hopf algebras having a dense big cell we discuss some axioms that ensure that a hopf algebra has its simple comodules classified using an analogue of the borel weil construction more precisely we show that a hopf algebra having a dense big cell satisfies the above requirement this method has its roots in the work of parshall and wang in the case of q deformed quantum groups gl and sl here we examine the example of universal cosovereign hopf algebras for which the weight group is the free group on two generators

super cocommutative hopf algebras of finite representation type abstract given an algebraically closed field k of characteristic p 5 we classify the finite algebraic k supergroups whose algebras of measures are of finite representation type let g be such a supergroup and g the largest ordinary algebraic k group determined by g we show that both g and u lie g the restricted enveloping algebra of lie superalgebra of g are of finite representation type moreover only some special representation finite algebraic k groups of dimension zero are shown to appear if g g the structure of g is almost determined by g and u lie g the auslander reiten quivers are determined by showing that they are nakayama algebras

clifford correspondence for finite dimensional hopf algebras abstract let b h be a crossed product algebra over an algebraically closed field with h a finite dimensional hopf algebra we give an explicit equivalence between the category of finite dimensional b h modules whose restriction to b is a direct sum of copies of a stable irreducible b module and the category of modules for a twisted product of h with the field this describes all finite dimensional irreducible b h modules containing a stable irreducible b submodule and thus generalizes the classical stable clifford correspondence for groups in case h is cocommutative we extend this correspondence to the nonstable case

universal deformation formulas and braided module algebras abstract we study formal deformations of a crossed product s v f g of a polynomial algebra with a group induced from a universal deformation formula introduced by witherspoon these deformations arise from braided actions of hopf algebras generated by automorphisms and skew derivations we show that they are non trivial in the characteristic free context even if g is infinite by showing that their infinitesimals are not coboundaries for this we construct a new complex which computes the hochschild cohomology of s v f g

on quiver theoretic description for quasitriangularity of hopf algebras abstract this paper is devoted to the study of the quasitriangularity of hopf algebras via hopf quiver approaches we give a combinatorial description of the hopf quivers whose path coalgebras give rise to coquasitriangular hopf algebras with a help of the quiver setting we study general coquasitriangular pointed hopf algebras and obtain a complete classification of the finite dimensional ones over an algebraically closed field of characteristic 0

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

hopf algebras of dimension pq ii abstract let h be a hopf algebra of dimension pq over an algebraically closed field of characteristic zero where p q are odd primes with p q 4 p 11 we prove that h is semisimple and thus isomorphic to a group algebra or the dual of a group algebra

on semisimple hopf algebras of dimension 2q3 abstract let q be a prime number k an algebraically closed field of characteristic 0 and h a semisimple hopf algebra of dimension 2 q 3 this paper proves that h is always semisolvable that is such hopf algebras can be obtained by a number of extensions from group algebras or duals of group algebras

quantum subgroups of glα β n abstract let c 0 and l n be odd with l 3 we determine all hopf algebra quotients of the quantized coordinate algebra o gl n when 1 is a primitive l th root of unity and satisfy certain mild conditions and we characterize all finite dimensional quotients when 1 is not a root of unity as a byproduct we find a new family of non semisimple and non pointed hopf algebras with non pointed duals which are quotients of o gl n

complexity periodicity and one parameter subgroups using the variety of infinitesimal one parameter subgroups introduced in 55 56 by suslin friedlander bendel we define a numerical invariant for representations of an infinitesimal group scheme g for an indecomposable g module m of complexity 1 this number which may also interpreted as the height of a vertex um g is related to the period of m in the context of the frobenius category of grt modules associated to a smooth reductive group g and a maximal torus t g our methods give control over the behavior of the heller operator of such modules as well as precise values for the periodicity of their restrictions to gr applications include the structure of stable auslander reiten components of grt modules as well as the distribution of baby verma modules

the tame infinitesimal groups of odd characteristic abstract given an infinitesimal group g over an algebraically closed field k of characteristic p 3 we provide criteria for the principal block b 0 g of its algebra of distributions to be of tame representation type these are employed in conjunction with galois coverings to determine the structure of g modulo its multiplicative center as well as the quiver and the relations of the algebra b 0 g

on the theory of frobenius extensions and its application to lie superalgebras by using an approach to the theory of frobenius extensions that emphasizes notions related to associative forms we obtain results concerning the trace and corestriction mappings and transitivity these are employed to show that the extension of enveloping algebras determined by a subalgebra of a lie superalgebra is a frobenius extension and to study certain questions in representation theory

application of particle swarm optimization based on chks smoothing function for solving nonlinear bilevel programming problem particle swarm optimization pso is a new optimization technique originating from artificial life and evolutionary computation it completes optimization through following the personal best solution of each particle and the global best value of the whole swarm pso can be used to solve nonlinear programming problems for global optimal solutions efficiently so a novel approach based on particle swarm optimization is proposed to solve nonlinear bilevel programming problem nblp in the proposed approach applying karush kuhn tucker kkt condition to the lower level problem we transform the nblp into a regular nonlinear programming with complementary constraints which is sequentially smoothed by chen harker kanzow smale chks smoothing function the pso approach is then applied to solve the smoothed nonlinear programming for getting the approximate optimal solution of the nblp problem simulations on 5 benchmark problems and practical example about watershed water trading decision making problem are made and the results demonstrate the effectiveness of the proposed method for solving nblp

semisimple hopf algebras of dimension 60 abstract we determine the isomorphism classes of semisimple hopf algebras of dimension 60 which are simple as hopf algebras

spectra of quantized hyperalgebras we describe the prime and primitive spectra for quantized enveloping algebras at roots of 1 in characteristic zero in terms of the prime spectrum of the underlying enveloping algebra our methods come from the theory of hopf algebra crossed products for primitive ideals we obtain an analogue of duflo s theorem which says that every primitive ideal is the annihilator of a simple highest weight module this depends on an extension of lusztig s tensor product theorem

on the ell 2 betti numbers of universal quantum groups we show that the first ell 2 betti number of the duals of the free unitary quantum groups is one and that all ell 2 betti numbers vanish for the duals of the quantum automorphism groups of full matrix algebras

substructures of bi frobenius algebras abstract bi frobenius algebras or bf algebras were recently introduced by the author and takeuchi these are both frobenius algebras and frobenius coalgebras and satisfy some compatibility conditions the concept generalizes finite dimensional hopf algebras in section 1 we give conditions for finite dimensional algebras and coalgebras to be bf algebras in section 2 we discuss substructures quotient structures of bf algebras section 3 is devoted a study of morphisms and we deduce some results in koppinen s theory

an approach to quasi hopf algebras via frobenius coordinates abstract we study quasi hopf algebras and their subobjects over certain commutative rings from the point of view of frobenius algebras we introduce a type of radford formula involving an anti automorphism and the nakayama automorphism of a frobenius algebra then view several results in quantum algebras from this vantage point in addition separability and strong separability of quasi hopf algebras are studied as frobenius algebras

the brauer group of a cocommutative coalgebra we construct a mayer vietoris type exact sequence for the brauer group of a cocommutative irreducible coalgebra c as an application the brauer group of c and its universal connected coalgebra r c are related

primitive cohomology of hopf algebras abstract primitive cohomology of a hopf algebra is defined by using a modification of the cobar construction of the underlying coalgebra among many of its applications two classifications are presented firstly we classify all non locally pi pointed hopf algebra domains of gelfand kirillov dimension two and secondly we classify all pointed hopf algebras of rank one the first classification extends some results of brown goodearl and others in an ongoing project to understand all hopf algebras of low gelfand kirillov dimension the second generalizes results of krop radford and wang you chen which classified hopf algebras of rank one under extra hypothesis properties and algebraic structures of the primitive cohomology are discussed

braided module and comodule algebras galois extensions and elements of trace 1 let k be a field and let h be a rigid braided hopf k algebra in this paper we continue the study of the theory of braided hopf crossed products began in j a guccione j j guccione theory of braided hopf crossed products j algebra 261 2003 54 101 first we show that to have an h braided comodule algebra is the same that to have an h braided module algebra where h is a variant of h and then we study the maps and that appear in the morita context introduced in the above cited paper

bifurcations of beam beam like maps the bifurcations of a class of mappings including the beam beam map are examined these maps are asymptotically linear at infinity where they exhibit invariant curves and elliptic periodic points the dynamical behaviour is radically different with respect to the henon like polynomial maps whose stability boundary dynamic aperture is at a finite distance rather than the period doubling bifurcations exhibited by the henon like maps we observe a systematic appearance of tangent bifurcations and in phase space one observes the disappearance of chains of islands born from the origin and coming from infinity this behaviour has relevant consequences on the transport process

rigid dualizing complexes over quantum homogeneous spaces abstract a quantum homogeneous space of a hopf algebra is a right coideal subalgebra over which the hopf algebra is faithfully flat it is shown that the auslander gorenstein property of a hopf algebra is inherited by its quantum homogeneous spaces if the quantum homogeneous space b of a pointed hopf algebra h is as gorenstein of dimension d then b has a rigid dualizing complex b d the nakayama automorphism is given by ad g s 2 where ad g is the inner automorphism associated to some group like element g h and is the algebra map determined by the left integral of b the quantum homogeneous spaces of u q sl 2 are classified and all of them are proved to be auslander regular as regular and cohen macaulay

galois correspondences for hopf bigalois extensions abstract lethbe a hopf algebra with bijective antipode over a commutative ringk a righth galois extension ofkis a righth comodule algebraasuch thatk aco hand a certain canonical mapa a a his a bijection we investigate galois connections for hopf galois extensions that can be formulated with the help of an additional hopf algebralover whichais also a leftl galois extension ofk and anl h bicomodule such an additional hopf algebra always exists and is unique up to isomorphism the galois connection between quotient coalgebras and left modules oflandh subcomodule algebras ofainduces a bijection between those quotients over whichlis faithfully coflat and those subalgebras over whichais faithfully flat as a consequence the lattices of hopf subalgebras oflandhover which these are faithfully flat are isomorphic

on modules associated to coalgebra galois extensions abstract for a given entwining structure a c involving an algebraa a coalgebrac and an entwining map c a a c a categorymca of right a c modules is defined and its structure analysed in particular the notion of a measuring of a c to a c is introduced and certain functors betweenmca andmca induced by such a measuring are defined it is shown that these functors are inverse equivalences iff they are exact or one of them faithfully exact and the measuring satisfies a certain galois type condition next left moduleseand right modules associated to ac galois extensionaofbare defined these can be thought of as objects dual to fibre bundles with coalgebracin the place of a structure group and a fibrev cross sections of such associated modules are defined as module mapse bor b it is shown that they can be identified with suitably equivariant maps from the fibre toa also it is shown that ac galois extension is cleft if and only ifa b cas leftb modules and rightc comodules the relationship between the moduleseand is studied in the case whenvis finite dimensional and in the case when the canonical entwining map is bijective

a multi level non linear multi objective decision making under fuzziness this paper studies a three level non linear multi objective decision making tln modm problem with linear or non linear constraints and in which the objective function at every level are non linear functions which are to be maximized this paper makes an extension work of abo sinna j operat res soc india opsearch 38 5 2001 484 495 which deal with a bi level non linear multi objective decision making problem under fuzziness the three level programming tlp problem whether from the stand point of the three planner stackelberg behavior or from the interactive organizational behavior is a very practical problem and encountered frequently in actual practice this paper proposes a three planner multi objective decision making model and solution method for solving this problem this method uses the concepts of tolerance membership function and multi objective optimization at each level to develop a fuzzy max min decision model for generating pareto optimal satisfactory solution for tln modm problem the first level decision maker fldm specifies his her objective functions and decisions with possible tolerances which are described by membership functions of fuzzy set theory then the second level decision maker sldm specifies his her objective functions and decisions in the view of the fldm with possible tolerances which are described by membership functions of fuzzy set theory finally the third level decision maker tldm uses the preference information for the fldm and sldm to solves his her problem subject to the two upper level decision makers restrictions an illustrative numerical example is given to demonstrate the obtained results

on the structure of tame graded basic hopf algebras ii abstract in this paper firstly we give a complete list of tame local graded frobenius algebras then by the method of bosonization and this list we determine the structure of tame radically graded basic hopf algebras completely

support varieties ar components and good filtrations let g be a reductive group defined over the galois field mathbb f p with p being good for g using support varieties and covering techniques based on grt modules we determine the position of simple modules and baby verma modules within the stable auslander reiten quiver s gr of the rth frobenius kernel of g in particular we show that the almost split sequences terminating in these modules usually have an indecomposable middle term concerning support varieties we introduce a reduction technique leading to isomorphisms mathcal v g r z r lambda cong mathcal v g r d z r d mu for baby verma modules of certain highest weights lambda mu in x t which are related by the notion of depth

application of particle swarm optimization algorithm for solving bi level linear programming problem bi level linear programming is a technique for modeling decentralized decision it consists of the upper level and lower level objectives this paper attempts to develop an efficient method based on particle swarm optimization pso algorithm with swarm intelligence the performance of the proposed method is ascertained by comparing the results with genetic algorithm ga using four problems in the literature and an example of supply chain model the results illustrate that the pso algorithm outperforms ga in accuracy

polyhedral groups mckay quivers and the finite algebraic groups with tame principal blocks given an algebraically closed field k of characteristic p 3 we classify the finite algebraic k groups whose algebras of measures afford a principal block of tame representation type the structure of such a group mathcal g is largely determined by a linearly reductive subgroup scheme hat mathcal g of sl 2 with the mckay quiver of hat mathcal g relative to its standard module being the gabriel quiver of the principal block mathcal b 0 mathcal g the graphs underlying these quivers are extended dynkin diagrams of type tilde a tilde d or tilde e and the tame blocks are morita equivalent to generalizations of the trivial extensions of the radical square zero tame hereditary algebras of the corresponding type

theory of braided hopf crossed products abstract we define a type of crossed product over braided hopf algebras which generalizes the ones introduced by blattner cohen and montgomery and doi and takeuchi and we study some of its properties for instance we prove maschke s theorem for these new crossed products and we construct a natural morita context which extends the one obtained by cohen fischman and montgomery

hopf module duality applied to x outer galois theory we work over a fixed ground field k in arbitrary characteristic let r be a prime algebra and let q denote the symmetric martindale quotient ring of r we know from 10 section 6 4 that to study group actions on r it is fundamental to consider the extended actions on q and those actions which are x outer or outer as automorphisms of q have nice properties which could be generalized for x outer in some appropriate sense hopf algebra actions to recall kharchenko s galois type correspondence theorem let g be a finite group of x outer automorphisms of r it was proved in 2 theorem 3 10 2 11 theorem b that

galois objects over generalized drinfeld doubles with an application to uq sl2 abstract ralf gunther has determined all the cleft extensions over the finite quotient hopf algebra uq s l 2 of the quantized universal enveloping algebra of s l 2 at a root of unity r gunther ph d thesis universitat munchen 1999 his techniques applications of the diamond lemma are similar to those used by a masuoka comm algebra22 1994 4537 4559 for the two generator taft algebras in the present paper we give another proof of a special case of gunther s classification namely the case of cleft galois extensions of the base field the idea is that uq s l 2 is the quotient of the drinfeld double of a taft algebra by a normal hopf subalgebra we use techniques that allow us to calculate all galois objects of such a composed hopf algebra

skew calabi yau algebras and homological identities abstract a skew calabi yau algebra is a generalization of a calabi yau algebra which allows for a non trivial nakayama automorphism we prove three homological identities about the nakayama automorphism and give several applications the identities we prove show i how the nakayama automorphism of a smash product algebra a h is related to the nakayama automorphisms of a graded skew calabi yau algebra a and a finite dimensional hopf algebra h that acts on it ii how the nakayama automorphism of a graded twist of a is related to the nakayama automorphism of a and iii that the nakayama automorphism of a skew calabi yau algebra a has trivial homological determinant in case a is noetherian connected graded and koszul

a probabilistic bi level linear multi objective programming problem to supply chain planning abstract bi level programming a tool for modeling decentralized decisions consists of the objective s of the leader at its first level and that is of the follower at the second level three level programming results when second level is itself a bi level programming by extending this idea it is possible to define multi level programs with any number of levels in most of the real life problems in mathematical programming the parameters are considered as random variables the branch of mathematical programming which deals with the theory and methods for the solution of conditional extremum problems under incomplete information about the random parameters is called stochastic programming supply chain planning problems are concerned with synchronizing and optimizing multiple activities involved in the enterprise from the start of the process such as procurement of the raw materials through a series of process operations to the end such as distribution of the final product to customers enterprise wide supply chain planning problems naturally exhibit a multi level decision network structure where for example one level may correspond to a local plant control scheduling planning problem and another level to a corresponding plant wide planning network problem such a multi level decision network structure can be mathematically represented by using multi level programming principles in this paper we consider a probabilistic bi level linear multi objective programming problem and its application in enterprise wide supply chain planning problem where 1 market demand 2 production capacity of each plant and 3 resource available to all plants for each product are random variables and the constraints may consist of joint probability distributions or not this probabilistic model is first converted into an equivalent deterministic model in each level to which fuzzy programming technique is applied to solve the multi objective nonlinear programming problem to obtain a compromise solution

multiobjective bilevel optimization in this work nonlinear non convex multiobjective bilevel optimization problems are discussed using an optimistic approach it is shown that the set of feasible points of the upper level function the so called induced set can be expressed as the set of minimal solutions of a multiobjective optimization problem this artificial problem is solved by using a scalarization approach by pascoletti and serafini combined with an adaptive parameter control based on sensitivity results for this problem the bilevel optimization problem is then solved by an iterative process using again sensitivity theorems for exploring the induced set and the whole efficient set is approximated for the case of bicriteria optimization problems on both levels and for a one dimensional upper level variable an algorithm is presented for the first time and applied to two problems a theoretical example and a problem arising in applications

triviality theorems for yetter drinfel d hopf algebras abstract under suitable assumptions on the base field we prove that a commutative semisimple yetter drinfel d hopf algebra over a finite abelian group is trivial i e is an ordinary hopf algebra if its dimension is relatively prime to the order of the finite abelian group furthermore we prove that a finite dimensional cocommutative cosemisimple yetter drinfel d hopf algebra contains a trivial yetter drinfel d hopf subalgebra of dimension greater than one at least if the yetter drinfel d hopf algebra itself has dimension greater than one

krull relations in hopf galois extensions lifting and twisting this paper continues our study begun in ms of the relationship between the prime ideals of an algebra a and of a subalgebra r such that r a is a faithfully flat h galois extension for some finite dimensional hopf algebra h in that paper we defined three basic krull relations incomparability inc t lying over t lo and going up gu analogous to the classical krull relations for prime ideals we also defined three new dual krull relations we say that h itself is said to have one of the krull relations if the relation holds for all faithfully flat h galois extensions we showed in ms that h has one of the three dual krull relations if and only if the dual hopf algebra h of h has the original relation hence the name an important example of hopf galois extensions is given by hopf crossed products a r h moreover galois extensions can be useful in studying crossed products since they satisfy a transitivity property which crossed products lack that is if k is a normal hopf subalgebra of h with hopf quotient h then in general one cannot write a r h r k h by an example of s2 another basic example of a hopf galois extension is given by a hopf algebra a with a normal hopf subalgebra r of finite index such that a is faithfully flat over r for then r a is faithfully flat h galois

models of quasiprojective homogeneous spaces for hopf algebras given an affine group scheme g of finite type over a field k a homogeneous space for g is a scheme x over k containing a rational point x such that g operates transitively on x assuming that g operates on the right we may identify x with the quotient k g of g by the left action of the stabilizer k of x in g the representation theoretic significance ofk g is that the induction functor indk from k modules to g modules factors as a category equivalence

quantum geometry of algebra factorisations and coalgebra bundles we develop the noncommutative geometry bundles connections etc associated to algebras that factorise into two subalgebras an example is the factorisation of matrices m2 2 2 we also further extend the coalgebra version of theory introduced previously to include frame resolutions and corresponding covariant derivatives and torsions as an example we construct q monopoles on all the podle quantum spheres s2q s

depth two hopf subalgebras of a semisimple hopf algebra abstract a depth two hopf subalgebra k of a semisimple hopf algebra h is shown to be a normal hopf subalgebra on the one hand we prove this using galois theory of quantum groupoids on the other hand we give a second proof using character theory of semisimple hopf algebras

infinite bifurcations gaps and bubbles in hamiltonian systems abstract we study the bifurcations gaps and bubbles along the characteristics of families of periodic orbits in hamiltonian systems of two degrees of freedom bubbles closed characteristics appear when the rotation number along a family does not vary monotonically there are infinite bifurcations appearing when the rotation number along a family is rational n m the most important bifurcations have m 1 or m 2 equal period bnd double period bifurcations in a rotating system the central family of periodic orbits is the family of direct circular orbits of the axisymmetric case at each bifurcation n 1 we have either a change of stability if n odd or a gap if n even of the central family this phenomenon we explain theoretically in cases of infinite transitions from stability to instability along the same family the successive intervals between bifurcations do not follow the universal feigenbaum ratio but we found also several cases of infinite pitchfork period doubling bifurcations and in all cases the feigenbaum ratio is consistent with the universal value 8 72 all feigenbaum sequences in rotating systems are followed by inverse feigenbaum sequences i e they form infinite bubbles an explanation of this effect is given we followed the evolution of the bubbles in some cases for small perturbations no bubbles of a certain type exist as increases one bubble appears and grows then this bubble generates a second order bubble etc until an infinite sequence is formed we found also sequences of infinite bubbles in non rotating systems and their evolution some bubbles are floating i e they are not connected to other families an interesting example of the generation of such a floating bubble is presented

on the depth 2 condition for group algebra and hopf algebra extensions let r 0 be a commutative ring and let h be a subgroup of finite index in a group g we prove that the group ring rg is a ring extension of the group ring rh of depth two if and only if h is a normal subgroup of g we also show that under suitable additional hypotheses an analogous result holds for extensions of hopf algebras over r

extensions of locally compact quantum groups and the bicrossed product construction abstract in the framework of locally compact quantum groups we study cocycle actions we develop the cocycle bicrossed product construction starting from a matched pair of locally compact quantum groups we define exact sequences and establish a one to one correspondence between cocycle bicrossed products and cleft extensions in this way we obtain new examples of locally compact quantum groups

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

ribbon transformations integrals and triangular decompositions abstract we study the theory of integrals in yetter drinfel d hopf algebras and use the results to determine the integrals of hopf algebras with triangular decomposition

non semisimple hopf algebras of dimension p2 abstract let h be a finite dimensional hopf algebra with antipode s of dimension pq over an algebraically closed field of characteristic 0 where p q are odd primes if h is not semisimple then the order of s 4 is p and tr s 2 p is an integer divisible by p 2 in particular if dim h p 2 we prove that h is isomorphic to a taft algebra this completes the classification for the hopf algebras of dimension p 2

the relative modular object and frobenius extensions of finite hopf algebras abstract for a certain kind of tensor functor f c d we define the relative modular object f d as the difference between a left adjoint and a right adjoint of f our main result claims that if c and d are finite tensor categories then f can be written in terms of a categorical analogue of the modular function on a hopf algebra applying this result to the restriction functor associated to an extension a b of finite dimensional hopf algebras we recover the result of fischman montgomery and schneider on the frobenius type property of a b we also apply our results to obtain a braided version and a bosonization version of the result of fischman et al

projectivity and freeness over comodule algebras let h be a hopf algebra and a an h simple right h comodule algebra it is shown that under certain hypotheses every h a hopf module is either projective or free as an a module and a is either a quasi frobenius or a semisimple ring as an application it is proved that every weakly finite in particular every finite dimensional hopf algebra is free both as a left and a right module over its finite dimensional right coideal subalgebras and the latter are frobenius algebras similar results are obtained for h simple h module algebras

on hopf algebras of dimension 4p abstract in this paper we prove that a non semisimple hopf algebra h of dimension 4 p with p an odd prime over an algebraically closed field of characteristic zero is pointed provided h contains more than two group like elements in particular we prove that non semisimple hopf algebras of dimensions 20 28 and 44 are pointed or their duals are pointed and this completes the classification of hopf algebras in these dimensions

period doubling bifurcations and universality in conservative systems we have studied numerically a sequence of period doubling bifurcations of periodic orbits of henon s conservative two dimensional mapping in agreement with other recent studies we also find evidence that such sequences possess universality properties similar to the ones observed for dissipative systems the corresponding universal constants however have significantly different values in each case we suggest that a possible explanation for this discrepancy may be the fact that conservative systems owing to their measure preserving property do not become asymptotically one dimensional as dissipative systems do

co frobenius hopf algebras and the coradical filtration we prove that a hopf algebra with a finite coradical filtration is co frobenius we also characterize co frobenius hopf algebras with coradical a hopf subalgebra let h be a hopf algebra whose coradical is a hopf algebra let gr h be the associated graded coalgebra and let r be the diagram of h c f 2 then the following are equivalent 1 h is co frobenius 2 gr h is co frobenius 3 r is finite dimensional 4 the coradical filtration of h is finite this theorem allows us to give systematically examples of co frobenius hopf algebras and opens the way to the classification of ample classes of such hopf algebras

calculations of some groups of hopf algebra extensions w let us recall the theory of group extensions with abelian kernels mc x chap iv let g be a group and m a g module the equivalence classes of group extensions m a e a g giving rise to the g module action on m form a group with respect to the baer sum furthermore this group is 2 isomorphic to the cohomology group h g m the theory is extended to the theory of hopf algebra extensions which w x are abelian in some sense h s we restrict our interests to hopf algebra extensions over a field k which are of the form

the structure of hopf algebras with a projection the smash product algebra and the smash coproduct coalgebra are well known in the context of hopf algebras 1 5 and these notions can be viewed as being motivated by the semidirect product construction in the theory of groups and in the theory of afiine group schemes respectively let

on the flatness and the projectivity over hopf subalgebras of hopf algebras over dedekind rings abstract we study the flatness and the projectivity of hopf algebras defined over a dedekind ring over their hopf subalgebras we give a criterion for the faithful flatness and use it to show the faithful flatness of an arbitrary flat hopf algebra upon its finite normal hopf subalgebras for the projectivity over hopf subalgebras of a hopf algebra we need some finiteness condition in terms of the module of integrals in particular we show that the module of integrals is projective of rank one

isomorphism classes of finite dimensional connected hopf algebras in positive characteristic abstract we classify all finite dimensional connected hopf algebras with large abelian primitive spaces we show that they are hopf algebra extensions of restricted enveloping algebras of certain restricted lie algebras for any abelian matched pair associated with these extensions we construct a cohomology group which classifies all the extensions up to equivalence moreover we present a 1 1 correspondence between the isomorphism classes and a group quotient of the cohomology group deleting some exceptional points where the group respects the automorphisms of the abelian matched pair and the exceptional points represent those restricted lie algebra extensions

frobenius extensions of subalgebras of hopf algebras we consider when extensions s r of subalgebras of a hopf algebra are frobenius that is frobenius of the second kind given a hopf algebra h we show that when s r are hopf algebras in the yetter drinfeld category for h the extension is frobenius provided r is finite over s and the extension of biproducts s h r h is cleft more generally we give conditions for an extension to be frobenius in particular we study extensions of integral type and consider when the frobenius property is inherited by the subalgebras of coinvariants we apply our results to extensions of enveloping algebras of lie coloralgebras thus extending a result of bell and farnsteiner for lie superalgebras 0 introduction in this paper we consider when various extensions s r of subalgebras of a hopf algebra are frobenius such extensions generalize the usual notion of frobenius extensions by having the module action on one side twisted by an automorphism of s it was already known that any extension of finite dimensional hopf algebras is frobenius a result of the third author sch 92 as is any finite extension u k u l of enveloping algebras of lie superalgebras a result of bell and farnsteiner bf note that u l is not an ordinary hopf algebra but rather a hopf algebra in the category of z2 graded modules these results were an important motivation for this paper and raised the question as to when an extension of hopf algebras in a category was frobenius one of the main results of this paper is that an extension s r of hopf algebras of finite index in the yetter drinfeld category hyd for a given hopf algebra h is frobenius provided that the associated extension of hopf algebras s h r h the biproducts of s and r with h has a normal basis theorem 5 6 this will happen whenever r and h are finite dimensional or when r h is pointed as an application we are able to generalize the bf result to lie coloralgebras if u k u l is a finite extension of enveloping algebras of lie coloralgebras then it is frobenius moreover we give an explicit description of the automorphism of u k and of the frobenius homomorphism f u l u k corollary 6 3 along the way we prove a number of other results about frobenius extensions and conditions that ensure that an extension is frobenius in section 1 we give a short direct proof of the fact that an extension b a of finite dimensional hopf algebras is always frobenius moreover we give explicit received by the editors december 10 1995 1991 mathematics subject classification primary 16w30 secondary 17b35 17b37 c 1997 american mathematical society

a class of drinfeld doubles that are ribbon algebras andruskiewitsch and schneider classify a large class of pointed hopf algebras with abelian coradical the drinfeld double of each such hopf algebra is investigated the drinfeld doubles of a family of hopf algebras from the above classification are ribbon hopf algebras

resonant normal forms interpolating hamiltonians and stability analysis of area preserving maps abstract the geometrical and dynamical properties of area preserving maps in the neighborhood of an elliptic fixed point are analyzed in the framework of resonant normal forms the interpolating flow is not obtained from a map tangent to the identity but from the normal form of the given map and a time independent interpolating hamiltonian h is introduced on this hamiltonian the local stability properties of the fixed point and the geometric structure of the orbits are transparent numerical agreement between the level lines of h and the orbits of the map suggests that the perturbative expansion of h is asymptotic this is confirmed by a rigorous error analysis based on majorant series the error for the normal form expansion grows as n while the truncation error for h also has a factorial growth and in a disc of radius r can be made exponentially small with 1 r the boundary of the global stability domain is considered for the quadratic map the identification with the inner envelope of the homoclinic tangle of the hyperbolic fixed point is strongly suggested by numerical evidence

an algorithm based on particle swarm optimization for multiobjective bilevel linear problems improved mopso for multiobjective at the upper level bilevel linear problems a hybrid scheme for the selection of the global best particles is proposed an adaptive mutation is also introduced the incorporation of these mechanisms led to significantly better results this paper presents an improved multiple objective particle swarm optimization mopso algorithm to solve bilevel linear programming problems with multiple objective functions at the upper level the algorithm aims to produce a good approximation of the entire pareto front of the problem we have previously designed a mopso algorithm for the same class of problems in which several techniques for the global best selection were tested including a new one the algorithm revealed a good convergence towards the pareto front but the diversity of the solutions was a drawback the algorithm we propose herein uses a hybrid strategy for the global best selection and an adaptive mutation operator the incorporation of these mechanisms led to an improved algorithm which also showed better overall performance than considering alternative options usually employed in mopso algorithms the algorithm and computational results are presented