Ostrik 26, who proved that every semisimple rigid monoidal category with. The following is a semisimple algebra that appears not to be of this form. We provide examples of reflection groups and reflection hopf algebras for noncommutative artinschelter algebras. The treatment of these two cases compact lie groupssemisimple lie algebras depends. Pdf on the existence of orders in semisimple hopf algebras.
For example, hopf algebras of dimension p over k are the group algebra kz p and hence there is only one type of hopf algebras of dimension p zhu94. Pdf let h be a finitedimensional semisimple hopf algebra over a field k we usually assume that k is algebraically closed of characteristic. The group of hopf algebra automorphisms for a finitedimensional semisimple cosemisimple hopf algebra over a field k was considered by radford and waterhouse. The book provides a detailed account of basic coalgebra and hopf algebra theory with emphasis on hopf algebras which are pointed, semisimple, quasitriangular, or are of certain other quantum groups. Representations of semisimple lie algebras in this chapter we study a special type of representations of semisimple lie algberas. Ng journal of algebra 255 2002 182197 183 kof dimension p2 is isomorphic to a taft algebra. In these notes we discuss the representation theory of. The set hom hopf h 1, h 2 of all hopf algebra maps from a finitedimensional semisimple hopf algebra h 1 to a finitedimensional cosemisimple hopf algebra h 2 is finite. Integrals in quasihopf algebras mit opencourseware.
All the maps in the diagrams are algebra homomorphisms given our hy. The planar algebra of a semisimple and cosemisimple hopf. Throughout, h denotes a finitedimensional hopf algebra over a field iof characteristic p g 0, and s is the antipode of h. Note that the hopf algebras we consider are not semisimple hopf algebras. Weak hopf algebras have quite interesting representation theory. Hence, the hopf algebras over kof dimension p2 can be completely classi. We prove that, over an algebraically closed field of characteristic zero, a semisimple.
The classification of finitedimensional triangular hopf algebras over an algebraically closed field of characteristic 0. This terminology is motivated by the notion of a unimodular lie group, which is a lie group on which a left invariant. If j is a nonzero nilpotent ideal in b, then its preimage under the natural projection map is a nilpotent ideal in a which is strictly larger than rada, a contradiction. Let h be a finitedimensional semisimple hopf algebra over a field k we usually assume that k is algebraically closed of characteristic zero. If h is a semisimple hopf algebra of dimension p2 over k, then h is isomorphic to kz p2 or kz. In this paper, the groups of hopf algebra automorphisms for two classes of pointed hopf algebras are determined. It was also shown by etingof, nikshych and ostrik that any fusion category is equivalent to a category of modules over a weak hopf algebra. From nonsemisimple hopf algebras to correlation functions. Ais an hmodule, and the multiplication and unit maps of aare hmorphisms. If h is a semisimple hopf algebra then all hopf subalgebras are semisimple as well mo, corollary 3. For semisimple quasitriangular hopf algebras the problem is still widely open. Let abe a semisimple hopf algebra of dimension pq2 over k.
For primes p and q it is known that semisimple hopf algebras of dimension p 27, 2p 18, p2 19, and pq 9, 11 are trivial. In the following we show that h itself is a twisted group algebra in the sense of dr2. The question whether the taft algebras are the only non semisimple hopf. Semisimple lie algebras and the killing form this section follows procesis book on lie groups. Noncommutative, noncocommutative semisimple hopf algebras arise from finite abelian groups siuhung ng mathematics department, towson university, towson, md 21252. Some properties of finitedimensional semisimple hopf. Some properties of finitedimensional semisimple hopf algebras. On the existence of orders in semisimple hopf algebras. Every semisimple lie algebra over an algebraically closed field of characteristic 0 is a direct sum of simple lie algebras by definition, and the finitedimensional simple lie algebras fall in four families a n, b n, c n, and d n with five exceptions e 6, e 7, e 8, f 4, and g 2. It is thus natural to focus on the problem of classi. In establishing a more general version of the mckay correspondence, we prove auslanders theorem for actions of semisimple hopf algebras h on noncommutative artinschelter regular algebras a of global dimension two, where a is a graded hmodule algebra, and the hopf action on a is inner faithful with trivial homological determinant. We also prove that a complex semisimple hopf algebra satisfies kaplanskys sixth conjecture if and only if. An algebra a is called semisimple if all finite dimensional amodules are completely reducible.
Trivial examples of semisimple hopf algebras are the group algebra kg and its dual. Semisimple triangular hopf algebras and tannakian categories. Hacts on ainner faithfully if there is not an induced action of hion afor any nonzero hopf ideal iof h. If p q and are odd primes, whether there is a non semisimple hopf algebra of dimension pqother than the taft algebras is still in question. By the main theorem in that paper it is then a galois extension with action of semisimple weak hopf algebra also regular and possessing haar integral. Semisimple hopf algebras via geometric invariant theory. Hopf algebras of dimension 2 department of mathematics. This is just the underlying algebra of sweedlers hopf algebra. This article explores the frobenius property, and some consequences thereof, for hopf algebras and for certain algebras that are closely related to hopf algebras without generally being hopf algebras themselves.
Reflection hopf algebras department of mathematics. The planar algebra of a semisimple and cosemisimple hopf algebra 5 xx h t1 t2 figure 6. Every semisimple hopf algebra is of frobenius type. We show that semisimple hopf algebras of dimension p2q2 can be constructed either from group algebras and their duals by means of extensions, or. Mckay correspondence for semisimple hopf actions on.
We develop the theory of semisimple weak hopf algebras and obtain analogues of a number of classical results for ordinary semisimple hopf algebras. The representation theory of a hopf algebra is particularly nice. Masuoka, freeness of hopf algebras over coideal subalgebras, communications in algebra 20 1992, 5373. Classification of semisimple hopf algebras sciencedirect. Introduction to hopf algebras and representations aalto math. The smallest example of a semisimple hopf algebra which is not semisolvable is a cocycle twist of a group of order 36 gn. We note that a semisimple hopf algebra over k must be finitedimensional by 6, corollary 2. Classification of semisimple hopf algebras request pdf. Abstractwe develop the theory of semisimple weak hopf algebras and obtain analogues of a number of classical results for ordinary semisimple hopf algebras. We prove a criterion for semisimplicity and analyze the square of the antipode s 2 of a semisimple weak hopf algebra a.
This planar algebra encodes the notion of temperleylieb algebra. In mathematics, a hopf algebra, named after heinz hopf, is a structure that is simultaneously an unital associative algebra and a counital coassociative coalgebra, with these structures compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property. Here, p is pnilpotent if some power of the pmap p is 0 on p. Semisimple hopf algebras of dimension 60 sciencedirect. The terms simple module and irreducible representation seem. In this paper we develop an approach for studying finite dimensional semisimple hopf algebra over an algebraically closed field of character istic. Algebra, 241996,307329 let h be a semisimple hopf algebra of dimension 2p2, where p is an odd integer. January 29, 2002 abstract given any nontrivial alternating tricharacter f on a.
Suppose that a is a semisimple hopf algebra over a field of characteristic 0, or that a is a semisimple cosemisimple involutory hopf algebra over a field k of characteristic p dim a. Pdf representation theory of semisimple hopf algebras. Let h be a finitedimensional hopf algebra which is semisimple. It is intended to be a graduate text as well as a research monograph. So the answer to question 1 is negative, and it can only be expected to be a. In other words, the hopf action does not factor through a smaller hopf. Graded semisimple algebras are symmetric sciencedirect. If a is generated as an algebra by a set b, to prove that a is a bialgebra we only need to check the diagrams in 1. Hopf algebra emerged slowly from the work of the topologists in the 1940s. Pdf representations of finitedimensional hopf algebras. The classification of finitedimensional triangular hopf. Hopf algebra that has a nontrivial selfdual simple module must have even. Representations of finitedimensional hopf algebras. Radford, semisimple hopf algebras, journal of algebra, to appear.
For the hopf algebras h of dimension p2, the only known non semisimple hopf algebras of dimension p2 are the taft algebras taf71 cf. Volume 324, issue 11, 1 december 2010, pages 30173034. Using the commuting pair established in 7, we obtain an analogue of the class equation for when h is semisimple and cosemisimple. On the quasitriangular structures of a semisimple hopf algebra. From non semisimple hopf algebras to correlation functions for logarithmic cft. Suppose that h is a finitedimensional hopf algebra. Kac, showing that a semisimple hopf algebra a of dimension pn,w here pis a prime and n 0, over. We prove a criterion for semisimplicity and analyze the square of the antipode s2 of a semisimple weak hopf algebra a. First, the classification of semisimple hopf algebras of dimension p2 p3. Ifpd2 and ais nontrivial, then ais isomorphic to either of the hopf algebras b 0 or b.
We show that in this noncommutative context there exist notions of the jacobian, reflection arrangement, and discriminant that extend the definitions used for reflection groups actions on polynomial algebras. Semisimple hopf algebras of dimension 6, 8 springerlink. A semisimple and cosemisimple hopf algebra over an algebraically closed field is encoded in a planar algebra defined by generators and relations, and corresponds up to isomorphism to a connected, irreducible, spherical, non degenerate planar algebra with non. Representations of hopf algebras 477 here are the main results of the article. Induced modules of semisimple hopf algebras algebra. Automorphism groups of pointed hopf algebras springerlink.
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