ideals and quotients in crossed products of hopf algebras let h be a hopf algebra over a field k acting on the k algebra r with action twisted by a cocycle t such that the crossed product algebra r h can be constructed this paper is concerned with the relationship between the ideals of r hand those of r and with computing the extended cen ter and symmetric quotient ring of r h in terms of the extended center c and symmetric quotient ring q of r our best results are obtained when h is an irreducible hopf algebra or more generally when h is of the form k kg where k is irreducible and kg is the group algebra of a group g some examples of irreducible hopf algebras are enveloping algebras of lie algebras u l along with their restricted counterparts u l in characteristic

prime ideals in group algebras oe polycyclic by finite groups index such that the prime ideals of k g0 ca n de described in a particularly nice manner furthermore he showed that this type of description does not in general apply to k g in this paper we offer a slightly different somewhat more complicated formulation which does indeed describe the primes of k g observe that the group algebra k g can be written as k g k g0 g go a crossed product of the finite group h g go over the ring r k g0 furthermore a recent paper 5 of the authors studies the general crossed product situation r h with h finite and offers a rather complicated scheme for describing the prime ideals of

structure theorems of h4 azumaya algebras abstract let k be a field and h 4 be sweedler s 4 dimensional algebra over k it is well known that h 4 has a family of triangular structures r t indexed by the ground field k and each triangular structure r t makes the h 4 module category m h 4 a braided monoidal category denoted m r t h 4 in this paper we study the azumaya algebras in the categories m r t h 4 we obtain the structure theorems for azumaya algebras in each braided monoidal category m r t h 4 t k utilizing the structure theorems we obtain a scalar invariant for each azumaya algebra in the aforementioned categories

weak c cleft extensions weak entwining structures and weak hopf algebras abstract we formulate the concept of weak cleft extension for a weak entwining structure in a braided monoidal category c with equalizers and coequalizers we prove that if a is a weak c cleft extension then there is an isomorphism of algebras between a and a subobject of the tensor product of a c and c where a c is a subalgebra of a also we prove the corresponding dual results and linking the information of this two parts we obtain a general property for a pair morphisms f c a and g a c of algebras and coalgebras satisfying certain conditions finally as particular instances we get the results of fernandez and rodriguez the theorems of radford majid and bespalov in the case of hopf algebras with projection and the ones obtained by alonso and gonzalez for weak hopf algebras living in a symmetric category with split idempotents for example the weak theorem of blattner cohen and montgomery for weak hopf algebras with coalgebra splitting is one of them

a duality theorem for hopf module algebras let g be a locally compact abelian group acting continuously on a von neumann algebra a via a homomorphism ci of g into aut a in 1973 takesaki showed how an action oi of the dual group g could be naturally defined on the von neumann algebra crossed product d x g and proved that the double crossed product 4 x a g x ti g was naturally isomorphic to where 6 is the ring of all bounded operators on l g this duality theorem for group actions was extended to arbitrary locally compact groups g independently by nakagami landstad and stratilavoiculescu zsido in the mid 1970s this extension required replacing the action of g on x r g by a coaction of g on x a g indeed coactions of g are defined in general and it is proved that jl e x rr g x d g n y 0 6 for any action c of g on and corresponding coaction oi of g on x g a similar theorem is proved reversing the roles of c1 and oi an exposition of this theorem may be found in nakagami takesaki 7 where complete references to the literature may also be found nakagami and takesaki phrase things in a dual fashion to handle the topological subtleties more easily in 2 cohen and montgomery recognized that for finite groups g a coaction of g on an f algebra a where f is a field is just a g grading of a it was known that a g grading of an algebra a is the same as an fg module algebra action of fg the hopf algebra dual to fg on a given these facts they were able to prove the following duality theorems

cocommutative calabi yau hopf algebras and deformations the calabi yau property of cocommutative hopf algebras is discussed by using the homological integral a recently introduced tool for studying infinite dimensional as gorenstein hopf algebras it is shown that the skew group algebra of a universal enveloping algebra of a finite dimensional lie algebra g with a finite subgroup g of automorphisms of g is calabi yau if and only if the universal enveloping algebra itself is calabi yau and g is a subgroup of the special linear group sl g the noetherian cocommutative calabi yau hopf algebras of dimension not larger than 3 are described the calabi yau property of sridharan enveloping algebras of finite dimensional lie algebras is also discussed we obtain some equivalent conditions for a sridharan enveloping algebra to be calabi yau and then partly answer a question proposed by berger we list all the nonisomorphic 3 dimensional calabi yau sridharan enveloping algebras

hopf algebra actions abstract finite groups acting on rings by automorphisms and group graded rings are instances of hopf algebras h acting on h module algebras a we study such actions let a h a a h a e h a all h h the ring of h invariants and form the smash product a h we study the ring extensions ah a a h we prove a maschke type theorem for a h modules we form an associated morita context ah a a a h and use these to get connections between the various rings

on hopf algebras with cocommutative coradicals 1 zf k is a right coideal subalgebra of h such that s k k h is faithfully flat as a right or left k module 2 zf x h n h is a quotient left h module coalgebra h is faithfully coflat as a right or left x h comodule 3 k h nk and 7 h h establish a 1 1 correspondence between the right coideal subalgebras k of h such that s k ko and the quotient left h module coalgebras n h n h 4 suppose k x in 3 there is a left x h comodule and right k module isomorphism h n x h q k if either a h is pointed or b k is finite dimensional over k

quantum groups on fibre bundles it is shown that the principle of locality and noncommutative geometry can be connected by a sheaf theoretical method in this framework quantum spaces are introduced and examples in mathematical physics are given within the language of quantum spaces noncommutative principal and vector bundles are defined and their properties are studied important constructions in the classical theory of principal fibre bundles like associated bundles and differential calculi are carried over to the quantum case at the endq deformed instanton models are introduced for every integral index

extensions of hopf algebras and lie bialgebras let f g be finite dimensional lie algebras over a field of characteristic zero regard f and g the dual lie coalgebra of g as lie bialgebras with zero cobracket and zero bracket respectively suppose that a matched pair f g of lie bialgebras is given which has structure maps then it induces a matched pair u f ug of hopf algebras where u f is the universal envelope of f and ug is the hopf dual of ug we show that the group opext u f ug of cleft hopf algebra extensions associated with u f ug is naturally isomorphic to the group opext f g of lie bialgebra extensions associated with f g an exact sequence involving either of these groups is obtained which is a variation of the exact sequence due to g i kac if g g g there follows a bijection between the set ext u f ug of all cleft hopf algebra extensions of u f by ug and the set ext f g of all lie bialgebra extensions of f by g

efficient calculation of the normalizing constant of the autologistic and related models on the cylinder and lattice motivated by the autologistic model for the analysis of spatial binary data on the two dimensional lattice we develop efficient computational methods for calculating the normalizing constant for models for discrete data defined on the cylinder and lattice because the normalizing constant is generally unknown analytically statisticians have developed various ad hoc methods to overcome this difficulty our aim is to provide computationally and statistically efficient methods for calculating the normalizing constant so that efficient likelihood based statistical methods are then available for inference we extend the so called transition method to find a feasible computational method of obtaining the normalizing constant for the cylinder boundary condition to extend the result to the free boundary condition on the lattice we use an efficient path sampling markov chain monte carlo scheme the methods are generally applicable to association patterns other than spatial such as clustered binary data and to variables taking three or more values described by for example potts models

morphisms of relative hopf modules smash products and duality the aim of this paper is to extend a result of h j schneider on endomorphism rings of relative hopf modules 17 theorem 3 2 to the case of projective hopf algebras with bijective antipode and to show the connection between this extension and several duality theorems for hopf algebras a consequence of this generalization is the following if h is a co frobenius hopf algebra over the field k a b is a hopf galois extension and m is a right a h hopf module then considering the usual ring embeddings

crossed products for weak hopf algebras with coalgebra splitting in this paper we prove that if g b h is a morphism of weak hopf algebras which is split as a coalgebra morphism then there exists a subalgebra bh of b morphisms bh h bh bh bh h h bh and an isomorphism of algebras and right h comodules bb b bh h being bh h a subobject of bh h with its algebra structure defined by a crossed product also we obtain the dual results and as a consequence we prove the radford s theorem for weak hopf algebras

oscillation results for emden fowler type differential equations abstract the third order nonlinear differential equation x a t x b t f x 0 is considered we present oscillation and nonoscillation criteria which extend and improve previous results existing in the literature in particular some results recently stated by m gregus and m gregus jr j math anal appl 181 1994 575 585 in addition contributions to the classification of solutions are given the techniques used are based on a transformation which reduces to a suitable disconjugate form to this aim auxiliary results on the asymptotic behavior of solutions of a second order linear differential equation associated to are stated they are presented in an independent form because they may be applied also to simplify and improve other qualitative problems concerning differential equations with quasiderivatives

a duality theorem for crossed products of hopf algebras several variations have appeared recently see l 5 8 101 among the best are 2 2 1 3 2 2 but the former is restricted to the case of a trivial cocycle 0 and the latter to the case of finite dimensional h the main purpose of this paper is to prove a common generalization of these two theorems in section 2 we first recall preliminaries about crossed products then we introduce some new algebra constructions and we prove related useful results in particular we define an algebra a h op u that will replace a h u in the main theorem and we show a relationship between the two algebras in the new construction u is allowed to be more general for example we may take u h section 3 introduces a subalgebra h of h with a comodule structure co h h h they are our tools to formulate and handle certain awkward assumptions in the main theorem these can be regarded as finiteness conditions and they correspond to the u local finiteness and rl condition of 2 section 4 contains the duality theorem and its proof in section 5 we draw some corollaries in particular we show that 2 2 1 3 2 2 do follow from our theorem 153 0021 8693 92 3 00

integrals quantum galois extensions and the affineness criterion for quantum yetter drinfel d modules abstract in this paper we shall generalize the notion of an integral on a hopf algebra introduced by sweedler by defining the more general concept of an integral of a threetuple h a c where h is a hopf algebra coacting on an algebra a and acting on a coalgebra c we prove that there exists a total integral c hom c a of h a c if and only if any representation of h a c is injective in a functorial way as a corepresentation of c in particular the quantum integrals associated to yetter drinfel d modules are defined let now a be an h bicomodule algebra h y d a the category of quantum yetter drinfel d modules and b a a s 1 a 1 a 1 a 0 1h a the subalgebra of coinvariants of the verma structure a h y d a we shall prove the following affineness criterion if there exists h hom h a a total quantum integral and the canonical map a b a h a a b b s 1 b 1 b 1 ab 0 is surjective i e a b is a quantum homogeneous space then the induction functor b a m b h y d a is an equivalence of categories the affineness criteria proven by cline parshall and scott and independently by oberst for affine algebraic groups schemes and schneider in the noncommutative case are recovered as special cases

actions of divided power hopf algebras abstract we study prime ideals in enveloping algebra smash products and use a duality construction to obtain results on prime ideals in rings on which divided power hopf algebras act these actions correspond to higher derivations first we consider chains of prime ideals in an enveloping algebra smash product over an arbitrary ring where the lie algebra is assumed to be finite dimensional abelian over a field of positive characteristic we give a bound on the length of such a chain where the ideals all have the same intersection with the coefficient ring then using an explicit construction of a duality theorem of blattner and montgomery in this context we are able to apply results on enveloping algebra smash products to study the invariant ideals of prime ideals in a ring under a locally nilpotent divided power hopf algebra action