hopf algebras over groups of prime order that are commutative and semisimple. In this section, most of the technical work is done; in particular, we find a linear and colinear character of order p that induces action and coaction on the other characters. This is the key step in the proof of the structure theorem in the next section. In Section 7, we dualize the situation and consider nontrivial YetterDrinfel'd Hopf algebras over groups of prime order that are cocommutative and cosemisimple. From the previous section, we know that there exists an invariant, coinvariant grouplike element of order p that induces action and coaction on the other grouplike elements. We then pass to a YetterDrinfel'd Hopf algebra quotient in which this grouplike element is equal to the unit. Now action and coaction in this quotient are trivial, and therefore this quotient is an ordinary Hopf algebra, which must, since it is cocommutative, be a group ring. It is easy to see that the initial YetterDrinfel'd Hopf algebra is a cleft comodule algebra over this group ring, and therefore the structure theorem stated above follows. As an application of the structure theorem, we then prove that the dimension of the YetterDrinfel'd Hopf algebra is divisible by p^{2} if it is also commutative, and classify YetterDrinfel'd Hopf algebras of this type in dimension p^{2}. In Section 8, these results are applied to semisimple Hopf algebras for the first time. For the prime p under consideration, we show that every semisimple Hopf algebra of dimension p^{3} is a Radford biproduct of a commutative, cocommutative YetterDrinfel'd Hopf algebra of dimension p^{2}, which were classified in the preceding section, with the group ring of the cyclic group of order p. This gives a new proof of Masuoka's classification of semisimple Hopf algebras of dimension p^{3} (cf. [50]). In contrast, Masuoka proves that every such Hopf algebra is a Hopf algebra extension of the group ring of a cyclic group of order p and the dual group ring of the elementary abelian group of order p^{2}. Of course, it follows again from the structure theorem that both approaches are related; the precise relation has been worked out in Section 5. In Section 9, we apply the results to semisimple Hopf algebras of dimension pq, where p and q are distinct prime numbers. Although the tools developed in the preceding sections were initially forged to deal with this problem, the results presented here were obtained independently and slightly earlier by P. Etingof, S. Gelaki, and S. Westreich. We prove that a semisimple Hopf algebra of dimension pq is commutative or cocommutative if it and its dual contain nontrivial grouplike elements. In Section 10, we find conditions that assure the existence of such grouplike elements. These results look slightly more general than the result proved in [21], where they were proved under the assumption that the dimensions of the simple modules divide the dimension of the Hopf algebra, i. e., are of dimension 1 or p, if p is smaller than q. However, Etingof and Gelaki later proved that this is in fact always the case (cf. [18]). Their proof has already been simplified by Y. Tsang and Y. Zhu (cf. [85]), and H.J. Schneider (cf. [74]); in addition, S. Natale has, in her investigation of semisimple Hopf algebras of dimension pq^{2}, given an extensiontheoretic proof of these results (cf. [59]). In the sequel, K denotes a field. All vector spaces that we consider are defined over K, and all tensor products without subscripts are taken over K. The multiplicative group K\{0} of K is denoted by K^{×}. Although most of the results mentioned above and proved in the following need the assumption that K is algebraically closed and of characteristic zero, some of the results hold in greater generality. The precise assumptions on K will be stated at the beginning of every section. Likewise, p denotes a natural number that in most cases, but not always, will be prime. The precise assumptions on p will also be stated at the beginning of every section. The transpose of a linear map f is denoted by f*. Following the conventions in group theory, we denote by Z_{n} := Z/nZ the cyclic group of order n, and therefore Z_{p} should not be confused with the set of padic integers. Propositions, definitions, and similar items are referenced by the paragraph in which they occur; they are only numbered separately if this reference is ambiguous. The paper is organized as follows: In Section 2, we recall the notion of a YetterDrinfel'd bialgebra and review some of their elementary properties that will be needed in the sequel. In Section 3, we carry out the first construction which leads to a Hopf algebra which has a twosided cosmash product as coalgebra structure. We show that many Hopf algebras with triangular decomposition are of this form. As special cases, we obtain Radford's biproduct and Majid's double crossproduct. In Section 4, we carry out the second construction which applies to a pair of YetterDrinfel'd Hopf algebras which are in a sense dual to each other. In Section 5, we explain how Lusztig's algebra ´f which corresponds to the nilpotent part of a semisimple Lie algebra is a YetterDrinfel'd Hopf algebra and how the second construction can be used to construct deformed enveloping algebras. 
