yetter drinfeld modules and projections of weak hopf algebras abstract in this paper we prove that if g b h is a morphism of weak hopf algebras which is split as an algebra coalgebra morphism then the subalgebra of coinvariants b h of b is a hopf algebra in the category of yetter drinfeld modules associated to h

root n consistent semiparametric regression on considere une variable aleatoire x z dans 7b r p 7b r q on construit un estimateur generalisant l estimateur des moindres carres ordinaires en inserant des estimateurs non parametriques de la regression dans la projection orthogonale non lineaire sur z

on the jacobians of plane cubics abstract let s be a scheme and f a ternary cubic form whose ten coefficients are sections of o s without common zero the equation f 0 defines a family of plane cubic curves x p s 2 s parametrized by s we prove that the family of generalized jacobians of those cubic curves is a group scheme j s which is the locus of smoothness of a scheme f 0 where f is a weierstrass cubic form f f x y z y 2 z a 1 xyz a 2 yz 2 x 3 a 2 x 2 z a 4 xz 2 a 6 z 3 in which the coefficient a i is a homogeneous polynomial with integral coefficients of degree i in the ten coefficients of f which we give explicitly a key ingredint of the proof is a characterization over sufficiently nice bases of group algebraic spaces which can be described by such a weierstrass equation

an approach to inference following model selection with applications to transformation based and adaptive inference abstract a decision theory approach is formulated for inference following model selection inference here refers to either testing point estimation or confidence estimation the loss function includes components for model selection as well as for inference and allows for flexibility in emphasis on one or the other if such emphasis is desired a general prescription for bayes and generalized bayes procedures is given a procedure consists of model selection and inference the general formulation is applied to transformation based inference where model selection is equated to choice of transformation hinkley and runger 1984 did transformation based inference that has aroused controversy the formulation here is directed to some of these issues in this approach we explicitly define the quantity of interest for which an inference is desired furthermore we evaluate procedures properly an example is given where one is interested in estimating the mean of the model selected and the choice of models is

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

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

a new invariant for finite dimensional leibniz lie algebras abstract for an n dimensional leibniz lie algebra h over a field k we introduce a new invariant a h called the universal algebra of h as a quotient of the polynomial algebra k x i j i j 1 n through an ideal generated by n 3 polynomials we prove that a h admits a unique bialgebra structure which makes it an initial object among all commutative bialgebras coacting on h the new object a h is the key tool in answering two open problems in lie algebra theory first we prove that the automorphism group aut l b z h of h is isomorphic to the group u g a h o of all invertible group like elements of the finite dual a h o secondly for an abelian group g we show that there exists a bijection between the set of all g gradings on h and the set of all bialgebra homomorphisms a h k g based on this all g gradings on h are explicitly classified and parameterized a h is also used to prove that there exists a universal commutative hopf algebra associated to any finite dimensional leibniz algebra h

model selection and akaike s information criterion aic the general theory and its analytical extensions during the last fifteen years akaike s entropy based information criterion aic has had a fundamental impact in statistical model evaluation problems this paper studies the general theory of the aic procedure and provides its analytical extensions in two ways without violating akaike s main principles these extensions make aic asymptotically consistent and penalize overparameterization more stringently to pick only the simplest of the true models these selection criteria are called caic and caicf asymptotic properties of aic and its extensions are investigated and empirical performances of these criteria are studied in choosing the correct degree of a polynomial model in two different monte carlo experiments under different conditions

hom nijienhuis operators and t extensions of hom lie superalgebras abstract in this paper we study hom nijienhuis operators and t extensions of hom lie superalgebras we show that a linear map between hom lie superalgebras is a morphism if and only if its graph is a hom subalgebra and show that the infinitesimal deformation generated by a hom nijienhuis operator is trivial moreover we introduce the definition of t extensions of hom lie superalgebras and show that t extensions preserve many properties such as nilpotency solvability and decomposition in some sense in particular we discuss the equivalence of t extensions using cohomology

pbw bases of coideal subalgebras and a freeness theorem let h be a character hopf algebra every right coideal subalgebra u that contains the coradical has a pbw basis which can be extended up to a pbw basis of h if additionally u is a bosonization of an invariant with respect to the left adjoint action subalgebra then h is a free left and right u module with a free pbw basis over u these results remain valid if h is a braided hopf algebra generated by a categorically ordered subset of primitive elements if the ground field is algebraically closed the results are still true provided that h is a pointed hopf algebra with commutative coradical and is generated over the coradical by a direct sum of finite dimensional yetter drinfeld submodules of skew primitive elements

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

integrals for monoidal hom hopf algebras and their applications integrals of monoidal hom hopf algebras are introduced and the existence and uniqueness of integrals for finite dimensional monoidal hom hopf algebras are investigated first then integrals are applied to the maschke type theorem for monoidal hom hopf algebras controlling the semisimplicity and separability of monoidal hom hopf algebras further monoidal hom algebras are characterized with additional frobenius property and the question when finite dimensional monoidal hom hopf algebras are frobenius is studied as applications of integrals the maschke type theorem for hom smash product is given and the morita context in the hom category h mk is constructed

yetter drinfeld modules for hom bialgebras the aim of this paper is to define and study yetter drinfeld modules over hom bialgebras a generalized version of bialgebras obtained by modifying the algebra and coalgebra structures by a homomorphism yetter drinfeld modules over a hom bialgebra with bijective structure map provide solutions of the hom yang baxter equation the category yhhd of yetter drinfeld modules with bijective structure maps over a hom bialgebra h with bijective structure map can be organized in two different ways as a quasi braided pre tensor category if h is quasitriangular respectively coquasitriangular the first respectively second quasi braided pre tensor category yhhd contains as a quasi braided pre tensor subcategory the category of modules respectively comodules with bijective structure maps over h

universal enveloping algebras of poisson hopf algebras abstract for a poisson algebra a by exploring its relation with lie rinehart algebras we prove a poincare birkhoff witt theorem for its universal enveloping algebra a e some general properties of the universal enveloping algebras of poisson hopf algebras are studied given a poisson hopf algebra b we give the necessary and sufficient conditions for a poisson polynomial algebra b x p to be a poisson hopf algebra we also prove a structure theorem for b e when b is a pointed poisson hopf algebra namely b e is isomorphic to b h b the crossed product of b and h b where h b is the quotient hopf algebra b e b e b

on the even powers of the antipode of a finite dimensional hopf algebra abstract the trace of powers of the square of the antipode s 2 of a finite dimensional hopf algebra a over a field k is studied it is shown in many cases that the trace function vanishes on s 2 m when s 2 m 1 a finer properties of the antipode are related to this phenomenon

relative hom hopf modules and total integrals let h be a monoidal hom hopf algebra and a a right h hom comodule algebra we first investigate the criterion for the existence of a total integral of a in the setting of monoidal hom hopf algebras also we prove that there exists a total integral h a if and only if any representation of the pair h a is injective in a functorial way as a corepresentation of h which generalizes doi s result finally we define a total quantum integral h hom h a and prove the following affineness criterion if there exists a total quantum integral and the canonical map a ba a h a bb 1 a b 0 b 1 is surjective then the induction functor a b k b k ah is an equivalence of categories

on semisimple hopf algebras of dimension pq2 let p and q be distinct prime numbers we prove a result on the existence of nontrivial group like elements in a certain class of semisimple hopf algebras of dimension pqr we conclude the classification of semisimple hopf algebras a of dimension pq2 over an algebraically closed field k of characteristic zero such that both a and a are of frobenius type we also complete the classification of semisimple hopf algebras of dimension pq2 100

quantum differentials on cross product hopf algebras abstract we construct canonical strongly bicovariant differential graded algebra structures on all four flavours of cross product hopf algebras namely double cross products a a h h double cross coproducts biproducts and bicrossproducts on the assumption that the factors have strongly bicovariant calculi a h or a braided version b we use super versions of each of the constructions moreover the latter three quantum groups all coact canonically on one of their factors and we show that this coaction is differentiable in the case of the drinfeld double d a h a op h where a is dually paired to h we show that its canonical actions on a h are differentiable examples include a canonical c q g l 2 c 2 for the quantum group of affine transformations of the quantum plane and c poin c 1 1 for the bicrossproduct poincare quantum group in 2 dimensions we also show that c q g l 2 itself is uniquely determined by differentiability of the canonical coaction on the quantum plane and of the determinant subalgebra

radford s biproducts and yetter drinfeld modules for monoidal hom hopf algebras let h be a monoidal hom bialgebra and b be a left h hom module algebra and also a left h hom comodule coalgebra then in this paper we first introduce the notion of a hom smash coproduct which is a monoidal hom coalgebra second we find sufficient and necessary conditions for the hom smash product algebra structure and the hom smash coproduct coalgebra structure on b h to afford b h a monoidal hom bialgebra structure generalizing the well known radford s biproduct where the conditions are equivalent to that b is a bialgebra in the category of hom yetter drinfeld modules hydhh finally we illustrate the category of hom yetter drinfeld modules hydhh and prove that the category hydhh is a braided monoidal category

support varieties and representation types for basic classical lie superalgebras abstract let be an algebraically closed field of characteristic p 3 and g a restricted lie superalgebra over we introduce the definition of restricted cohomology for g and show its cohomology ring is finitely generated provided g is a basic classical lie superalgebra as a consequence we show that the restricted enveloping algebra of a basic classical lie superalgebra g is always wild except g sl 2 or g osp 1 2 or g c 2

hom l r smash products hom diagonal crossed products and the drinfeld double of a hom hopf algebra abstract we introduce the hom analogue of the l r smash product and use it to define the hom analogue of the diagonal crossed product when h is a finite dimensional hom hopf algebra with bijective antipode and bijective structure map we define the drinfeld double of h its algebra structure is a hom diagonal crossed product and it has all expected properties namely it is quasitriangular and modules over it coincide with left right yetter drinfeld modules over h

hopf bimodules coquasibialgebras and an exact sequence of kac abstract based on the ideas of tannaka krein reconstruction we present a categorical construction that assigns to any cleft hopf algebra inclusion k h a coquasibialgebra having k as a hopf subalgebra as a special case the construction gives an intrinsic connection between the bismash product k q and the double cross product q k constructed from the same combinatorial data a cocommutative coquasibialgebra is the same as a cocommutative bialgebra equipped with a sweedler three cocycle thus our construction assigns to every bicrossproduct or hopf algebra extension of a commutative and a cocommutative factor a corresponding cocommutative double crossproduct equipped with a sweedler three cocycle based on this observation we use the construction to prove generalizations of kac s exact sequence for the group of hopf algebra extensions of a group algebra by a dual group algebra

low lying zeros of families of elliptic curves the random matrix model predicts that many statistics associated to zeros of a family of l functions can be modeled or predicted by the distribution of eigenval ues of large random matrices in one of the classical linear groups if the statistics of a family of l functions are modeled by the eigenvalues of the group g then we say that g is the symmetry group or symmetry type associated to the family the statistic of interest to us in this work is the density of zeros near the central point also known as the 1 level density the random matrix model predicts that the distribution of these zeros should be modeled by the eigenvalues nearest 1 for one of the symmetry types g unitary symplectic and orthogonal all of the different groups g have distinct behavior in this regard therefore computing the 1 level density gives a theoretical way to predict the symmetry type of a family the 1 level density has been studied for a wide variety of families of l functions see r ksi ils mil for example it is standard to assume the generalized riemann hypothesis grh to study the 1 level density and we do so throughout this work in particular it is necessary to use grh for the application of obtaining a bound on the average analytic rank from a density theorem in some cases the use of grh improves the range of the density theorem which translates to an improved bound on the average rank besides these crucial applications of grh we have freely assumed grh even when it could be removed with extra work since it simplifies arguments in some non essential places it is especially interesting to investigate the 1 level density for families of l functions attached to elliptic curves over the rationals since zeros at the central point have important arithmetic information by the conjecture of birch and swinnerton dyer these investigations have been the main focus of this work

on hom algebras with surjective twisting abstract a hom associative structure is a set a together with a binary operation and a self map such that an twisted version of associativity is fulfilled in this paper we assume that is surjective we show that in this case under surprisingly weak additional conditions on the multiplication the binary operation is a twisted version of an associative operation as an application an earlier result fregier and gohr 1 on weakly unital hom algebras is recovered with a different proof in the second section consequences for the deformation theory of hom algebras with surjective twisting map are discussed

deformations of lie algebras using σ derivations in this article we develop an approach to deformations of the witt and virasoro algebras based on sigma derivations we show that sigma twisted jacobi type identity holds for generators of such deformations for the sigma twisted generalization of lie algebras modeled by this construction we develop a theory of central extensions we show that our approach can be used to construct new deformations of lie algebras and their central extensions which in particular include naturally the q deformations of the witt and virasoro algebras associated to q difference operators providing also corresponding q deformed jacobi identities c 2005 elsevier inc all rights reserved

quantum group gauge theory on quantum spaces we construct quantum group valued canonical connections on quantum homogeneous spaces including aq deformed dirac monopole on the quantum sphere of podles with quantum differential structure coming from the 3d calculus of woronowicz onsu q 2 the construction is presented within the setting of a general theory of quantum principal bundles with quantum group hopf algebra fibre associated quantum vector bundles and connection one forms both the base space spacetime and the total space are non commutative algebras quantum spaces

constructing new braided t categories over monoidal hom hopf algebras let autmhh h denote the set of all automorphisms of a monoidal hopf algebra h with bijective antipode in the sense of caenepeel and goyvaerts monoidal hom hopf algebras commun algebra 39 2216 2240 2011 and let g be a crossed product group autmhh h autmhh h the main aim of this paper is to provide new examples of braided t category in the sense of turaev crossed group categories arabian j sci eng sect c 33 2c 483 503 2008 for this purpose we first introduce a class of new categories mhydhh a b of a b yetter drinfeld hom modules with a b autmhh h then we construct a category mhyd h mhydhh a b a b g and show that such category forms a new braided t category generalizing the main constructions by panaite and staic generalized anti yetter drinfel d modules as components of a braided t category isr j math 158 349 366 2007 finally we compute an explicit new example of such braided t categories

symmetries and the u condition in hom yetter drinfeld categories let h s be a monoidal hom hopf algebra and hydhh the hom yetter drinfeld category over h then in this paper we first find sufficient and necessary conditions for hydhh to be symmetric and pseudosymmetric respectively second we study the u condition in hydhh and show that the hom yetter drinfeld module h adjoint resp h m coadjoint satisfies the u condition if and only if s2 id finally we prove that hydhh over a triangular resp cotriangular hom hopf algebra contains a rich symmetric subcategory

projections and yetter drinfel d modules over hopf co quasigroups abstract in this paper we introduce the notions of yetter drinfel d module and strong projection over a hopf quasigroup h if the antipode of h is invertible we show that the category of left left yetter drinfel d modules yd h h is braided and there exists a categorical equivalence between the category of hopf quasigroups in yd h h and the category of strong hopf algebra projections associated to h

weak hopf algebras with projection and weak smash bialgebra structures abstract in this paper we study weak hopf algebras with projection if f h b g b h are morphisms of weak hopf algebras such that g f idh we prove that it is possible to find an object bh in the new category of weak yetter drinfeld modules that verifies similar conditions to the ones include in the definition of weak hopf algebra finally we define weak smash bialgebra structures and prove that under central and cocentral conditions bh and h determine an example of them

on defining ideals and differential algebras of nichols algebras abstract this paper is devoted to understanding the defining ideal of a nichols algebra from the decomposition of specific elements in the group algebra of braid groups a family of primitive elements is found and algorithms are proposed to prove the main result the differential algebra of a nichols algebra is constructed moreover another point of view on serre relations is provided

weak hopf algebras and weak yang baxter operators abstract in this paper we introduce the notions of weak yang baxter operator and weak braided hopf algebra we prove that it is possible to obtain examples of these notions working with yetter drinfeld modules associated to a weak hopf algebra h with invertible antipode finally we complete the study of the structure of weak hopf algebras with a projection obtaining a categorical equivalence between the category of weak hopf algebra projections associated to h and the category of hopf algebras in the non strict braided monoidal category of left left yetter drinfeld modules over h

the iwasawa main conjectures for gl2 we prove the one two and three variable iwasawa greenberg main conjectures for a large class of modular forms that are ordinary with respect to an odd prime p the method of proof involves an analysis of an eisenstein ideal for ordinary hida families for gu 2 2

a classification result on prime hopf algebras of gk dimension one abstract in this paper we classify all prime hopf algebras h of gk dimension one satisfying the following two conditions 1 h has a 1 dimensional representation of order pi deg h and 2 the invariant components of h with respect to this 1 dimensional representation are domains see section 2 for related definitions as consequences 1 a number of new hopf algebras of gk dimension one are found and some of them are not pointed 2 we give a partial answer to a question posed in 12 and 3 the question 1 1 is solved

average ranks of elliptic curves tension between data and conjecture rational points on elliptic curves are the gems of arithmetic they are to diophantine geometry what units in rings of integers are to algebraic number theory what algebraic cycles are to algebraic geometry a rational point in just the right context at one place in the theory can inhibit and control thanks to ideas of kolyvagin kol88 the existence of rational points and other mathematical structures elsewhere despite all that we know about these objects the initial mystery and excitement that drew mathematicians to this arena in the first place remains in full force today we have a network of heuristics and conjectures regarding rational points and we have massive data accumulated to exhibit instances of the phenomena generally we would expect that our data support our conjectures and if not we lose faith in our conjectures but here there is a somewhat more surprising interrelation between data and conjecture they are not exactly in open conflict one with the other but they are no great comfort to each other either we discuss various aspects of this story including recent heuristics and data that attempt to resolve this mystery we shall try to convince the reader that despite seeming discrepancy data and conjecture are in fact in harmony

the invariants of a genus one curve it was first pointed out by weil that we can use classical invariant theory to compute the jacobian of a genus one curve the invariants required for curves of degree n 2 3 4 were already known to the nineteenth century invariant theorists we have succeeded in extending these methods to curves of degree n 5 where although the invariants are too large to write down as explicit polynomials we have found a practical algorithm for evaluating them

on the birch swinnerton dyer quotients modulo squares let a be an abelian variety over a number field k an identity between the l functions l a k i s for extensions k i of k induces a conjectural relation between the birch swinnerton dyer quotients we prove these relations modulo finiteness of lii and give an analogous statement for selmer groups based on this we develop a method for determining the parity of various combinations of ranks of a over extensions of k as one of the applications we establish the parity conjecture for elliptic curves assuming finiteness of iii e k e 2 6 and some restrictions on the reduction at primes above 2 and 3 the parity of the mordell weil rank of e k agrees with the parity of the analytic rank as determined by the root number we also prove the p parity conjecture for all elliptic curves over q and all primes p the parities of the p selmer rank and the analytic rank agree

geometric representations of graded and rational cherednik algebras abstract we provide geometric constructions of modules over the graded cherednik algebra h gr and the rational cherednik algebra h rat attached to a simple algebraic group g together with a pinned automorphism these modules are realized on the cohomology of affine springer fibers of finite type that admit c actions in the rational cherednik algebra case the standard grading on these modules is derived from the perverse filtration on the cohomology of affine springer fibers coming from its global analog hitchin fibers when is trivial we show that our construction gives the irreducible finite dimensional spherical modules l triv of h gr and of h rat we give a formula for the dimension of l triv and give a geometric interpretation of its frobenius algebra structure the rank two cases are studied in further details

quantum quasi shuffle algebras ii we establish some properties of quantum quasi shuffle algebras they include the necessary and sufficient condition for the construction of the quantum quasi shuffle product the universal property and the commutativity condition as an application we use the quantum quasi shuffle product to construct a linear basis of t v for a special kind of yang baxter algebras v m

admissible variable selection procedures when fitting regression models by least squares for prediction given data with a single dependent variable arising from a normal linear regression model a variable selection procedure completely specifies a least squares fit by choosing the subset of the independent variables to include in the model for loss equal to squared error of prediction we prove that all variable selection procedures are admissible for choosing among least squares fits of the regression model

ternary cubic forms having bounded invariants and the existence of a positive proportion of elliptic curves having rank 0 we prove an asymptotic formula for the number of sl3 z equivalence classes of integral ternary cubic forms having bounded invariants we use this result to show that the average size of the 3 selmer group of all elliptic curves when ordered by height is equal to 4 this implies that the average rank of all elliptic curves when ordered by height is less than 1 17 combining our counting techniques with a recent result of dokchitser and dokchitser we prove that a positive proportion of all elliptic curves have rank 0 assuming the niteness of the tate shafarevich group we also show that a positive proportion of elliptic curves have rank 1 finally combining our counting results with the recent work of skinner and urban we show that a positive proportion of elliptic curves have analytic rank 0 i e a positive proportion of elliptic curves have nonvanishing l function at s 1 it follows that a positive proportion of all elliptic curves satisfy bsd

on the classification of finite dimensional pointed hopf algebras we classify finite dimensional complex hopf algebras a which are pointed that is all of whose irreducible comodules are one dimensional and whose group of group like elementsg a is abelian such that all prime divisors of the order of g a are 7 since these hopf algebras turn out to be deformations of a natural class of generalized small quantum groups our result can be read as an axiomatic description of generalized small quantum groups

finite dimensional simple pointed hopf algebras for some q g k 0 with the advent of quantum groups these relations took on added importance in the theory of hopf algebras the quantized w x enveloping algebras see 2 5 for example are generated by pairs of such elements a and x which satisfy these relations quantized enveloping algebras are examples of pointed hopf algebras a natural question to ask about pointed hopf algebras is which ones are simple in an appropriate sense in this paper we give a definition of simple pointed hopf algebra and refer to such hopf algebras as simplepointed we describe the structure of simple pointed hopf algebras in the class of pointed hopf algebras which are generated by pairs a and x which satisfy the relations above when k is algebraically closed in the finite dimensional characteristic 0 case we characterize the coalgebra structure of the duals of these simple pointed hopf algebras as well many finite dimensional hopf algebras a over a field k are non trivial biproducts a necessary and sufficient condition for a to be a biproduct is the existence of a hopf algebra projection a a h from a onto a sub hopf algebra h of a the associated biproduct realization of a has

representations of pointed hopf algebras and their drinfel d quantum doubles abstract we study representations of nontrivial liftings of nilpotent type of quantum linear spaces and their drinfel d quantum doubles we construct a family of verma type modules in both cases and prove a parametrization theorem for simple modules we compute the loewy and socle series of verma modules under a mild restriction on the datum of a lifting we find bases and dimensions of simple modules

a new method to discriminate between enzyme kinetic models abstract a novel statistical procedure s zwanzig math opsforsch statist ser statist 11 23 47 1980 is applied to discriminate between kinetic models of enzyme reactions two examples are elucidated on the basis of numerically simulated data 1 a discrimination between a michaelis menten model and a hill equation 2 b discrimination between competitive and noncompetitive inhibition of a two substrate reaction next the proposed method is applied to assess a kinetic model of the allosteric enzyme phosphofructokinase from the malarial parasite plasmodium berghei it is demonstrated that the novel procedure allows a more sensitive discrimination as it is obtained with alternative statistical tests

binary quartic forms having bounded invariants and the boundedness of the average rank of elliptic curves we prove a theorem giving the asymptotic number of binary quartic forms having bounded invariants this extends to the quartic case the classical results of gauss and davenport in the quadratic and cubic cases respectively our techniques are quite general and may be applied to counting integral orbits in other representations of algebraic groups we use these counting results to prove that the average rank of elliptic curves over q when ordered by their heights is bounded in particular we show that when elliptic curves are ordered by height the mean size of the 2 selmer group is 3 this implies that the limsup of the average rank of elliptic curves is at most 1 5

cocycle deformations for hopf algebras with a coalgebra projection abstract let h be a hopf algebra over a field k of characteristic 0 and let a be a bialgebra or hopf algebra such that h is isomorphic to a sub hopf algebra of a and there is an h bilinear coalgebra projection from a to h which splits the inclusion then a r h where r is the pre bialgebra of coinvariants in this paper we study the deformations of a by an h bilinear cocycle if is a cocycle for a then can be restricted to a cocycle r for r and a r r h as examples we consider liftings of b v k where is a finite abelian group v is a quantum plane and b v is its nichols algebra and explicitly construct the cocycle which twists the radford biproduct into the lifting

hom lie hopf algebras abstract we studied both the double cross product and the bicrossproduct constructions for the hom hopf algebras of general type this allows us to consider the universal enveloping hom hopf algebras of hom lie algebras which are of id type we show that the universal enveloping hom hopf algebras of a matched pair of hom lie algebras form a matched pair of hom hopf algebras we observe also that the semi dualization of a double cross product hom hopf algebra is a bicrossproduct hom hopf algebra in particular we apply this result to the universal enveloping hom hopf algebras of a matched pair of hom lie algebras to obtain hom lie hopf algebras

admissible selection of an accurate and parsimonious normal linear regression model let mo be a normal linear regression model and let m1 mk be distinct proper linear submodels of mo let k e 0 k be a model selection rule based on observed data from the true model given k let the unknown parameters of the selected model ms be fitted by the maximum likelihood method a loss function is introduced which depends additively on two parts i a measure of the difference between the fitted model ms and the true model and ii a measure cs of the complexity of the selected model a natural model selection rule k which minimizes an empirical version of this loss is shown to be admissible and very nearly bayes

the average analytic rank of elliptic curves all the results in this paper are conditional on the riemann hypothesis for the l functions of elliptic curves under this assumption we show that the average analytic rank of all elliptic curves over mathbb q is at most 2 thereby improving a result of brumer 2 we also show that the average within any family of quadratic twists is at most 3 2 improving a result of goldfeld 3 a third result concerns the density of curves with analytic rank at least r and shows that the proportion of such curves decreases faster than exponentially as r grows the proofs depend on an analogue of weil s explicit formula

qed hopf algebras on planar binary trees abstract in this paper we describe the hopf algebras on planar binary trees used to renormalize the feynman propagators of quantum electrodynamics and the coaction which describes the renormalization procedure both structures are related to some semi direct coproduct of hopf algebras

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

a monoidal approach to splitting morphisms of bialgebras the main goal of this paper is to investigate the structure of hopf algebras with the property that either its jacobson radical is a hopf ideal or its coradical is a subalgebra let us consider a hopf algebra a such that its jacobson radical j is a nilpotent hopf ideal and h a j is a semisimple algebra we prove that the canonical projection of a on h has a section which is an h colinear algebra map furthermore if h is cosemisimple too then we can choose this section to be an h h bicolinear algebra morphism this fact allows us to describe a as a generalized bosonization of a certain algebra r in the category of yetter drinfeld modules over h as an application we give a categorical proof of radford s result about hopf algebras with projections we also consider the dual situation let a be a bialgebra such that its coradical is a hopf sub bialgebra with antipode then there is a retraction of the canonical injection of h into a which is an h linear coalgebra morphism furthermore if h is semisimple too then we can choose this retraction to be an h h bilinear coalgebra morphism then also in this case we can describe a as a generalized bosonization of a certain coalgebra r in the category of yetter drinfeld modules over h

quantum lines over non cocommutative cosemisimple hopf algebras the recognition of the importance of braided tensor categories came by the remarkable recent developments of quantum group theory an especially important example of such a category is the braided tensor category hyd of yetter drinfeld modules over a hopf algebra h with bijective antipode it is interesting and delightful to see that old results on ordinary hopf algebras such as those by nichols 18 1978 and by radford 19 1985 are reproduced in terms of the new notion of braided hopf algebra in hyd given a braided hopf algebra r in hyd one can construct an ordinary hopf algebra r h of biproduct this naturally forms a triple r h called a hopf algebra triple over h such that h r h and r h h are hopf algebra maps with id if we define h 1 h x h e x h it was essentially proved by radford 19 and then reproduced by majid 13 that r r h gives a category equivalence called bosonisation from the category of braided hopf algebras in hhyd to the category of hopf algebra triples over h see proposition 1 1

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

the category of yetter drinfel d hom modules and the quantum hom yang baxter equation in this paper we introduce the category of yetter drinfel d hom modules which is a braided monoidal category and show that the category of yetter drinfel d hom modules is a full monoidal subcategory of the left center of left hom module category also we study the equivalent relationship between the category of yetter drinfel d hom modules and the category of hom modules over the drinfel d double finally the faddeev reshetikhin takhtajan frt type theorem for the quantum hom yang baxter equation is investigated

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

hom novikov algebras we study a twisted generalization of novikov algebras called hom novikov algebras in which the two defining identities are twisted by a linear map it is shown that hom novikov algebras can be obtained from novikov algebras by twisting along any algebra endomorphism all algebra endomorphisms on complex novikov algebras of dimensions 2 or 3 are computed and their associated hom novikov algebras are described explicitly another class of hom novikov algebras is constructed from hom commutative algebras together with a derivation generalizing a construction due to dorfman and gel fand two other classes of hom novikov algebras are constructed from hom lie algebras together with a suitable linear endomorphism generalizing a construction due to bai and meng

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

characteristic classes for cohomology of split hopf algebra extensions we introduce characteristic classes for the spectral sequence associated to a split short exact sequence of hopf algebras these classes can be seen as obstructions for the vanishing of differentials in the spectral sequence we give a decomposition theorem and interpret our results in the settings of group and lie algebra extensions as applications we derive several results concerning the collapse of the lyndon hochschild serre spectral sequence and the order of characteristic classes

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

physics for algebraists non commutative and non cocommutative hopf algebras by a bicrossproduct construction abstract the initial part of this paper presents physics for algebraists in the context of quantum mechanics combined with gravity such physical notions as the yang baxter equations position observables momentum space momentum and position quantization etc are described many readers may wish to just read this initial part of the paper the physics leads to the search for self dual algebraic structures and finally to non commutative and non cocommutative hopf algebras by a bicrossproduct construction the entire paper contains numerous examples the non commutative and non cocommutative hopf algebras are obtained as a simultaneous smash product and smash coproduct and denoted h1 h2 among the examples is one obtained by modifying the weyl algebra we also give the context in which the compatibility requirements on the structure maps reduce to the classical yang baxter equations and an example related to drinfel d s double hopf algebra d h

cross product bialgebras part ii abstract this is the central article of a series of three papers on cross product bialgebras we present a universal theory of bialgebra factorizations or cross product bialgebras with cocycles and dual cocycles we also provide an equivalent co modular co cyclic formulation all known examples as for instance bi or smash doublecross and bicross product bialgebras as well as double biproduct bialgebras and bicrossed or cocycle bicross product bialgebras are now united within a single theory furthermore our construction yields various novel types of cross product bialgebras

on a factorization of graded hopf algebras using lyndon words abstract we find a generalization of the restricted pbw basis for pointed hopf algebras over abelian groups constructed by kharchenko we obtain a factorization of the hilbert series for a wide class of graded hopf algebras these factors are parametrized by lyndon words and they are the hilbert series of certain graded hopf algebras