Irreducible characters 9 1.3. The semisimplicity and cosemisimplicity assumptions imply that both H and H∗ are multi-matrix algebras and the dimensions, say n,ofH, as well as those of its irreducible representations, are ‘non-zero in k’. Google Scholar. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): 0. Semisimple Hopf algebras Viacheslav A. Artamonov (Moscow State University, Russia) viacheslav.artamonov@gmail.com Suppose that H is a nite dimensional semisimple Hopf algebra over an alge-braically closed eld whose characteristic does not divide the dimension of H. We shall assume that for any positive integer d > 1 any two irreducible H-modules of dimension d are isomorphic. In order to do that two main directions were followed until now. It follows that the … We prove that a depth two Hopf subalgebra K of a semisimple Hopf algebra H is normal (where the ground field k is algebraically closed of characteristic zero). The … Nobuyuki Fukuda. We study finite dimensional Hopf algebra actions on so-called filtered Artin-Schelter regular algebras of dimension n, particularly on those of dimension 2. It is intended to be a graduate text as well as a research monograph. Semisimple and cosemisimple Hopf algebras Webeginbyrecallingwell-knownfactsaboutsuchanH,theproofsofwhichmaybefound in [TngGlk] and [LrsRdf]. Other examples of Frobenius extensions are pairs of group algebras associated to a subgroup of finite index, Hopf subalgebras of a semisimple Hopf algebra, Galois extensions and certain von Neumann algebra subfactors of finite index. We prove a criterion for semisimplicity and analyze the square of the antipode S2 of a semisimple weak Hopf algebra A. Algebra structure 8 1.2. Title: Some Properties of Finite-Dimensional Semisimple Hopf Algebras. The first Weyl algebra is an example of such on algebra with n=2, for instance. We also make use of the classification of semisimple Hopf algebras in dimension p and p q [26, 14, 6, 9]. We study Hopf algebras via tools from geometric invariant theory. Self-dual Modules of Semisimple arXiv:math/0106254v2 [math.RA] 3 Jul 2001 Hopf Algebras Yevgenia Kashina Yorck Sommerh¨auser Yongchang Zhu Abstract We prove that, over an algebraically closed field of characteristic zero, a semisimple Hopf algebra that … Contents Introduction and Main Results 1 Conventions and Notation. One of the methods consists of using the theory of Hopf algebra extensions (see Section 1.5) which started with the devel-opments of Blattner, Montgomery, Schneider and others. Results on the Gorenstein condition and on the global dimension of the corresponding fixed subrings are also provided. (c) There exists a nontrivial semisimple Hopf algebra which is simple in dimension 36. 7 Chapter 1. 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. NORMAL HOPF SUBALGEBRAS OF SEMISIMPLE HOPF ALGEBRAS SEBASTIAN BURCIU Abstract. Semisimple Hopf algebras deserve to be considered as "quantum" analogue of finite groups, but even so, the problem remains extremely hard (even in low dimensions) and very little is known. are either group algebras or the dual of group algebras). A great boost to the theory of Hopf algebras was given by Drinfeld who invented the fundamental class of (quasi)triangular Hopf algebras (A,R). G. Mason, Siu-Hung Ng, Central invariants and Frobenius–Schur indicators for semisimple quasi-Hopf algebras, preprint, arXiv:math.QA/0303213. 198 1221 View the article online for updates and enhancem Coinvariants It is known that a semisimple Hopf algebra of dimension p n is always semisolvable [13, 12]. Semisimple Hopf algebras deserve to be considered as "quantum" analogue of finite groups, but even so, the problem remains extremely hard (even in low dimensions) and very little is known. for semisimple Hopf algebras. This concerns an open question raised by S. Mont-gomery; see [1, Question 4.17]. [LR] R. Larson and D. Radford,Semisimple Hopf algebras, Journal of Algebra, to appear. January 29, 2002 Abstract Given any nontrivial alternating tri-character f on a finite abelian group G, one can construct a finite dimensional non-commutative and non-cocommutative semisimple Hopf algebra H. The group of … 2, Cambridge University Press, Cambridge, 1995, pp. … 1.2 Thesis Organization The thesis is organized as follows: Chapter 1 introduces related background and devel-opment of finite dimensional Hopf algebras. In dimension < 60 this is the only possible such Hopf algebra, by [16]. Introduction. We show that all the invariants we get can be constructed using the integrals of the Hopf algebra and its dual together with the multiplication and the comultiplication, and that these invariants determine the isomorphism class of the Hopf algebra. In Chapter 2, we give some basic definitions, important theorems and useful lemmas of Hopf algebras for remainder discussion. Full-text: Open access . Let H be a finite dimensional non-semisimple Hopf algebra over an algebraically closed field k of characteristic 0. Semisimple Hopf Algebras of dimension 32 In this talk we will discuss classification of semisimple Hopf algebras of dimension 32. Semisimple Hopf algebras and their depth two Hopf subalgebras Item Preview remove-circle Share or Embed This Item. The twist contains a scalar fraction which makes impossible the definability of such Hopf algebras over number rings. G. Mason, The quantum double of a finite group and its role in conformal field theory, in; Groups ’93 Galway/St. We develop the theory of semisimple weak Hopf algebras and obtain analogues of a number of classical results for ordinary semisimple Hopf algebras. Math. zbMATH CrossRef MathSciNet Google Scholar [MD] A. Masuoka and Y. Doi,Generalization of cleft comodule algebras, Communications in Algebra20 (1992), 3703–3721. EMBED (for wordpress.com hosted blogs and … Masuoka proved that for a prime p, semisimple Hopf algebras of dimension 2p over an algebraically closed field k of characteristic 0, are trivial (i.e. Introduction. semisimple Hopf algebras over the eld of complex numbers, with an additional condition on the existence of an involution), of dimension 3p over k,wherepis prime, are trivial [IK]. The purpose of this paper is to begin to lay the foundations of a general theory of actions of Hopf algebras on noncommutative algebras. In particular, every normal Hopf subalgebra of a semisimple Hopf algebra H is the kernel of a representation of H. The maximal normal Hopf … Any semisimple Hopf algebra of dimension pq over k,wherepand q are distinct prime numbers, is trivial. Semisimple Hopf algebra Semisolvability Radford biproduct Character Drinfelddouble Let q be a prime number, k an algebraically closed field of char-acteristic 0, and H a semisimple Hopf algebra of dimension 2q3. Similar properties to those of the kernel of a group representation are pro They are Drinfel'd twists of certain group algebras. N2 - We show that there is a family of complex semisimple Hopf algebras that do not admit a Hopf order over any number ring. We prove that a depth two Hopf subalgebra K of a semisimple Hopf algebra H is normal (where the ground field k is algebraically closed of characteristic zero). A great boost to the theory of Hopf algebras was given by Drinfeld who invented the fundamental class of (quasi)triangular Hopf algebras (A,R). December 2008; Algebras and Representation Theory 15(3) DOI: 10.1007/s10468-010-9252 … Semisimple Hopf Algebras 8 1.1. Authors: Pavel Etingof, Shlomo Gelaki (Submitted on 11 Dec 1997) Abstract: Kaplansky conjectured that if H is a finite-dimensional semisimple Hopf algebra over an algebraically closed field k of characteristic 0, then H is of Frobenius type (i.e. 405–417. Semisimple Triangular Hopf Algebras and Tannakian Categories Item Preview remove-circle Share or Embed This Item. Similar properties to the kernel of a group representation are proved in some special cases. We will fix an abelian group of grouplike elements of order 16 and describe all nonisomorphic semisimple Hopf algebras of dimension 32 with this group of grouplike elements. SEMISIMPLE HOPF ALGEBRAS AND THEIR DEPTH TWO HOPF SUBALGEBRAS LARS KADISON Abstract. if V is an irreducible representation of H then dimV divides dimH). Thus, a natural conjecture is: Conjecture 1. EMBED (for wordpress.com hosted blogs and … classify semisimple Hopf algebras in terms of group algebras and their duals, and twists of them. Abstract. Non-commutative, Non-cocommutative Semisimple Hopf Algebras arise from Finite Abelian Groups Siu-Hung Ng Mathematics Department, Towson University, Towson, MD 21252. M. MügerFrom … Thispaperprovesthat H isalwayssemisolvable.Thatis,suchHopf algebras can be obtained by (a number of) extensions from group OpenURL . Finally, if H is a finite dimensional semisimple Hopf algebra, we consider when semisimplicity or semiprimeness of A implies that of A #„ H; in particular this is true if the (weak) action of H is inner. This example gives a negative answer to [11, Question, pp. Algebras and Representation Theory features carefully refereed papers relating, in its broadest sense, to the structure and representation theory of algebras, including Lie algebras and superalgebras, rings of differential operators, group rings and algebras, C*-algebras and Hopf algebras, with particular emphasis on quantum groups. MIKHAIL KOTCHETOV, Memorial University of Newfoundland Graded modules over simple … On the other hand, we prove that if the group G is nilpotent then any twisting of k G is semisolvable. This implies that non-semisimple Hopf algebras of dimensions 20, 28 and 44 are pointed or dual-pointed. Sbornik: Mathematics 6HPLVLPSOHILQLWH GLPHQVLRQDO+RSIDOJHEUDV To cite this article: Vyacheslav A Artamonov 2007 Sb. EMBED. The notion of kernel of a representation of a semisimple Hopf algebra is introduced. EMBED. A coalgebra, C, over a field k is co-semi-simple if it is the sum of simple subcoalgebras [5, page 290]. @MISC{Kadison08semisimplehopf, author = {Lars Kadison}, title = {Semisimple Hopf algebras and their depth two Hopf subalgebras }, year = {2008}} Share. 269]. Andrews, Vol. On Normal Hopf Subalgebras of Semisimple Hopf Algebras. Semisimple Hopf algebras of dimension 12. Google Scholar [M] A. Masuoka,Freeness of Hopf algebras over coideal subalgebras, Communications in Algebra20 (1992), 1353–1373. Google Scholar. This means on the one hand that a Hopf subalgebra is normal when inducing (then restricting) modules several times as opposed to one time … The notion of kernel of a representation of a semisimple Hopf algebra is introduced.
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