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Classification of finitedimensional triangular Hopf algebras with the Chevalley property
 Mathematical Research Letters
"... Abstract. A fundamental problem in the theory of Hopf algebras is the classification and construction of finitedimensional quasitriangular Hopf algebras over C. Quasitriangular Hopf algebras constitute a very important class of Hopf algebras, introduced by Drinfeld. They are the Hopf algebras whose ..."
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Abstract. A fundamental problem in the theory of Hopf algebras is the classification and construction of finitedimensional quasitriangular Hopf algebras over C. Quasitriangular Hopf algebras constitute a very important class of Hopf algebras, introduced by Drinfeld. They are the Hopf algebras whose representations form a braided tensor category. However, this intriguing problem is extremely hard and is still widely open. Triangular Hopf algebras are the quasitriangular Hopf algebras whose representations form a symmetric tensor category. In that sense they are the closest to group algebras. The structure of triangular Hopf algebras is far from trivial, and yet is more tractable than that of general Hopf algebras, due to their proximity to groups. This makes triangular Hopf algebras an excellent testing ground for general Hopf algebraic ideas, methods and conjectures. A general classification of triangular Hopf algebras is not known yet. However, the problem was solved in the semisimple case, in the minimal triangular pointed case, and more generally for triangular Hopf algebras with the Chevalley property. In this paper we report on all of this, and explain in full details the mathematics and ideas involved in this theory. The classification in the semisimple case relies on Deligne’s theorem on Tannakian categories and on Movshev’s theory in an essential way. We explain Movshev’s theory in details, and refer to [G5] for a detailed discussion of the first aspect. We also discuss the existence of grouplike elements in quasitriangular semisimple Hopf algebras, and the representation theory of cotriangular semisimple Hopf algebras. We conclude the paper with a list of open problems; in particular with the question whether any finitedimensional triangular Hopf algebra over C has the Chevalley property. 1.
A FrobeniusSchur theorem for Hopf algebras
 Alg. Rep. Theory
"... In this note we prove a generalization of the FrobeniusSchur theorem for finite groups for the case of semisimple Hopf algebra over an algebraically closed field of characteristic 0. A similar result holds in characteristic p> 2 if the Hopf algebra is also cosemisimple. In fact we show a more gener ..."
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In this note we prove a generalization of the FrobeniusSchur theorem for finite groups for the case of semisimple Hopf algebra over an algebraically closed field of characteristic 0. A similar result holds in characteristic p> 2 if the Hopf algebra is also cosemisimple. In fact we show a more general version
Grouptheoretical properties of nilpotent modular categories, eprint arXiv:0704.0195v2 [math.QA
"... Abstract. We characterize a natural class of modular categories of prime power FrobeniusPerron dimension as representation categories of twisted doubles of finite pgroups. We also show that a nilpotent braided fusion category C admits an analogue of the Sylow decomposition. If the simple objects o ..."
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Cited by 13 (3 self)
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Abstract. We characterize a natural class of modular categories of prime power FrobeniusPerron dimension as representation categories of twisted doubles of finite pgroups. We also show that a nilpotent braided fusion category C admits an analogue of the Sylow decomposition. If the simple objects ofC have integral FrobeniusPerron dimensions then C is grouptheoretical in the sense of [ENO]. As a consequence, we obtain that semisimple quasiHopf algebras of prime power dimension are grouptheoretical. Our arguments are based on a reconstruction of twisted group doubles from Lagrangian subcategories of modular categories (this is reminiscent to the characterization of doubles of quasiLie bialgebras in terms of Manin pairs given in [Dr]). 1. introduction In this paper we work over an algebraically closed field k of characteristic 0. By a fusion category we mean a klinear semisimple rigid tensor category C with finitely many isomorphism classes of simple objects, finite dimensional spaces of morphisms, and such that the unit object 1 of C is simple. We refer the reader to [ENO] for a general theory of such categories. A fusion category is pointed if all its simple objects are invertible. A pointed fusion category is equivalent to Vec ω G, i.e., the category of Ggraded vector spaces with the associativity constraint given by some cocycle ω ∈ Z 3 (G, k × ) (here G is a finite group). 1.1. Main results. Theorem 1.1. Any braided nilpotent fusion category has a unique decomposition into a tensor product of braided fusion categories whose FrobeniusPerron dimensions are powers of distinct primes. The notion of nilpotent fusion category was introduced in [GN]; we recall it in Subsection 2.2. Let us mention that the representation category Rep(G) of a finite group G is nilpotent if and only if G is nilpotent. It is also known that fusion categories of prime power FrobeniusPerron dimension are nilpotent [ENO]. On the other hand, Vec ω G is nilpotent for any G and ω. Therefore it is not true that any nilpotent fusion category is a tensor product of fusion categories of prime power dimensions.
Computing the FrobeniusSchur indicator for abelian extensions of Hopf algebras
"... Let H be a finitedimensional semisimple Hopf algebra. Recently it was shown in [LM] that a version of the FrobeniusSchur theorem holds for Hopf algebras, and thus that the Schur indicator ν(χ) of the character χ of a simple Hmodule is welldefined; this fact for the special case of Kac algebras w ..."
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Let H be a finitedimensional semisimple Hopf algebra. Recently it was shown in [LM] that a version of the FrobeniusSchur theorem holds for Hopf algebras, and thus that the Schur indicator ν(χ) of the character χ of a simple Hmodule is welldefined; this fact for the special case of Kac algebras was shown in [FGSV]. In this paper we
Duals of pointed Hopf algebras
"... algebra of V = ⊕iV χi gi k[Γ] In this paper, we study the duals of some finite dimensional pointed Hopf algebras working over an algebraically closed field k of characteristic 0. In particular, we study pointed Hopf algebras with coradical k[Γ] for Γ a finite abelian group, and with associated grade ..."
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Cited by 8 (3 self)
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algebra of V = ⊕iV χi gi k[Γ] In this paper, we study the duals of some finite dimensional pointed Hopf algebras working over an algebraically closed field k of characteristic 0. In particular, we study pointed Hopf algebras with coradical k[Γ] for Γ a finite abelian group, and with associated graded Hopf algebra of the form B(V)#k[Γ] where B(V) is the Nichols k[Γ] ∈ YD. As a corollary to a general theorem on duals of coradically graded Hopf algebras, we have that the dual of B(V)#k[Γ] is B(W)#k [ ˆ Γ] where W = ⊕iW gi χi ∈k[ ˆ Γ] YD. This description of the dual is used to explicitly describe k[ˆΓ] the Drinfel’d double of B(V)#k[Γ]. We also show that the dual of a nontrivial lifting A of B(V)#k[Γ] which is not itself a Radford biproduct, is never pointed. For V a quantum linear space of dimension 1 or 2, we describe the duals of some liftings of B(V)#k[Γ]. We conclude with some examples where we determine all the irreducible finitedimensional representations of a lifting of B(V)#k[Γ] by computing the matrix coalgebras in the coradical of the dual. 1
Hopf Algebra Extensions and Cohomology
"... Abstract. This is an expository paper on ‘abelian ’ extensions of (quasi) Hopf algebras, which can be managed by the abelian cohomology, with emphasis on the author’s recent results which are motivated by an exact sequence due to George Kac. The cohomology plays here an important role in constructi ..."
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Abstract. This is an expository paper on ‘abelian ’ extensions of (quasi) Hopf algebras, which can be managed by the abelian cohomology, with emphasis on the author’s recent results which are motivated by an exact sequence due to George Kac. The cohomology plays here an important role in constructing and classifying those extensions, and even their cocycle deformations. We see also a strong connection of Hopf algebra extensions arising from a (matched) pair of Lie algebras with Lie bialgebra extensions.
PRODUCTS IN HOCHSCHILD COHOMOLOGY AND GROTHENDIECK RINGS OF GROUP CROSSED PRODUCTS
, 2002
"... Abstract. We give a general construction of rings graded by the conjugacy classes of a finite group. Some examples of our construction are the Hochschild cohomology ring of a finite group algebra, the Grothendieck ring of the Drinfel’d double of a group, and the orbifold cohomology ring for a global ..."
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Cited by 4 (1 self)
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Abstract. We give a general construction of rings graded by the conjugacy classes of a finite group. Some examples of our construction are the Hochschild cohomology ring of a finite group algebra, the Grothendieck ring of the Drinfel’d double of a group, and the orbifold cohomology ring for a global quotient. We generalize the first two examples by deriving product formulas for the Hochschild cohomology ring of a group crossed product and for the Grothendieck ring of an abelian extension of Hopf algebras. Our results account for similarities in the product structures among these examples. 1991 Mathematics Subject Classification. Primary: 16E40, 16E20 1.
Hopf algebra deformations of binary polyhedral groups
"... Abstract. We show that semisimple Hopf algebras having a selfdual faithful irreducible comodule of dimension 2 are always obtained as abelian extensions with quotient Z2. We prove that nontrivial Hopf algebras arising in this way can be regarded as deformations of binary polyhedral groups and descr ..."
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Cited by 2 (1 self)
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Abstract. We show that semisimple Hopf algebras having a selfdual faithful irreducible comodule of dimension 2 are always obtained as abelian extensions with quotient Z2. We prove that nontrivial Hopf algebras arising in this way can be regarded as deformations of binary polyhedral groups and describe its category of representations. We also prove a strengthening of a result of Nichols and Richmond on cosemisimple Hopf algebras with a 2dimensional irreducible comodule in the finite dimensional context. Finally, we give some applications to the classification of certain classes of semisimple Hopf algebras. 1. Introduction and
and
, 2002
"... Let H be a finite dimensional nonsemisimple Hopf algebra over an algebraically closed field k of characteristic 0. If H has no nontrivial skewprimitive elements, we find some bounds for the dimension of H1, the second term in the coradical filtration of H. Using these results, we are able to show ..."
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Let H be a finite dimensional nonsemisimple Hopf algebra over an algebraically closed field k of characteristic 0. If H has no nontrivial skewprimitive elements, we find some bounds for the dimension of H1, the second term in the coradical filtration of H. Using these results, we are able to show that every Hopf algebra of dimension 14, 55 and 77 is semisimple and thus isomorphic to a group algebra or the dual of a group algebra. We also have some partial results in the classification problem for dimension 16. 0