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

extended fuzzy logic sets and systems the concepts of sets and approximate reasoning within extended fuzzy logic fle provide a systematic procedure for transforming unprecisiated knowledge into a nonlinear mapping over what we define here as f sets an f set differs from a fuzzy set in that it is associated with the restriction of validity in addition to that of possibility therefore by f set we can simultaneously deal with two different types of uncertainties one that is related to ill known objects represented by incomplete information information with its one or more aspects being imprecise vague partial nonspecific undetermined and another that is related to truth values considering gradualness here we define new concepts of vartheta cuts and alpha vartheta cuts introduce the f extension principle and consider arithmetic computations within fle we then address other aspects of the proposed fle system such as fuzzification and validification operations in input processing stage set conversion and defuzzification in output processing stage and inferencing in fact in this paper we intend to develop fle theoretically and practically from the stands of sets and systems to extend the concept of approximate reasoning as a consequence of this development we assert that considering the validity degree of methods and information can lead to more reasonable and trustworthy results through capturing more uncertainty

liftings of nichols algebras of diagonal type iii cartan type g2 abstract we complete the classification of hopf algebras whose infinitesimal braiding is a principal yetter drinfeld realization of a braided vector space of cartan type g 2 over a cosemisimple hopf algebra we develop a general formula for a class of liftings in which the quantum serre relations hold we give a detailed explanation of the procedure for finding the relations based on the recent work of andruskiewitsch angiono and rossi bertone

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

finite dimensional pointed hopf algebras over finite simple groups of lie type i non semisimple classes in psln q abstract we show that nichols algebras of most simple yetter drinfeld modules over the projective special linear group over a finite field corresponding to non semisimple orbits have infinite dimension we spell out a new criterium to show that a rack collapses

linkable dynkin diagrams abstract in this article we develop some aspects of the construction of new hopf algebras found recently by andruskiewitsch and schneider there the authors classified under some slight restrictions all pointed finite dimensional hopf algebras with coradical z p s we contribute to this work by giving a closer description of the possible exotic linkings

competitive lotka volterra population dynamics with jumps abstract this paper considers competitive lotka volterra population dynamics with jumps the contributions of this paper are as follows a we show that a stochastic differential equation sde with jumps associated with the model has a unique global positive solution b we discuss the uniform boundedness of the p th moment with p 0 and reveal the sample lyapunov exponents c using a variation of constants formula for a class of sdes with jumps we provide an explicit solution for one dimensional competitive lotka volterra population dynamics with jumps and investigate the sample lyapunov exponent for each component and the extinction of our n dimensional model

logarithmic widehat s ell 2 cft models from nichols algebras i we construct chiral algebras that centralize rank 2 nichols algebras with at least one fermionic generator this gives logarithmic w algebra extensions of a fractional level algebra we discuss crucial aspects of the emerging general relation between nichols algebras and logarithmic conformal field theory cft models i the extra input beyond the nichols algebra proper needed to uniquely specify a conformal model ii a relation between the cft counterparts of nichols algebras connected by weyl groupoid maps and iii the common double bosonization u x of such nichols algebras for an extended chiral algebra candidates for its simple modules that are counterparts of the u x simple modules are proposed as a first step toward a functorial relation between u x and w algebra representation categories

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

actions of quantum linear spaces on quantum algebras abstract we study actions of bosonizations of quantum linear spaces on quantum algebras under mild conditions we classify actions on quantum affine spaces and quantum matrix algebras in the former case it is shown that all actions of generalized taft algebras are trivial extensions of actions on quantum planes in both cases we achieve bounds on the rank of the bosonization acting on the algebra

abelian and non abelian second cohomologies of quantized enveloping algebras abstract for a class of pointed hopf algebras including the quantized enveloping algebras we discuss cleft extensions cocycle deformations and the second cohomology we present such a non standard method of computing the abelian second cohomology that derives information from the non abelian second cohomology classifying cleft extensions as a sample computation a quantum analogue of whitehead s second lemma for lie algebra cohomology is proved

yetter drinfeld modules under cocycle twists abstract we give an explicit formula for the correspondence between simple yetter drinfeld modules for certain finite dimensional pointed hopf algebras h and those for cocycle twists h of h this implies an equivalence between modules for their drinfeld doubles to illustrate our results we consider the restricted two parameter quantum groups u r s sl n under conditions on the parameters guaranteeing that u r s sl n is a drinfeld double of its borel subalgebra we determine explicit correspondences between u r s sl n modules for different values of r and s and provide examples where no such correspondence can exist our examples were obtained via the computer algebra system singular plural

some topics on frobenius lusztig kernels i abstract an alternative description of the frobenius lusztig kernels and restricted specializations is presented and used to find their coradical filtrations automorphisms and imbeddings moreover factor hopf algebras of the frobenius lusztig kernels are determined

modular quantizations of lie algebras of cartan type k via drinfeld twists of jordanian type we construct explicit drinfel d twists of jordanian type for the generalized cartan type k lie algebras in characteristic 0 and obtain the corresponding quantizations especially their integral forms by making modular reductions including modulo p and modulo p restrictedness reduction and base changes we derive certain modular quantizations of the restricted universal enveloping algebra u k 2n 1 1 for the restricted simple lie algebra of cartan type k in characteristic p they are new families of noncommutative and noncocommutative hopf algebras of dimension pp2n 1 1 if 2n 4 0 modp or pp2n 1 if 2n 4 0 modp over a truncated p polynomials ring which also contain the well known radford algebras see 20 as hopf subalgebras some open questions are proposed

hopf cocycle deformations and invariant theory for a given finite dimensional hopf algebra h we describe the set of all equivalence classes of cocycle deformations of h as an affine variety using methods of geometric invariant theory we show how our results specialize to the universal coefficients theorem in the case of a group algebra and we also give examples from other families of hopf algebras including dual group algebras and bosonizations of nichols algebras in particular we use the methods developed here to classify the cocycle deformations of a dual pointed hopf algebra associated to the symmetric group on three letters we also give an example of a cocycle deformation over a dual group algebra which has only rational invariants but which is not definable over the rational field this differs from the case of group algebras in which every 2 cocycle is equivalent to one which is definable by its invariants

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

finite dimensional pointed or copointed hopf algebras over affine racks abstract we study the pointed or copointed liftings of nichols algebras associated to affine racks and constant cocycles for any finite group admitting a principal yd realization of these racks in the copointed case we complete the classification for the six affine racks whose nichols algebra is known to be of finite dimension in the pointed case our method allows us to finish four of them in all of the cases the hopf algebras obtained turn out to be cocycle deformations of their associated graded hopf algebras all of them are new examples of finite dimensional copointed or pointed hopf algebras over non abelian groups

lifting of quantum linear spaces and pointed hopf algebras of orderp3 abstract we propose the following principle to study pointed hopf algebras or more generally hopf algebras whose coradical is a hopf subalgebra given such a hopf algebraa consider its coradical filtration and the associated graded coalgebra gr a then gr ais a graded hopf algebra since the coradicala0ofais a hopf subalgebra in addition there is a projection gr a a0 letrbe the algebra of coinvariants of then by a result of radford and majid ris a braided hopf algebra and gr ais the bosonization or biproduct ofranda0 gr a r a0 the principle we propose to studyais first to studyr then to transfer the information to gr avia bosonization and finally to lift toa in this article we apply this principle to the situation whenris the simplest braided hopf algebra a quantum linear space as consequences of our technique we obtain the classification of pointed hopf algebras of orderp3 pan odd prime over an algebraically closed field of characteristic zero with the same hypothesis the characterization of the pointed hopf algebras whose coradical is abelian and has indexporp2 and an infinite family of pointed nonisomorphic hopf algebras of the same dimension this last result gives a negative answer to a conjecture of i kaplansky

pointed hopf actions on fields ii abstract this is a continuation of the authors study of finite dimensional pointed hopf algebras h which act inner faithfully on commutative domains as mentioned in part i of this work the study boils down to the case where h acts inner faithfully on a field these hopf algebras are referred to as galois theoretical in this work we provide classification results for finite dimensional pointed galois theoretical hopf algebras h of finite cartan type namely we determine when such h of type a 1 r and some h of rank two possess the galois theoretical property moreover we provide necessary and sufficient conditions for reshetikhin twists of small quantum groups to be galois theoretical

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

digital quantum groups we find and classify all bialgebras and hopf algebras or quantum groups of dimension 4 over the field f2 0 1 we summarize our results as a quiver where the vertices are the inequivalent algebras and there is an arrow for each inequivalent bialgebra or hopf algebra built from the algebra at the source of the arrow and the dual of the algebra at the target of the arrow there are 314 distinct bialgebras and among them 25 hopf algebras with at most one of these from one vertex to another we find a unique smallest noncommutative and noncocommutative one which is moreover self dual and resembles a digital version of uq sl2 we also find a unique self dual hopf algebra in one anyonic variable x4 0 for all our hopf algebras we determine the integral and associated fourier transform operator viewed as a representation of the quiver we also find all quasitriangular or universal r matrix structures on our hopf algebras these induce solutions of the yang baxter or braid relations in any representation

finite quantum groups and cartan matrices abstract we consider a class of hopf algebras having as an invariant a generalized cartan matrix for a hopf algebra of this kind we prove that it is finite dimensional if and only if its generalized cartan matrix is actually a finite cartan matrix under some mild hypothesis these results allow us to classify all the finite dimensional coradically graded pointed hopf algebras whose coradical has odd prime dimension p we also characterize coradically graded pointed hopf algebras of order p 4

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

tetrahedron equation and generalized quantum groups we construct families of solutions of the yang baxter equation from n products of three dimensional r and l operators satisfying the tetrahedron equation they are identified with the quantum r matrices for the hopf algebras known as generalized quantum groups depending on the number of r s and l s involved in the product the trace construction interpolates the symmetric tensor representations of and the antisymmetric tensor representations of whereas a boundary vector construction interpolates the q oscillator representation of and the spin representation of the intermediate cases are associated with an affinization of quantum superalgebras

non degeneracy conditions for braided finite tensor categories abstract for a braided finite tensor category c with unit object 1 c lyubashenko considered a certain hopf algebra f c endowed with a hopf pairing f f 1 to define the notion of a non semisimple modular tensor category we say that c is non degenerate if the hopf pairing is non degenerate in this paper we show that c is non degenerate if and only if it is factorizable in the sense of etingof nikshych and ostrik if and only if its muger center is trivial if and only if the linear map hom c 1 f hom c f 1 induced by the pairing is invertible as an application we prove that the category of yetter drinfeld modules over a hopf algebra in c is non degenerate if and only if c is

the weyl groupoid of a nichols algebra of diagonal type the theory of nichols algebras of diagonal type is known to be closely related to that of semi simple lie algebras in this paper the connection between both theories is made closer for any nichols algebra of diagonal type invertible transformations are introduced which remind one of the action of the weyl group on the root system associated to a semi simple lie algebra they give rise to the definition of a groupoid as an application an alternative proof of classification results of rosso andruskiewitsch and schneider is obtained without using any technical assumptions on the braiding

the calabi yau property of smash products let h be a semisimple hopf algebra and r a braided hopf algebra in the category of yetter drinfeld modules over h when r is a calabi yau algebra a necessary and sufficient condition for r h to be a calabi yau hopf algebra is given conversely when h is the group algebra of a finite group and the smash product r h is a calabi yau algebra we give a necessary and sufficient condition for the algebra r to be a calabi yau algebra