measurable kac cohomology for bicrossed products we study the kac cohomology for matched pairs of locally compact groups this cohomology theory arises from the extension theory of locally compact quantum groups we prove a measurable version of the kac exact sequence and provide methods to compute the cohomology we give explicit calculations in several examples using results of moore and wigner

twisted dual group algebras equivariant deformations of c0 g abstract we consider a family of twisted fourier algebras a g of a locally compact group g which in the case of a abelian group g are the fourier transforms of the usual twisted group algebras of over cap g o the corresponding c algebras c g o are deformations of c 0 g which are equivariant in the sense that g still acts by left translation the main examples come from cocycles on the dual of an abelian subgroup h of g we prove that for such cocycles the twisted dual group algebras c g o are induced from the twisted group algebras c h o and we give detailed formulas for the multiplication on a g which extend to larger dense subalgebras of c 0 g and c b g we anticipate that these larger subalgebras will be useful for constructing deformations of homogeneous spaces c 0 g

deformations of compact quantum groups via rieffel s quantization it is shown that compact quantum groups containing torus subgroups can be deformed into new compact quantum groups under rieffel s quantization this is applied to showing that the two classes of compact quantum groupskqu andkq studied by levendorkii and soibelman are strict deformation quantization of each other and that the quantum groupsau m have many deformations

admissibility conjecture and kazhdan s property t for quantum groups abstract we give a partial solution to a long standing open problem in the theory of quantum groups namely we prove that all finite dimensional representations of a wide class of locally compact quantum groups factor through matrix quantum groups admissibility conjecture for quantum group representations we use this to study kazhdan s property t for quantum groups with non trivial scaling group strengthening and generalising some of the earlier results obtained by fima kyed and soltan chen and ng daws skalski and viselter and brannan and kerr our main results are i all finite dimensional unitary representations of locally compact quantum groups which are either unimodular or arise through a special bicrossed product construction are admissible ii a generalisation of a theorem of wang which characterises property t in terms of isolation of finite dimensional irreducible representations in the spectrum iii a very short proof of the fact that quantum groups with property t are unimodular iv a generalisation of a quantum version of a theorem of bekka valette proven earlier for quantum groups with trivial scaling group which characterises property t in terms of non existence of almost invariant vectors for weakly mixing representations v a generalisation of a quantum version of kerr pichot theorem proven earlier for quantum groups with trivial scaling group which characterises property t in terms of denseness properties of weakly mixing representations

non semi regular quantum groups coming from number theory in this paper we study c algebraic quantum groups obtained through the bicrossed product construction examples using groups of adeles are given and they provide the first examples of locally compact quantum groups which are not semi regular the crossed product of the quantum group acting on itself by translations does not contain any compact operator we describe all corepresentations of these quantum groups and the associated universal c algebras on the way we provide several remarks on c algebraic properties of quantum groups and their actions

compact lie group action with the continuous rokhlin property abstract in this paper we study the continuous rokhlin property of c dynamical systems using techniques of equivariant kk theory and quantum group theory in particular we determine the kk equivalence class and give a classification of kirchberg g algebras when the g is a compact lie group with the hodgkin condition

rieffel deformation via crossed products we start from rieffel data a where a is a c algebra is an action of an abelian group on a and is a 2 cocycle on the dual group using landstad theory of crossed product we get a deformed c algebra a in the case of rn we obtain a very simple proof of invariance of k groups under the deformation in the general case we also get a very simple proof that nuclearity is preserved under the deformation we show how our approach leads to quantum groups and investigate their duality the general theory is illustrated by an example of the deformation of sl 2 c a description of it in terms of noncommutative coordinates is given

quantum symmetry groups of noncommutative spheres we show that the noncommutative spheres of connes and landi are quantum homogeneous spaces for certain compact quantum groups we give a general construction of homogeneous spaces which support noncommutative spin geometries

on locally compact quantum groups whose algebras are factors in this paper we are interested in examples of locally compact quantum groups m such that both von neumann algebras m and the dual m are factors there is a lot of known examples such that m m are respectively of type i1 i1 but there is no examples with factors of other types we construct new examples of type i1 ii1 ii1 ii1 and iii iii for each 2 0 1 also we show that there is no such example with m or m a nite factor a locally compact l c quantum group in the von neumann algebra setting see 6 7 is a pair m where m is a von neumann algebra and is a comultiplication on m with left and right invariant weight in a canonical way every l c group is a commutative l c quantum group and every von neu mann group algebra of a l c group is a cocommutative l c quantum group conversely every commutative or cocommutative l c quantum group is ob tained in this way our aim is to obtain new examples of l c quantum groups which are as far as possible from groups so we will be interested in the least commutative and cocommutative examples the formulation of the problem is the following given a pair of factors of a certain type x1 x2 is it possible to nd a l c quantum group m such that m is a type x1 factor and the dual algebra m is a type x2 factor in the sense of murray von neumann s and connes classication of factors there exists a lot of examples for the

equivariant deformations of homogeneous spaces abstract c algebraic deformations of homogeneous spaces g are constructed by completing dense subspaces of c 0 g in a different multiplication and c norm these deformations are equivariant in the sense that they still carry a natural action of g by left translation the motivating examples are the noncommutative heisenberg manifolds of rieffel but the construction given here is based on the authors twisted dual group algebras which are equivariant deformations of c 0 g the procedure is general enough to give other recently studied families of noncommutative spaces and some interesting new examples of simple c algebras

ergodic actions of compact groups on operator algebras i general theory we introduce some general machinery for studying ergodic actions of compact groups on operator algebras the most important tool being that of the multiplicity map this theory is used subsequently in 42 to prove that a von neumann algebra on which su 2 acts ergodically is necessarily of type i

free products of compact quantum groups we construct and study compact quantum groups from free products ofc algebras in this connection we discover two mysterious classes of natural compact quantum groups a u m anda o m thea u m s respectivelya o m s are non isomorphic to each other for differentm s and are not obtainable by the ordinary quantization method we also clarify some basic concepts in the theory of compact quantum groups

semidirect products of c quantum groups multiplicative unitaries approach c quantum groups with projection are the noncommutative analogues of semidirect products of groups radford s theorem about hopf algebras with projection suggests that any c quantum group with projection decomposes uniquely into an ordinary c quantum group and a braided c quantum group we establish this on the level of manageable multiplicative unitaries

examples of compact matrix pseudogroups arising from the twisting operation we construct multiparameter compact matrix pseudogroups of all types and show their representation theories are the same as their classical analogs

property t discrete quantum groups and subfactors with triangle presentations abstract kazhdan s property t has been studied for several discrete group like structures including standard invariants of jones subfactors and discrete quantum groups we prove a uk type spectral gap criterion for property t in this setting this allows us to construct a family of property t discrete quantum groups as well as subfactor standard invariants given by generators and a triangle presentation these are the first examples of property t discrete quantum groups that are not directly related to a discrete group with property t

tannaka krein duality for compact matrix pseudogroups twisted su n groups the notion of concrete monoidalw category is introduced and investigated a generalization of the tannaka krein duality theorem is proved it leads to new examples of compact matrix pseudogroups among them we have twistedsu n groups denoted bys u n it is shown that the representation theory fors u n is similar to that ofsu n irreducible representations are labeled by young diagrams and formulae for dimensions and multiplicity are the same as in the classical case

deformation of operator algebras by borel cocycles abstract assume that we are given a coaction of a locally compact group g on a c algebra a and a t valued borel 2 cocycle on g motivated by the approach of kasprzak to rieffel s deformation we define a deformation a of a among other properties of a we show that a k l 2 g is canonically isomorphic to a g g this together with a slight extension of a result of echterhoff et al implies that for groups satisfying the baum connes conjecture the k theory of a remains invariant under homotopies of

proper actions fixed point algebras and naturality in nonabelian duality abstract suppose a locally compact group g acts freely and properly on a locally compact hausdorff space x and let be the induced action on c 0 x we consider a category in which the objects are c dynamical systems a g for which there is an equivariant homomorphism of c 0 x into the multiplier algebra m a rieffel has shown that such systems are proper and saturated and hence have a generalized fixed point algebra a which is morita equivalent to a r g we show that the assignment a a is functorial and that rieffel s morita equivalence is natural in a suitable sense we then use our results to prove a categorical version of landstad duality which characterizes crossed products by coactions and to prove that mansfield imprimitivity for crossed products by homogeneous spaces is natural

deformation quantization for actions of qpd abstract the main objective of this article is to develop the theory of deformation of c algebras endowed with a group action from the perspective of non formal equivariant quantization this program initiated in 4 aims to extend rieffel s deformation theory 27 for more general groups than r d in 4 we have constructed such a theory for a class of non abelian lie groups in the present article we study the somehow opposite situation of abelian but non lie groups more specifically we construct here a deformation theory of c algebras endowed with an action of a finite dimensional vector space over a non archimedean local field of characteristic different from 2 at the root of our construction stands the p adic version of the weyl quantization introduced by haran 12 and further extended by bechata 1 and unterberger 34

2 cocycles and twisting of kac algebras we describe the twisting construction with the help of 2 cocycles on hopf von neumann and george kac algebras we show that twisted kac algebras are again kac algebras using this construction we give a wide class of new quantizations of the heisenberg group and describe several series of non trivial finite dimensional hopf algebras kac algebras of dimensions 4n and as twisting of finite groups

actions and coactions of finite quantum groupoids on von neumann algebras extensions of the matched pair procedure abstract in this work actions and coactions of finite c quantum groupoids are studied in an operator algebras context in particular we prove a double crossed product theorem and the existence of an universal von neumann algebra on which any finite groupoid acts outerly we give two actually different extensions of the matched pairs procedure in previous works n andruskiewitsch and s natale define for any matched pair of groupoids two c quantum groupoids in duality we give here an interpretation of them in terms of crossed products of groupoids using a single multiplicative partial isometry which gives a complete description of these structures the second extension deals only with groups to define an other type of finite c quantum groupoids

examples of non compact quantum group actions abstract we present two examples of actions of non regular locally compact quantum groups on their homogeneous spaces the homogeneous spaces are defined in a way specific to these examples but the definitions we use have the advantage of being expressed in purely c algebraic language we also discuss continuity of the obtained actions finally we describe in detail a general construction of quantum homogeneous spaces obtained as quotients by compact quantum subgroups

an analytical investigation of shape control of large space structures by applied temperatures an analytical procedure for the static shape control of flexible space structures subjected to thermal distortions is developed which is based on prescribing temperatures in control elements having much higher coefficients of thermal expansion than the main structure the temperatures at the control elements are defined so as to minimize the overall thermal distortion of the structure from its ideal shape and a matrix equation is obtained which can be solved for the set of optimum control temperatures a formulation of the procedure for continuous structures governed by differential equations and a formulation for discrete finite element modeled structures governed by matrix equations are presented the equations from the continuous formulation are employed for the shape control of a simple beam distorted by nonuniform heating and the discrete formulation is applied in a general purpose finite element structural analysis computer program for the shape control of a 750 m radiometer antenna reflector dish subjected to orbital heating a reduction in thermal distortion by a factor of nearly 50 was obtained with the use of only seven control elements results for four different sets of control locations for the antenna are presented in which reductions in distortion of up to a factor of four were obtained

bicrossproduct structure of the quantum weyl group abstract the quantum weyl group formula associated to a complex simple lie algebra g consists of the quantum group uq g with certain quantum simple reflections w i adjoined let kw be the group algebra of the standard covering w of the weyl group of g here k c we show that formula has the structure of a cocycle bicrossproduct formula kw uq g it consists as an algebra of a cocycle semidirect product by a cocycle action of kw on uq g defined with respect to a certain non abelian cocycle it consists as a coalgebra of an extension by a non abelian dual cocycle the dual of formula is also a bicrossproduct and consists as an algebra of an extension of the dual of uq g by the commutative algebra of functions on w via a cocycle

classifying complements for hopf algebras and lie algebras abstract let a e be a given extension of hopf respectively lie algebras we answer the classifying complements problem ccp which consists of describing and classifying all complements of a in e if h is a given complement then all the other complements are obtained from h by a certain type of deformation we establish a bijective correspondence between the isomorphism classes of all complements of a in e and a cohomological type object ha 2 h a where is the matched pair associated to h the factorization index e a f is introduced as a numerical measure of the ccp for two n th roots of unity we construct a 4 n 2 dimensional hopf algebra whose factorization index over the group algebra is arbitrary large

hopf von neumann algebra bicrossproducts kac algebra bicrossproducts and the classical yang baxter equations two groups g1 g2 are said to be a matched pair if each acts on the space of the other and these actions say obey a certain compatibility condition for every matched pair of locally compact groups g1 g2 we construct an associated coinvolutive hopf von neumann algebra m g1 l g2 by simultaneous cross product and cross coproduct for non trivial these bicrossproduct hopf von neumann algebras are non commutative and non cocommutative if the modules for the actions are also matched then these bicrossproducts are kac algebras in this case we show that the dual kac algebra is of the same form with the roles of g1 g2 and of interchanged examples exist with g1 a simply connected lie group and choices of g2 determined by suitable solutions of the classical yang baxter equations on the complexification of the lie algebra of g1

bicrossproducts of multiplier hopf algebras abstract in this paper we generalize majid s bicrossproduct construction we start with a pair a b of two regular multiplier hopf algebras we assume that b is a right a module algebra and that a is a left b comodule coalgebra the right action of a on b gives rise to the smash product a b the left coaction of b on a gives a possible coproduct on a b we discuss in detail the necessary compatibility conditions between the action and the coaction for to be a proper coproduct on a b the result is again a regular multiplier hopf algebra majid s construction is obtained when we have hopf algebras we also look at the dual case constructed from a pair c d of regular multiplier hopf algebras where now c is a left d module algebra while d is a right c comodule coalgebra we show that indeed these two constructions are dual to each other in the sense that a natural pairing of a with c and of b with d yields a duality between a b and the smash product c d we show that the bicrossproduct of an algebraic quantum group is again an algebraic quantum group i e a regular multiplier hopf algebra with integrals the algebra case is also considered some special cases are treated and they are related with other constructions available in the literature the basic example coming from a not necessarily finite group g with two subgroups h and k such that g k h and h k e where e is the identity of g is used to illustrate our theory more examples will be considered in forthcoming papers on the subject

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

metric aspects of noncommutative homogeneous spaces for a closed cocompact subgroup of a locally compact group g given a compact abelian subgroup k of g and a homomorphism k g satisfying certain conditions landstad and raeburn constructed equivariant noncommutative deformations c g of the homogeneous space g generalizing rieffel s construction of quantum heisenberg manifolds we show that when g is a lie group and g is connected given any norm on the lie algebra of g the seminorm on c g induced by the derivation map of the canonical g action defines a compact quantum metric furthermore it is shown that this compact quantum metric space depends on continuously with respect to quantum gromov hausdorff distances

hopf images in locally compact quantum groups abstract this manuscript is devoted to the study of the concept of a generating subset a k a hopf image of a morphism in the setting of locally compact quantum groups the aim of this paper is to provide an accurate description of the hopf image of a given morphism we extend and unify the previously existing approaches for compact and discrete quantum groups and present some results that can shed light on some local perspective in the theory of quantum groups in particular we provide a characterization of fullness of hopf image in the language of partial actions as well as in representation theoretic terms in the spirit of representation c categories extending some known results to the broader setting of non compact quantum groups

quantum symmetry groups of finite spaces we determine the quantum automorphism groups of finite spaces these are compact matrix quantum groups in the sense of woronowicz

unbounded elements affiliated with c algebras and non compact quantum groups the affiliation relation that allows to include unbounded elements operators into thec algebra framework is introduced investigated and applied to the quantum group theory the quantum deformation of the two fold covering of the group of motions of euclidean plane is constructed a remarkable radius quantization is discovered it is also shown that the quantumsu 1 1 group does not exist on thec algebra level for real value of the deformation parameter

equivariant poincare duality for quantum group actions we extend the notion of poincare duality in kk theory to the setting of quantum group actions an important ingredient in our approach is the replacement of ordinary tensor products by braided tensor products along the way we discuss general properties of equivariant kk theory for locally compact quantum groups including the construction of exterior products as an example we prove that the standard podle sphere is equivariantly poincare dual to itself

traces on noncommutative homogeneous spaces abstract we study properties of c algebraic deformations of homogeneous spaces g which are equivariant in the sense that they preserve the natural action of g by left translation the center is shown to be isomorphic to c g g 0 for a certain subgroup g 0 of g and there is a 1 1 correspondence between normalised traces and probability measures on g g 0 this makes it possible to represent the deformed algebra as operators over l 2 g applications to k theory are also mentioned

rieffel deformation of group coactions let g be a locally compact group gamma subset g an abelian subgroup and let be a continuous 2 cocycle on the dual group hat gamma let b be a c algebra and delta b in rm mor b b otimes rm c 0 g a continuous right coaction using rieffel deformation we can construct a quantum group rm c 0 g tilde psi otimes psi delta psi and the deformed c algebra b the aim of this paper is to present a construction of the continuous coaction delta b psi of the quantum group rm c 0 g tilde psi otimes psi delta psi on b the transition from the coaction b to its deformed counterpart delta b psi is nontrivial in the sense that delta b psi contains complete information about b in order to illustrate our construction we apply it to the action of the lorentz group on the minkowski space obtaining a c algebraic quantum minkowski space

operator equalities related to the quantum e 2 group the paper deals with normal operatorsr ands satisfying simple commutation relations that we encounter investigating the quantum deformation of thee 2 group we show thatr s admits a normal extension if and only ifr 1s satisfies a certain spectral condition a number of related formulae are derived in particular all the functionsf satisfying the character equationf r s f r f s are found

ergodic actions of compact groups on operator algebras iii classification for su 2 we prove that the only ergodic actions ofsu 2 are on type i von neumann algebras using the theory developed in 8

quantum group of orientation preserving riemannian isometries we formulate a quantum group analogue of the group of orientation preserving riemannian isometries of a compact riemannian spin manifold more generally of a possibly r twisted and of compact type spectral triple the main advantage of this formulation which is directly in terms of the dirac operator is that it does not need the existence of any good laplacian as in our previous works on quantum isometry groups several interesting examples including those coming from rieffel type deformation as well as the equivariant spectral triples on su 2 and s c2 are discussed

rieffel deformation of homogeneous spaces let g1 g be a closed subgroup of a locally compact group g and let x g g1 be the quotient space of left cosets let x c0 x x be the corresponding g c algebra where g c0 g suppose that is a closed abelian subgroup of g1 and let be a 2 cocycle on the dual group let g be the rieffel deformation of g using the results of the previous paper of the author we may construct g c algebra x the rieffel deformation of x on the other hand we may perform the rieffel deformation of the subgroup g1 obtaining the closed quantum subgroup g1 g which in turn by the results of s vaes leads to the g c algebra g g1 in this paper we show that g g1 x we also consider the case where g is not a subgroup of g1 for which we cannot construct the subgroup g1 then generically x cannot be identified with a quantum quotient what may be shown is that it is a g simple object in the category of g c algebras

a new approach to induction and imprimitivity results given a closed quantum subgroup of a locally compact quantum group we study induction of unitary corepresentations of the quantum subgroup to the ambient quantum group more generally we study induction given a coaction of the quantum subgroup on a c algebra we prove imprimitivity theorems that unify the existing theorems for actions and coactions of groups this means that we deflne quantum homogeneous spaces as c algebras and that we prove morita equivalence of crossed products and homogeneous spaces we essentially use von neumann algebraic techniques to prove these morita equivalences between c algebras the theory of induced representations of locally compact l c groups was introduced by mackey 21 who discovered the imprimitivity theorem characterizing induced representations through the presence of a covariant representation of a homogeneous space riefiel 26 provided a modern approach using the language of hilbert c modules after the work of riefiel several induction procedures and imprimitivity results have been obtained both for actions and for coactions of groups we shall brie y review them below the purpose of this paper is to develop such an induction and imprimitivity machinery in the setting of locally compact quantum groups 16 17 in this way we provide a unifled approach to several results on actions and coactions of groups on c algebras at the same time our proofs in the setting of l c quantum groups are simpler than the classical proofs dealing with coactions of groups another motivation comes from quantum group theory meyer and nest have undertaken a reformulation of the baum connes conjecture 24 in which induction and restriction play a crucial role the development of induction and imprimitivity in the current paper should play an equally important role in the formulation of a baum connes conjecture for quantum groups

embeddable quantum homogeneous spaces abstract we discuss various notions generalizing the concept of a homogeneous space to the setting of locally compact quantum groups on the von neumann algebra level we recall an interesting duality for such objects studied earlier by m izumi r longo s popa for compact kac algebras and by m enock in the general case of locally compact quantum groups a definition of a quantum homogeneous space is proposed along the lines of the pioneering work of vaes on induction and imprimitivity for locally compact quantum groups the concept of an embeddable quantum homogeneous space is selected and discussed in detail as it seems to be the natural candidate for the quantum analog of classical homogeneous spaces among various examples we single out the quantum analog of the quotient of the cartesian product of a quantum group with itself by the diagonal subgroup analogs of quotients by compact subgroups as well as quantum analogs of trivial principal bundles the former turns out to be an interesting application of the duality mentioned above

deformation of a kac algebra by an abelian subgroup some examples of quantum groups in literature arise as deformations of a locally compact group by a dual 2 cocycle we make this construction in the framework of kac algebras we show that these deformations are still kac algebras using this construction we give new quantizations of the heisenberg group from this point of view we analyse the dimension 8 non trivial example of kac and paljutkin and give a new example of non trivial dimension 12 semi simple hopf algebras a dimension 12 kac algebra

on a certain approach to quantum homogeneous spaces we propose a definition of a quantum homogeneous space of a locally compact quantum group we show that classically it reduces to the notion of homogeneous spaces giving rise to an operator algebraic characterization of the transitive group actions on the quantum level our definition goes beyond the quotient case providing a framework which besides the vaes quotient of a locally compact quantum group by its closed quantum subgroup our main motivation is also compatible with generically non quotient quantum homogeneous spaces of a compact quantum group studied by p podle as well as the rieffel deformation of g homogeneous spaces finally our definition rules out the paradoxical examples of the non compact quantum homogeneous spaces of a compact quantum group

cross product bialgebras i abstract the subject of this article is bialgebra factorizations or cross product bialgebras without cocycles we establish a theory characterizing cross product bialgebras universally in terms of projections and injections especially all known types of biproduct double cross product and bicross product bialgebras can be described by this theory furthermore the theory provides new families of cocycle free cross product bialgebras besides the universal characterization we find an equivalent co modular description of certain types of cross product bialgebras in terms of so called hopf data with the help of hopf data construction we recover again all known cross product bialgebras as well as new and more general types of cross product bialgebras we are working in the general setting of braided monoidal categories which allows us to apply our results in particular to the braided category of hopf bimodules over a hopf algebra majid s double biproduct is seen to be a twisting of a certain tensor product bialgebra in this category this resembles the case of the drinfel d double which can be constructed as a twist of a specific cross product

krein duality for compact quantum groups we define noncommutative krein algebras and extend the classical krein duality for compact groups to compact quantum groups

tensor products of operator spaces abstract in this paper we lay the foundations for a systematic study of tensor products of subspaces of c algebras to accomplish this various notions of duality are introduced and employed elementary proofs of the complete injectivity of the haagerup norm and of the extension theorem for completely bounded maps are given pisier s gamma norms are examined and found to be special cases of the haagerup norm we identify the greatest operator space cross norm and show that the spatial tensor norm is the least operator space cross norm in an appropriate sense indeed most of the elementary theory of banach space tensor norms generalizes to the category of operator spaces

rieffel type discrete deformation of finite quantum groups we introduce a discrete deformation of rieffel type for finite quantum groups using this we give an example of a finite quantum group a of order 18 such that neither a nor its dual can be expressed as a crossed product of the form c g1 g2 with g1 and g2 ordinary finite groups we also give a deformation of finite groups of lie type by using their maximal abelian subgroups

noncommutative manifolds the instanton algebra and isospectral deformations we give new examples of noncommutative manifolds that are less standard than the nc torus or moyal deformations of n they arise naturally from basic considerations of noncommutative differential topology and have non trivial global features

exact three dimensional analysis for static torsion of piezoelectric rods based on the theory of elasticity exact analytical and numerical solutions of piezoelectric rods under static torsion are studied in this paper direct solution method is used the main scope is to check the extension of validity of assumptions in previous papers that had been made based on linear distribution of electric potential through the cross section and their influences on deflection and the angle of rotation stress and electric induction functions are employed to obtain the exact solution of the static and electrostatic equilibrium equations under torsional loading it is shown that previous assumptions are valid only in some types of piezoelectric materials while in other types these assumptions lead to considerable deviations from accurate modeling the present analytical solutions are compared with three dimensional finite element analysis results and absolute agreements are found at the end of this article torsional rigidity shape effects on induced piezoelectric deformation and the range of valid region for linear distribution of electric potential assumption have been studied

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

symmetries of quantum spaces subgroups and quotient spaces of quantum su 2 and so 3 groups we prove that each action of a compact matrix quantum group on a compact quantum space can be decomposed into irreducible representations of the group we give the formula for the corresponding multiplicities in the case of the quotient quantum spaces we describe the subgroups and the quotient spaces of quantumsu 2 andso 3 groups

on cocycle twisting of compact quantum groups abstract we provide a class of examples of compact quantum groups and unitary 2 cocycles on them such that the twisted quantum groups are non compact but still locally compact quantum groups in the sense of kustermans and vaes this also gives examples of cocycle twists where the underlying c algebra of the quantum group changes

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

mapping ideals of quantum group multipliers abstract we study the dual relationship between quantum group convolution maps l 1 g l g and completely bounded multipliers of g for a large class of locally compact quantum groups g we completely isomorphically identify the mapping ideal of row hilbert space factorizable convolution maps with m c b l 1 g yielding a quantum gilbert representation for completely bounded multipliers we also identify the mapping ideals of completely integral and completely nuclear convolution maps the latter coinciding with l 1 b g where b g is the quantum bohr compactification of g for quantum groups whose dual has bounded degree we show that the completely compact convolution maps coincide with c b g our techniques comprise a mixture of operator space theory and abstract harmonic analysis including fubini tensor products the non commutative grothendieck inequality quantum eberlein compactifications and a suitable notion of inner co amenable quantum group that we introduce and for which we exhibit examples from the bicrossed product construction our main results are new even in the setting of group von neumann algebras v n g for quasi sin locally compact groups g

the unitary implementation of a locally compact quantum group action abstract in this paper we study actions of locally compact quantum groups on von neumann algebras and prove that every action has a canonical unitary implementation paralleling haagerup s classical result on the unitary implementation of a locally compact group action this result is an important tool in the study of quantum groups in action we will use it in this paper to study subfactors and inclusions of von neumann algebras when is an action of the locally compact quantum group m on the von neumann algebra n we can give necessary and sufficient conditions under which the inclusion n n m n is a basic construction here n denotes the fixed point algebra and m n is the crossed product when is an outer and integrable action on a factor n we prove that the inclusion n n is irreducible of depth 2 and regular giving a converse to the results of m enock and r nest 1996 j funct anal 137 466 543 1998 j funct anal 154 67 109 finally we prove the equivalence of minimal and outer actions and we generalize the main theorem of yamanouchi 1999 math scand 84 297 319 every integrable outer action with infinite fixed point algebra is a dual action

on a morita equivalence between the duals of quantum su 2 and quantum e 2 abstract let su q 2 and e q 2 be woronowicz s q deformations of respectively the compact lie group su 2 and the non trivial double cover of the lie group e 2 of euclidean transformations of the plane we prove that in some sense their duals are morita equivalent locally compact quantum groups in more concrete terms we prove that the von neumann algebraic quantum groups l su q 2 and l e q 2 are unitary cocycle deformations of each other

saint venant end effects in piezoceramic materials the stress decay rate and characteristic decay length for anisotropic materials and fiber composites have been intensively investigated however those for piezoelectric materials have not been studied in this paper we examine the saint venant end effects of piezoceramic materials by considering the problem of a semi infinite piezoceramic strip polarized in the thickness direction and with applied voltages on the upper and lower surfaces it is assumed that the gradient of electric potential in the axial direction is much smaller than that in the thickness direction thus the governing equations in terms of the airy stress function and electric potential function can be uncoupled the governing equation in terms of the airy stress function involves only two non dimensional parameters after non dimensionalization finally the stress decay rates and characteristic decay lengths for a variety of piezoceramic materials are computed numerically and their variation with the two non dimensional parameters is presented

an analogue of the thom isomorphism for crossed products of a c algebra by an action of r abstract in this paper we show the existence and uniqueness of a natural isomorphism oj of kj a with kj 1 a r j z 2 where a r is a c dynamical r system k is the functor of topological k theory and a r is the crossed product of a by the action of r the pimsner voiculescu exact sequence is obtained as a corollary we show that given an invariant trace on a with dual trace gt one has gto 1 u 1 2 i u u for any unitary u in the domain of the derivation of a associated to the action finally we show that the crossed product of c s3 continuous functions on the 3 sphere by a minimal diffeomorphism is a simple c algebra with no nontrivial idempotent

cohomology and coquasi bialgebra extensions associated to a matched pair of bialgebras abstract by using braid diagrams we explicitly reconstruct the cohomology associated to a matched pair of cocommutative bialgebras in order to give a method of constructing coquasi bialgebras which generalize bialgebras and classifying them up to monoidal equivalence of their comodule categories an alternative homological proof is given for schauenburg s generalized kac sequence involving the abelian group opext h k of bialgebra extensions we define an abelian group opext h k of coquasi bialgebra extensions associated to a singer pair h k of bialgebras and prove a variant of schauenburg s sequence which involves the group it is also proved that there is a natural isomorphism opext h 1 k opext h 2 k that preserves monoidal equivalence classes if h 1 k and h 2 k arise from such matched pairs that are shifts of each other

non compact quantum groups associated with abelian subgroups letg be a lie group for any abelian subalgebra mathfrak h of the lie algebra g ofg and any r in mathfrak h lambda mathfrak h the difference of the left and right translates ofr gives a compatible poisson bracket ong we show how to construct the corresponding quantum group in thec algebra setting the main tool used is the general deformation quantization construction developed earlier by the author for actions of vector groups onc algebras

ergodic coactions with large multiplicity and monoidal equivalence of quantum groups we construct new examples of ergodic coactions of compact quantum groups in which the multiplicity of an irreducible corepresentation can be strictly larger than the dimension of the latter these examples are obtained using a bijective correspondence between certain ergodic coactions on c algebras and unitary fiber functors on the representation category of a compact quantum group we classify these unitary fiber functors on the universal orthogonal and unitary quantum groups the associated c algebras and von neumann algebras can be defined by generators and relations but are not yet well understood

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

deformation of c algebras by cocycles on locally compact quantum groups given a c algebra a with a left action of a locally compact quantum group g on it and a unitary 2 cocycle on g we define a deformation a of a the construction behaves well under certain additional technical assumptions on the most important of which is regularity meaning that c0 g g is isomorphic to the algebra of compact operators on some hilbert space in particular then a is stably isomorphic to the iterated twisted crossed product g op g a also in good situations the c algebra a carries a left action of the deformed quantum group g and we have an isomorphism g a g a when g is a genuine locally compact group we show that the action of g on c0 g cr g is always integrable stronger assumptions of properness and saturation of the action imply regularity as an example we make a preliminary analysis of the cocycles on the duals of some solvable lie groups recently constructed by bieliavsky et al and discuss the relation of our construction to that of bieliavsky and gayral

closed quantum subgroups of locally compact quantum groups abstract we investigate the fundamental concept of a closed quantum subgroup of a locally compact quantum group two definitions one due to s vaes and one due to s l woronowicz are analyzed and relations between them discussed among many reformulations we prove that the former definition can be phrased in terms of quasi equivalence of representations of quantum groups while the latter can be related to an old definition of podle from the theory of compact quantum groups the cases of classical groups duals of classical groups compact and discrete quantum groups are singled out and equivalence of the two definitions is proved in the relevant context a deep relationship with the quantum group generalization of the herz restriction theorem from classical harmonic analysis is also established in particular in the course of our analysis we give a new proof of the herz restriction theorem

note evasive properties of sparse graphs and some linear equations in primes we give an unconditional version of a conditional on the extended riemann hypothesis result of babai banerjee kulkarni and naik 2010 1 on the evasiveness of sparse graphs on n nodes provided that n is large enough we also obtain a substantially stronger estimate that holds for almost all n our approach is based on some rather deep tools from analytic number theory the bombieri vinogradov theorem a result of balog and sarkozy on prime divisors of sum sets and a result baker and harman on large prime divisors of shifted primes

an algebraic framework for group duality abstract a hopf algebra is a pair a whereais an associative algebra with identity and a homomorphism formatoa asatisfying certain conditions if we drop the assumption thatahas an identity and if we allow to have values in the so called multiplier algebram a a we get a natural extension of the notion of a hopf algebra we call this a multiplier hopf algebra the motivating example is the algebra of complex functions with finite support on a group with the comultiplication defined as dual to the product in the group also for these multiplier hopf algebras there is a natural notion of left and right invariance for linear functionals called integrals in hopf algebra theory we show that if such invariant functionals exist they are unique up to a scalar and faithful for a regular multiplier hopf algebra a i e with invertible antipode with invariant functionals we construct in a canonical way the dual it is again a regular multiplier hopf algebra with invariant functionals it is also shown that the dual of is canonically isomorphic with the original multiplier hopf algebra a it is possible to generalize many aspects of abstract harmonic analysis here one can define the fourier transform one can prove plancherel s formula because any finite dimensional hopf algebra is a regular multiplier hopf algebra and has invariant functionals our duality theorem applies to all finite dimensional hopf algebras then it coincides with the usual duality for such hopf algebras but our category of multiplier hopf algebras also includes in a certain way the discrete quantum groups and the compact quantum groups our duality includes the duality between discrete quantum groups and compact quantum groups in particular it includes the duality between compact abelian groups and discrete abelian groups one of the nice features of our theory is that we have an extension of this duality to the non abelian case but within one category this is shown in the last section of our paper where we introduce the algebras of compact type and the algebras of discrete type we prove that also these are dual to each other we treat an example that is sufficiently general to illustrate most of the different features of our theory it is also possible to construct the quantum double of drinfel d within this category this provides a still wider class of examples so we obtain many more than just the compact and discrete quantum within this setting

relative matched pairs of finite groups from depth two inclusions of von neumann algebras to quantum groupoids in this work we give a generalization of matched pairs of finite groups to describe a general class of depth two inclusions of factor von neumann algebras and the c quantum groupoids associated with using double groupoids

quantum groups with projection on von neumann algebra level abstract we introduce an axiomatization of the notion of a semidirect product of locally compact quantum groups and study their properties our approach is slightly different from the one introduced in the thesis of s roy 16 and unlike the investigations of roy we work within the von neumann algebraic picture this allows the use of powerful techniques related to crossed products by actions of locally compact quantum groups in particular we show existence of a braided comultiplication on the algebra spanned by slices of braided multiplicative unitary

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

a new quantum deformation of ax b group the paper is devoted to locally compact quantum groups that are related to the classical ax b group we discuss in detail the quantization of the deformation parameter assumed with no justification in the previous paper next we construct on the c level a larger family of quantum deformations of the ax b group corresponding to the deformation parameter q2 running over an interval in the unit circle to this end beside the reflection operator known from the previous paper we use a new unitary generator w it commutes with a b and w ssgn b w where s s1 is a new deformation parameter related to q2 at the end we discuss the groups at roots of unity

classifying complements for associative algebras abstract for a given extension a e of associative algebras we describe and classify up to an isomorphism all a complements of e i e all subalgebras x of e such that e a x and a x 0 let x be a given complement and a x the canonical matched pair associated with the factorization e a x we introduce a new type of deformation of the algebra x by means of the given matched pair and prove that all a complements of e are isomorphic to such a deformation of x several explicit examples involving the matrix algebra are provided