We show that there is a family of complex semisimple Hopf algebras that do not admit a Hopf order over any number ring. They are Drinfel'd twists of certain group algebras. The twist contains a scalar fraction which makes impossible the definability of such Hopf algebras over number rings. We also prove that a complex semisimple Hopf algebra satisfies Kaplansky's sixth conjecture if and only if it admits a weak order, in the sense of Rumynin and Lorenz, over the integers.Given a twist J for H a new Hopf algebra H J , called Drinfel'd twist of H, can be constructed as follows: H J = H as an algebra, the counit is that of H, and the new comultiplication and antipode are:Here U J := J (1) S(J (2) ) and, if we write J −1 = J −(1) ⊗ J −(2) , its inverse is U −1 J = S(J −(1) )J −(2) . We next describe the particular example of twist for the group algebra that we will use. This construction was used several times in the past, see for example, [6], [8] or [18] and the references therein. Let M be a finite abelian group and M its group of characters. Consider the group algebra KM and its dual Hopf algebra (KM ) * . We identify (KM ) * and K M as Hopf algebras. Assume that char K ∤ |M |. There is an isomorphism of Hopf algebras (KM ) * ≃ KM induced by an isomorphism of groups M ≃ M . Let {ϑ m } m∈M ⊂ (KM ) * be the dual basis of {m} m∈M . If ω : M × M → K × is a normalized 2-cocycle, then J = m,m ′ ∈M ω(m, m ′ )ϑ m ⊗ ϑ m ′ ∈ (KM ) * ⊗ (KM ) *
ABSTRACT. Harish-Chandra induction and restriction functors play a key role in the representation theory of reductive groups over finite fields. In this paper, extending earlier work of Dat, we introduce and study generalisations of these functors which apply to a wide range of finite and profinite groups, typical examples being compact open subgroups of reductive groups over non-archimedean local fields. We prove that these generalisations are compatible with two of the tools commonly used to study the (smooth, complex) representations of such groups, namely Clifford theory and the orbit method. As a test case, we examine in detail the induction and restriction of representations from and to the Siegel Levi subgroup of the symplectic group Sp 4 over a finite local principal ideal ring of length two. We obtain in this case a Mackey-type formula for the composition of these induction and restriction functors which is a perfect analogue of the well-known formula for the composition of Harish-Chandra functors. In a different direction, we study representations of the Iwahori subgroup In of GLnpFq, where F is a non-archimedean local field. We establish a bijection between the set of irreducible representations of In and tuples of primitive irreducible representations of smaller Iwahori subgroups, where primitivity is defined by the vanishing of suitable restriction functors.
Abstract. We study Hopf algebras via tools from geometric invariant theory. 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. We then define certain canonical subspaces Inv i,j of tensor powers of H and H * , and use the invariant theory to prove that these subspaces satisfy a certain non-degeneracy condition. Using this non-degeneracy condition together with results on symmetric monoidal categories, we prove that the spaces Inv i,j can also be described as (A , where A is the group of Hopf automorphisms of H. As a result we prove that the number of possible Hopf orders of any semisimple Hopf algebra over a given number ring is finite. we give some examples of these invariants arising from the theory of Frobenius-Schur Indicators, and from Reshetikhin-Turaev invariants of three manifolds. We give a complete description of the invariants for a group algebra, proving that they all encode the number of homomorphisms from some finitely presented group to the group. We also show that if all the invariants are algebraic integers, then the Hopf algebra satisfies Kaplansky's sixth conjecture: the dimensions of the irreducible representations of H divide the dimension of H.
Abstract. Let W be a finite dimensional algebraic structure (e.g. an algebra) over a field K of characteristic zero. We study forms of W by using Deligne's Theory of symmetric monoidal categories. We construct a category C W , which gives rise to a subfield K 0 ⊆ K, which we call the field of invariants of W . This field will be contained in any subfield of K over which W has a form. The category C W is a K 0 -form of RepK(Aut(W )), and we use it to construct a generic form W over a commutative K 0 -algebra B W (so that forms of W are exactly the specializations of W ). This generalizes some generic constructions for central simple algebras and for H-comodule algebras. We give some concrete examples arising from associative algebras and H-comodule algebras. As an application, we also explain how one can use the construction to classify two-cocycles on some finite dimensional Hopf algebras.
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