an r matrix approach to the quantization of the euclidean group e 2 the r matrices for two different deformations of the euclidean group e 2 calculated in a two dimensional representation are used to determine the deformed hopf algebra of the representative functions the duality of the latter with the initial quantum algebras is explicitly proved and the relationship between the two quantum groups is discussed and clarified

functional distribution of _ with real characters and denseness of quadratic class numbers we investigate the functional distribution of l functions l s d with real primitive characters d on the region 1 2 res 1 as d varies over fundamental discriminants actually we establish the so called universality theorem for l s d in the d aspect from this theorem we can of course deduce some results concerning the value distribution and the non vanishing as another corollary it follows that for any fixed a b with 1 2 a b 1 and positive integers r m there exist infinitely many d such that for every r 1 2 r the r th derivative l r s d has at least m zeros on the interval a b in the real axis we also study the value distribution of l s d for fixed s with res 1 and variable d and obtain the denseness result concerning class numbers of quadratic fields

classification of bicovariant differential calculi on quantum groups suppose thatq is not a root of unity we classify all bicovariant differential calculi of dimension greater than one on the quantum groupsgl q n o q n andsp q n for which the differentials du of the matrix entriesu generate the left module of first order forms our first classification theorem asserts that there are precisely two one parameter families of such calculi ongl q n forn 3 in the limitq 1 only two of these calculi give the ordinary differential calculus ongl n our second main theorem states that apart from finitely manyq there exist precisely two differential calculi with these properties ono q n andsp q n forn 4 this strengthens the corresponding result proved in our previous paper ss2 there are four such calculi ono q 3 we introduce two new 4 dimensional bicovariant differential calculi ono q 3

selberg s orthonormality conjecture and joint universality of l functions in the paper we introduce a new method how to use only an orthonormality relation of coefficients of dirichlet series defining given l functions from the selberg class to prove joint universality

joint universality theorems for pairs of automorphic zeta functions in this paper we establish the joint universality theorems for pairs consisting of the riemann zeta function automorphic l functions associated with holomorphic hecke eigen cusp forms rankin selberg l functions and symmetric square l functions

algebraic structure of multiparameter quantum groups let g be a connected semi simple complex lie group we define and study the multi parameter quantum group cq p g in the case where q is a complex parameter that is not a root of unity using a method of twisting bigraded hopf algebras by a cocycle 2 we develop a unified approach to the construction of cq p g and of the multi parameter drinfeld double dq p using a general method of deforming bigraded pairs of hopf algebras we construct a hopf pairing between these algebras from which we deduce a peterweyl type theorem for cq p g we then describe the prime and primitive spectra of cq p g generalizing a result of joseph in the one parameter case this description was conjectured and established in the sl n case by the first and second authors in 15 16 it was proved in the general case by joseph in 18 19 in particular the orbits in prim cq p g under the natural action of the maximal torus h are indexed as in the one parameter case by the elements of the double weyl group w w unlike the one parameter case there is not in general a bijection between symp g and prim cq p g however in the case when

the compact quantum group uq 2 i corresponding to hayashi s r matrix we first follow woronowicz s method to construct an other concrete compact quantum group uq 2 q c 0 then we discuss its irreducible representations and we show that these representations correspond to those of irrational algebras b and in the final we also give the formula of haar measure of uq 2

k amenability of hnn extensions of amenable discrete quantum groups abstract we construct hnn extensions of discrete quantum groups we study their representation theory and we show that an hnn extension of amenable discrete quantum groups is k amenable

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

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

quantum deformation of lorentz group a one parameter quantum deformations l 2 ofsl 2 is introduced and investigated an analog of the iwasawa decomposition is proved the compact part of this decomposition coincides withs u 2 whereas the solvable part is identified as a pontryagin dual ofs u 2 it shows thats l 2 is the result of the dual version of drinfeld s double group construction applied tos u 2 the same construction applied to any compact quantum groupgc is discussed in detail in particular the explicit formulae for the haar measures on the pontryagin dualgd ofgc and on the double groupg are given we show that there exists remarkable 1 1 correspondence between representations ofg and bicovariant bimodules tensor bundles overgc the theory of smooth representations ofs l 2 is the same as that ofsl 2 clebsh gordon coefficients are however modified the corresponding tame bicovariant bimodules ons u 2 are classified an application to 4d differential calculus is presented the nonsmooth case is also discussed

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

the hom yang baxter equation hom lie algebras and quasi triangular bialgebras we study a twisted version of the yang baxter equation called the hom yang baxter equation hybe which is motivated by hom lie algebras three classes of solutions of the hybe are constructed one from hom lie algebras and the others from drinfeld s dual quasi triangular bialgebras each solution of the hybe can be extended to operators that satisfy the braid relations assuming an invertibility condition these operators give a representation of the braid group

the free wreath product of a compact quantum group by a quantum automorphism group in this thesis we define and study the free wreath product of a compact quantum group by a quantum automorphism group and in this way we generalize the previous notion of free wreath product by the quantum symmetric group introduced by bichon our investigation is divided into two part in the first we define the free wreath product of a discrete group by a quantum automorphism group we show how to describe its intertwiners by making use of decorated noncrossing partitions and from this thanks to a result of lemeux we deduce the irreducible representations and the fusion rules then we prove some properties of the operator algebras associated to this compact quantum group such as the simplicity of the reduced c algebra and the haagerup property of the von neumann algebra the second part is a generalization of the first one we start by defining the notion of free wreath product of a compact quantum group by a quantum automorphism group we generalize the description of the spaces of the intertwiners obtained in the discrete case and by adapting a monoidal equivalence result of lemeux and tarrago we find the irreducible representations and the fusion rules then we prove some stability properties of the free wreath product operation in particular we find under which conditions two free wreath products are monoidally equivalent or have isomorphic fusion semirings we also establish some analytic and algebraic properties of the dual quantum group and of the operator algebras associated to a free wreath product as a last result we prove that the free wreath product of two quantum automorphism groups can be seen as the quotient of a suitable quantum automorphism group

on semi groups arising from quantum groups abstract letgqbe a quantum group in this paper we introduce and completely characterize a semi group and its borel sub semi groups consisting of algebra homomorphisms from the coordinate algebrak gq ofgq also we present invariant elements for the semi group which are closely connected to hopf ideals ofk gq finally based on the theory of structures of the semi group we prove the existence of a frobenius morphism overgqwhenever the parameterqis a root of unity

on the equivariant tamagawa number conjecture for tate motives let l be a finite abelian extension of and let k be any subfield of l for each integer r with r 0 we prove the equivariant tamagawa number conjecture for the pair h0 spec l r gal l k

the iwasawa conjecture for totally real fields let f be a totally real number field let p be a prime number and for any integer n let fun denote the group of nth roots of unity let 41 be a p adic valued artin character for f and let f be the extension of f attached to 4 i e so that 4 is the character of a faithful representation of gal f f we will assume that f is also totally real for a number field k let k denote the cyclotomic zp extension of k following greenberg we say that 4 is of type s if f n fc f and of type w if 4 is one dimensional with f p c fcc deligne and ribet in dr following kubota and leopoldt for the case f q have proved the existence of a p adic l function associated to a one dimensional artin character 4 with f totally real this function lp s 4 is continuous for s e zp 1 and even at s 1 if 4 is not trivial and satisfies the following interpolation property

coribbon hopf face algebras generated by lattice models abstract by studying points of the underlying quantum groups of coquasitriangular hopf face algebras we construct ribbon categories for each lattice models without spectral parameter of both vertex and face type also we give a classification of the braiding and the ribbon structure on quantized classical groups and modular tensor categories closely related to quantum su n l invariants of 3 manifolds

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

conjugation in braided c categories and orthogonal quantum groups we consider braided c categories with conjugates and the notion of dimension if the braid is a birman wenzl murakami symmetry then we characterize the compact orthogonal quantum groups

nonstandard comodules for quantum matrix bialgebras we carry out a construction of nonstandard matrix quantum groups comodules through a generalization of the coaction on tensor powers of a comodule this generalization amounts to consider the situation where quantum space coordinates and its endomorphism matrix entries belong to a noncommutative quadratic algebra

finite dimensional representations of the quantum lorentz group all finite dimensional irreducible representations of the quantum lorentz group sl q 2 are described explicitly and it is proved all finite dimensional representations of sl q 2 are completely reducible the conjecture of podle and woronowicz will be answered affirmatively

the hom yang baxter equation and hom lie algebras motivated by recent work on hom lie algebras a twisted version of the yang baxter equation called the hom yang baxter equation hybe was introduced by yau j phys a 42 165202 2009 in this paper several more classes of solutions of the hybe are constructed some of the solutions of the hybe are closely related to the quantum enveloping algebra of sl 2 the jones conway polynomial and yetter drinfel d modules under some invertibility conditions we construct a new infinite sequence of solutions of the hybe from a given one

weak bialgebras of fractions abstract we construct the algebra of fractions of a weak bialgebra relative to a suitable denominator set of group like elements that is almost central a condition we introduce in the present article which is sufficient in order to guarantee existence of the algebra of fractions and to render it a weak bialgebra the monoid of all group like elements of a coquasi triangular weak bialgebra for example forms a suitable set of denominators as does any monoid of central group like elements of an arbitrary weak bialgebra we use this technique in order to construct new weak bialgebras whose categories of finite dimensional comodules relate to sl 2 fusion categories in the same way as gl 2 relates to sl 2

relative k_ 0 annihilators fitting ideals and stickelberger phenomena when g is abelian and l is a prime we show how elements of the relative k group k 0 bf z l g bf q l give rise to annihilator fitting ideal relations of certain associated bf z g modules examples of this phenomenon are ubiquitous particularly we give examples in which g is the galois group of an extension of global fields and the resulting annihilator fitting ideal relation is closely connected to stickelberger s theorem and to the conjectures of coates and sinnott and brumer higher stickelberger ideals are defined in terms of special values of l functions when these vanish we show how to define fractional ideals generalising the stickelberger ideals with similar annihilator properties the fractional ideal is constructed from the borel regulator and the leading term in the taylor series for the l function en route our methods yield new proofs in the case of abelian number fields of formulae predicted by lichtenbaum for the orders of k groups and etale cohomology groups of rings of algebraic integers

double quantum groups and iwasawa decomposition abstract the double quantum groups c q d g c q g c q g are the hopf algebras underlying the complex quantum groups of which the simplest example is the quantum lorentz group they are nonstandard quantizations of the double groupg g we construct a corresponding quantized universal enveloping algebrauq d g and prove that the pairing between c q d g anduq d g is nondegenerate we analyze the representation theory of these c q d g give a detailed version of the iwasawa decomposition proved by podles and woronowicz for the quantum lorentz group and show that c q d g is noetherian finally we outline how to construct more general nonstandard quantum groups using quantum double groups and their generalizations

on the classification of quantum poincare groups using the general theory of 10 quantum poincare groups without dilatations are described and investigated the description contains a set of numerical parameters which satisfy certain polynomial equations for most cases we solve them and give the classification of quantum poincare groups each of them corresponds to exactly one quantum minkowski space the poincare series of these objects are the same as in the classical case we also classify possibler matrices for the fundamental representation of the group

quasitriangularity and enveloping algebras for inhomogeneous quantum groups coquasitriangular universal r matrices on quantum lorentz and quantum poincare groups are classified the results are extended under certain assumptions to inhomogeneous quantum groups enveloping algebras on those objects are described

face algebras and unitarity of su n l tqft using face algebras i e algebras of l operators of irf models we construct modular tensor categories with positive definite inner product whose fusion rules and s matrices are the same as or slightly different from those obtained by u q n at roots of unity also we obtain state sums of abf models on framed links which give quantum su 2 invariants of corresponding 3 manifolds

the isolated quantum group of the gl2 family abstract quantum analogs of gl 2 are parameterized by pgl 2 orbits in the commuting variety of the projective linear group pgl 2 the quantum group corresponding to the orbit constituting the second connected component of the commuting variety has previously received little attention this paper investigates ring theoretical properties of its coordinate ring

consistent multiparameter quantisation of gl n the author describes a manifold of quantum group structures on the vector space of the universal enveloping algebra of gl n and on its dual the space of polynomials in n2 variables the dimension of the manifold is n2 n 2 2

a pairing theorem between a braided bialgebra and its dual bialgebra abstract this paper constructs a dual bialgebra for the frt bialgebra related to a hayashi s one parameter r matrix by using the skew derivation method the natural pairing between them is non degenerate

woronowicz construction of compact quantum groups for functions on permutations classification result for n 3 abstract we provide a classification of compact quantum groups which can be obtained by the woronowicz construction when the arrays used in the twisted determinant condition are extensions of functions on permutations general properties of such quantum groups are revealed with the aid of operators intertwining tensor powers of the fundamental corepresentation two new families of quantum groups appear three versions of the quantum group su q 3 with complex q and a non commutative analogue of the semi direct product of two dimensional torus with the alternating group a 3

cylinder braiding for quantum linear groups this group is isomorphic to the group of topological braids with k strings in the cylinder c 0 0 1 it is the purpose of this paper to describe a general construction of tensor representations of zbk based on quantum linear groups a tensor representation of zbk is obtained from any four braid pair x f on a module v over a commutative ring k by definition this consists of a yangbaxter automorphism r matrix x v v v v which satisfies the braid relation x 1 1 x x 1 1 x x 1 1 x on v v v and an automorphism f v v such that the four braid relation

annihilating selmer modules we prove that for a wide class of motives the local and global non commutative tamagawa number conjectures of fukaya and kato imply explicit restrictions on the galois structure of selmer modules in the sense of bloch and kato we thereby obtain a variety of new concrete and general conjectures concerning the structures of selmer modules in several important cases we prove these conjectures

cocycle deformations of coordinate rings of quantum matrices abstract a negative answer is given to the natural question whether o q gl n the coordinate ring of quantum gl n is a cocycle deformation of o gl n it is shown that this occurs only ifq 1 the main results are expressed more generally in terms of two parameter quantizations o m n

the fusion rules of some free wreath product quantum groups and applications abstract in this article we find the fusion rules of the free wreath product quantum groups s n for any discrete group to do this we describe the spaces of intertwiners between basic corepresentations which allows us to identify the irreducible corepresentations we then apply the knowledge of the fusion rules to prove in most cases several operator algebraic properties of the associated reduced c algebras such as simplicity and uniqueness of the trace we also prove that the associated von neumann algebra is a full type ii 1 factor and that the dual of s n has the haagerup approximation property for all finite groups

quantum groups and quantum determinants abstract we construct function coordinate algebras of multi parameter quantum deformations of g sl n gl n so n o n and sp n when g is of rank l the obtained function algebra has 1 l l 1 2 continuous parameters and some discrete parameters whose values are roots of unity to do this we introduce a notion of co quasitriangular bialgebras and study their group like elements quantum determinants

quantum groups and quantum semigroups abstract we construct a functor from a certain category of quantum semigroups to a category of quantum groups which for example assigns fun mat q n to fun gl q n combining with a generalization of the faddeev reshetikhin takhtadzhyan construction we obtain quantum groups with universal r matrices for every constant solution of the yang baxter equation of both vertex and face type

duality for multiparametric quantum gl n we show that the hopf algebra uuq dual to the multiparameter matrix quantum group gluq n may be realized a la sudbery 1990 i e tangent vectors at the identity furthermore we give the cartan weyl basis of uuq and show that this is consistent with the duality we show that as a commutation algebra uuq equivalent to uu sl n c x uu z where z is one dimensional and uu z is a central algebra in uuq however as a co algebra uuq cannot be split in this way and depends on all parameters

tannaka duals in semisimple tensor categories abstract tannaka duals of finite dimensional hopf algebras inside semisimple tensor categories are used to construct orbifold tensor categories which are shown to include the tannaka dual of the dual hopf algebras the second orbifolds are then monoidally equivalent to the initial tensor categories in a canonical fashion