on the haar measure of the quantumsu n group we prove that the haar state associated to the compact matrix quantum groupsu n is faithful for 1 1 0 and anyn 2

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

finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple lie algebra let g be a complex simple lie algebra we show that whent is not a root of 1 all finite dimensional representations of the quantum analogu t g are completely reducible and we classify the irreducible ones in terms of highest weights in particular they can be seen as deformations of the representations of the classical u g

complementarity of suq 3 and uq 2 and q boson realization of the suq 3 irreducible representations the complementarity relationship also termed duality that arises between the irreps of the su 3 and u 2 lie algebras when their direct sum su 3 u 2 is embedded into a larger u 6 algebra and single row irreps of the latter are considered is extended to the corresponding q algebras suq 3 and uq 2 it is demonstrated by explicitly constructing the unique q boson state that is simultaneously maximal in suq 3 and uq 2 for a given u q 2 weight in addition the relations between the suq 3 and uq 2 casimir operators resulting from their complementarity are explicitly found together with the q bargmann representation of q boson operators the complementarity relationship is then used to construct a gel fand tseitlin basis for arbitrary suq 3 irreps in terms of q boson operators

wavelet galerkin schemes for boundary integral equations implementation and quadrature in the present paper we consider the fully discrete wavelet galerkin scheme for the fast solution of boundary integral equations in three dimensions it produces approximate solutions within discretization error accuracy offered by the underlying galerkin method at a computational expense that stays proportional to the number of unknowns we focus on algorithmical details of the scheme in particular on numerical integration of relevant matrix coefficients we illustrate the proposed algorithm by numerical results

q deformation of the lorentz group we describe a q deformation of the lorentz group in terms of a q deformation of the van der waerden spinor algebra

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

a quantum analog of the z algebra we define a natural quantum analog for the z algebra and which we refer to as the zq algebra by modding out the heisenberg algebra from the quantum affine algebra uq sl 2 with level k we discuss the representation theory of this zq algebra in particular we exhibit its reduction to a group algebra and to a tensor product of a group algebra with a quantum clifford algebra when k 1 and k 2 and thus we recover the explicit constructions of uq sl 2 standard modules as achieved by frenkel jing and bernard respectively moreover for arbitrary nonzero level k we show that the explicit basis for the simplest z generalized verma module as constructed by lepowsky and primc is also a basis for its corresponding zq module i e it is invariant under the q deformation for generic q we expect this zq algebra associated with uq sl 2 at level k to play the role of a dynamical symmetry in the off critical zk statistical models

toroidal and level 0 uq sln 1 actions on uq gln 1 modules 1 utilizing a braid group action on a completion of uq sln 1 an algebra homomorphism from the toroidal algebra uq sln 1 tor n 2 to a completion of uq gln 1 is obtained 2 the toroidal actions by saito induces a level 0 uq sln 1 action on level 1 integrable highest weight modules of uq sln 1 another level 0 uq sln 1 action was defined by jimbo et al in the case n 1 using the fact that the intertwiners of uq sln 1 modules are intertwiners of toroidal modules for an appropriate comultiplication the relation between these two level 0 uq sln 1 actions is clarified

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

on riesz minimal energy problems abstract in r n n 2 we study the constructive and numerical solution of minimizing the energy relative to the riesz kernel x y n where 1 n for the gauss variational problem considered for finitely many compact mutually disjoint boundaryless n 1 dimensional c manifolds l l l each l being charged with borel measures with the sign l 1 prescribed we show that the gauss variational problem over an affine cone of borel measures can alternatively be formulated as a minimum problem over an affine cone of surface distributions belonging to the sobolev slobodetski space h e 2 where e 1 and l l l this allows the application of simple layer boundary integral operators on and hence a penalty approximation a corresponding numerical method is based on the galerkin bubnov discretization with piecewise constant boundary elements wavelet matrix compression is applied to sparsify the system matrix to the discretized problem a gradient projection method is applied numerical results are presented to illustrate the approach

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 mechanics on profinite groups and partial order inverse limits and profinite groups are used in a quantum mechanical context two cases are considered a quantum system with positions in the profinite group and momenta in the group and a quantum system with positions in the profinite group and momenta in the group the corresponding schwatz bruhat spaces of wavefunctions and the heisenberg weyl groups are discussed the sets of subsystems of these systems are studied from the point of view of partial order theory it is shown that they are directed complete partial orders it is also shown that they are topological spaces with t0 topologies and this is used to define continuity of various physical quantities the physical meaning of profinite groups non archimedean metrics partial orders and t0 topologies in a quantum mechanical context is discussed

quantum deformation of brst algebra we investigate theq deformation of the brst algebra the algebra of the ghost matter and gauge fields on one spacetime point using the result of the bicovariant differential caculus there are two nilpotent operations in the algebra the brst transformation b and the derivatived we show that one can define the covariant commutation relations among the fields and their derivatives consistently with these two operations as well as the operation the antimultiplicativ e inner involution

on unitary representation theories of compact quantum groups categorical structure of unitary representation of compact quantum groups is studied with relation to a metrical structure encountered in the monoidal category of bimodules of finite jones index

le langage des espaces et des groupes quantiques we study the foundations of the differential calculus in quantum geometry the notions of differential quantum space and cone are introduced generalizing a construction of manin to a quantum cone we associate the quantum group of its linear automorphisms preserving the differentials and deduce a de rham complex on this group we give examples of differential calculi on quantum hyperplanes and quantum linear groups

compression techniques for boundary integral equations asymptotically optimal complexity estimates matrix compression techniques in the context of wavelet galerkin schemes for boundary integral equations are developed and analyzed that exhibit optimal complexity in the following sense the fully discrete scheme produces approximate solutions within discretization error accuracy offered by the underlying galerkin method at a computational expense that is proven to stay proportional to the number of unknowns key issues are the second compression which reduces the near field complexity significantly and an additional a posteriori compression the latter is based on a general result concerning an optimal work balance that applies in particular to the quadrature used to compute the compressed stiffness matrix with sufficient accuracy in linear time

differential calculus on compact matrix pseudogroups quantum groups the paper deals with non commutative differential geometry the general theory of differential calculus on quantum groups is developed bicovariant bimodules as objects analogous to tensor bundles over lie groups are studied tensor algebra and external algebra constructions are described it is shown that any bicovariant first order differential calculus admits a natural lifting to the external algebra so the external derivative of higher order differential forms is well defined and obeys the usual properties the proper form of the cartan maurer formula is found the vector space dual to the space of left invariant differential forms is endowed with a bilinear operation playing the role of the lie bracket commutator generalized antisymmetry relation and jacobi identity are proved

non integrable representations of the restricted quantum analogue of sl 3 the structure of irreducible representations of restricted at roots of unity is understood within the gelfand zetlin basis the latter needs a weakened definition for non integrable representations where the quadratic casimir operator of the quantum subalgebra is not completely diagonalized this is necessary in order to take into account the indecomposable modules that appear the set of redefined mixed states has a teepee shape inside the pyramid made with the whole representation

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

the full classification of orthogonal easy quantum groups we study easy quantum groups a combinatorial class of orthogonal quantum groups introduced by banica speicher in 2009 we show that there is a countable descending chain of easy quantum groups interpolating between bichon s free wreath product with the permutation group s n and a semi direct product of a permutation action of s n on a free product this reveals a series of new commutation relations interpolating between a free product construction and the tensor product furthermore we prove a dichotomy result saying that every hyperoctahedral easy quantum group is either part of our new interpolating series of quantum groups or belongs to a class of semi direct product quantum groups recently studied by the authors this completes the classification of easy quantum groups we also study combinatorial and operator algebraic aspects of the new interpolating series

simple compact quantum groups i the notion of simple compact quantum group is introduced as non trivial noncommutative and noncocommutative examples the following families of compact quantum groups are shown to be simple a the universal quantum groups bu q for q gl n c satisfying qq in n 2 b the quantum automorphism groups aaut b of finite dimensional c algebras b endowed with the canonical trace when dim b 4 including the quantum permutation groups aaut xn on n points n 4 c the standard deformations kq of simple compact lie groups k and their twists kqu as well as rieffel s deformation kj

bicovariant differential calculi on glp q 2 and quantum subgroups in a previous paper all bicovariant differential calculi on the two parameter quantum group glp q 2 were determined in this work the authors elaborate some of their properties and discuss relations with work by other authors furthermore they show that there are two different ways in which bicovariant differential calculi on glq 2 where p q induce corresponding calculi on the quantum subgroup slq 2 it is shown that on slq 2 there are only two different bicovariant differential calculi from these one obtains the 4d or calculi on suq 2 the classical limit of the two differential calculi on slq 2 is investigated revealing a relation with recent work on deformed differential calculi on commutative function algebras

characterization of su q l 1 equivariant spectral triples for the odd dimensional quantum spheres the quantum group suq l 1 has a canonical action on the odd dimensional sphere s 2l 1 q all odd spectral triples acting on the l2 space of s 2l 1 q and equivariant under this action have been characterized this characterization then leads to the construction of an optimum family of equivariant spectral triples having nontrivial k homology class these generalize the results of chakraborty pal for suq 2 ams subject classification no 58b34 46l87 19k33

the boundary of universal discrete quantum groups exactness and factoriality we study the c algebras and von neumann algebras associated with the universal discrete quantum groups they give rise to full prime factors and simple exact c algebras the main tool in our work is the study of an amenable boundary action yielding the akemann ostrand property finally this boundary can be identified with the martin or the poisson boundary of a quantum random walk

singular vectors of representations of quantum groups the author gives explicit formulae for singular vectors of verma modules over uq g they give the general formula for g an for some cases when g dn and for some rank two subalgebras of g not an dn for this he uses a special basis of uq g where g is the negative root subalgebra of g which was introduced in his earlier work on the case q 1 this basis seems more economical than the poincare birkhoff witt type of basis used by malikov feigin and fuchs 1986 for the construction of singular vectors of verma modules in the case q 1 furthermore this basis turns out to be part of a general basis introduced recently for other reasons by lusztig 1988 for uq b where b is a borel subalgebra of g

braided lie algebras and bicovariant differential calculi over co quasitriangular hopf algebras we show that if g is the quantum tangent space or quantum lie algebra in the sense of woronowicz of a bicovariant first order differential calculus over a co quasitriangular hopf algebra a r then a certain extension of it is a braided lie algebra in the category of a comodules this is used to show that the woronowicz quantum universal enveloping algebra u g is a bialgebra in the braided category of a comodules we show that this algebra is quadratic when the calculus is inner examples with this unexpected property include finite groups and quantum groups with their standard differential calculi we also find a quantum lie functor for co quasitriangular hopf algebras which has properties analogous to the classical one this functor gives trivial results on standard quantum groups oq g but reasonable ones on examples closer to the classical case such as the cotriangular jordanian deformations in addition we show that split braided lie algebras define generalized lie algebras in a different sense of deforming the adjoint representation we construct these and their enveloping algebras for oq sl n recovering the witten algebra for n 2

differential operators on quantized flag manifolds at roots of unity abstract the quantized flag manifold which is a q analogue of the ordinary flag manifold is realized as a non commutative scheme and we can define the category of d modules on it using the framework of non commutative algebraic geometry however when the parameter q is a root of unity lusztig s frobenius morphism allows us to handle d modules on the quantized flag manifold through modules over a certain sheaf of rings on the ordinary flag manifold in this paper we will show that this sheaf of rings on the ordinary flag manifold is an azumaya algebra over its center we also show that its restriction to certain subsets are split azumaya algebras these are analogues of some results of bezrukavnikov mirkovic rumynin on d modules on flag manifolds in positive characteristics

connected components of compact matrix quantum groups and finiteness conditions abstract we introduce the notion of identity component of a compact quantum group and that of total disconnectedness as a drawback of the generalized burnside problem we note that totally disconnected compact matrix quantum groups may fail to be profinite we consider the problem of constructing the identity component by introducing canonical approximating transfinite sequences of subgroups these sequences have lengths 1 in the classical case but can be countably infinite for duals of discrete groups we give examples of free product quantum groups where the identity component is not normal and the associated sequence has length 1 we give necessary and sufficient conditions for normality of the identity component in the sense of wang and finiteness or profiniteness of the quantum component group among them we introduce an ascending chain condition on the representation ring called lie property which characterizes lie groups in the classical case and reduces to group noetherianity of the discrete group dual in the cocommutative case it is weaker than ring noetherianity but ensures existence of a generating representation the lie property and ring noetherianity are inherited by quotient quantum groups we show that a u f is not of lie type we discuss an example arising from the compact real form of u q sl 2 for q 0

classification of bicovariant differential calculi on quantum groups of type a b c and d under the assumptions thatq is not a root of unity and that the differentialsduji of the matrix entries span the left module of first order forms we classify bicovariant differential calculi on quantum groupsan 1 bn cn anddn we prove that apart one dimensional differential calculi and from finitely many values ofq there are precisely2n such calculi on the quantum groupan 1 slq n forn 3 all these calculi have the dimensionn2 for the quantum groupsbn cn anddn we show that except for finitely manyq there exist precisely twon2 dimensional bicovariant calculi forn 3 wheren 2n 1 forbn andn 2n forcn dn the structure of these calculi is explicitly described and the corresponding ad invariant right ideals of ker e are determined in the limitq 1 two of the 2n calculi foran 1 and one of the two calculi forbn cn anddn contain the ordinary classical differential calculus on the corresponding lie group as a quotient

quantumr matrices related to the spin representations ofbn anddn we present the explicit form of the trigonometricr matrices related to the spin representations of the simple lie algebrasx n b n d n we conjecture that one dimensional configuration sums of the corresponding vertex models in the face formulation are the string functions ofx 1 modules

quantum deformations of certain simple modules over enveloping algebras soit f un corps soit f 0 un sous corps de f soit u l algebre enveloppante universelle sur f 0 de l algebre de lie de kac moody correspondant a la matrice a d entiers on montre qu un module integrable de poids le plus eleve v de u admet une deformation quantique

p adic models of ultrametric diffusion constrained by hierarchical energy landscapes we demonstrate that the p adic analysis is a natural basis for the construction of a wide variety of models of ultrametric diffusion constrained by hierarchical energy landscapes a general analytical description in terms of the p adic analysis is given for a class of models two exactly solvable examples i e the ultrametric diffusion constrained by the linear energy landscape and the ultrametric diffusion with a reaction sink are considered we show that such models can be applied to both the relaxation in complex systems and the rate processes coupled to rearrangement of the complex surrounding

replica symmetry breaking related to a general ultrametric space i replica matrices and functionals abstract a family of replica matrices related to general ultrametric spaces is introduced these matrices generalize the known parisi matrices some functionals of the replica approach are computed

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

quantum groups acting on 4 points we classify the compact quantum groups acting on 4 points these are the quantum subgroups of the quantum permutation group q4 our main tool is a new presentation for the algebra c q4 corresponding to an isomorphism of type q4 so 1 3 the quantum subgroups of q4 are subject to a mckay type correspondence that we describe at the level of algebraic invariants

replica symmetry breaking related to a general ultrametric space ii rsb solutions and the n 0 limit we study the replica symmetry breaking related to a general ultrametric space ii in particular the rsb solutions and the n to0 limit

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

complex quantum group dual algebra and bicovariant differential calculus the method used to construct the bicovariant bimodule in ref csww is applied to examine the structure of the dual algebra and the bicovariant differential calculus of the complex quantum group the complex quantum group fun q sl n c is defined by requiring that it contains fun q su n as a subalgebra analogously to the quantum lorentz group analyzing the properties of the fundamental bimodule we show that the dual algebra has the structure of the twisted product fun q su n fun q su n reg then the bicovariant differential calculi on the complex quantum group are constructed

bicovariant differential calculus on quantum groups su q n and so q n following woronowicz s proposal the bicovariant differential calculus on the quantum groupssu q n andso q n is constructed a systematic construction of bicovariant bimodules by using the hat r q matrix is presented the relation between the hopf algebras generated by the linear functionals relating the left and right multiplication of these bicovariant bimodules and theq deformed universal enveloping algebras is given imposing the conditions of bicovariance and consistency with the quantum group structure the differential algebras and exterior derivatives are defined as an application the maurer cartan equations and theq analogue of the structure constants are formulated

the yang baxter equation and invariants of links on considere le fait que l on peut construire des invariants d isotopie p et f pour des liens en utilisant certaines solutions des equations de yang baxter

coherent states for quantum compact groups coherent states are introduced and their properties are discussed for simple quantum compact groupsal bl cl anddl the multiplicative form of the canonical element for the quantum double is used to introduce the holomorphic coordinates on a general quantum dressing orbit the coherent state is interpreted as a holomorphic function on this orbit with values in the carrier hilbert space of an irreducible representation of the corresponding quantized enveloping algebra using gauss decomposition the commutation relations for the holomorphic coordinates on the dressing orbit are derived explicitly and given in a compactr matrix formulation generalizing this way theq deformed grassmann and flag manifolds the antiholomorphic realization of the irreducible representations of a compact quantum group the analogue of the borel weil construction is described using the concept of coherent state the relation between representation theory and non commutative differential geometry is suggested

harmonic analysis on gf pp part ii abstract harmonic analysis on gf p p is studied the heisenberg weyl hw g gf p p g group of displacements is shown to be a locally compact and totally disconnected topological group the formalism introduces algebraic concepts from the theory of galois fields into harmonic analysis for example a galois group of frobenius transformations on functions analogous to the galois group of frobenius transformations in galois theory is introduced into harmonic analysis a larger group which we call heisenberg weyl galois group is also discussed

a classification of unitary highest weight modules of the quantum analogue of the symmetric pair an an 1 abstract we study the irreducibility and unitarity of highest weight modules of the quantized enveloping algebra associated to s u n 1 we obtain a classification of these modules analogous to that of ehw in the classical context we also compute all commutation relations between root vectors in u q s u n 1

numerical quadrature for high dimensional singular integrals over parallelotopes we introduce and analyze a family of algorithms for an efficient numerical approximation of integrals of the form email protected c 1 c 2 f x y y x dydx where c 1 c 2 are d dimensional parallelotopes i e affine images of d hypercubes and f has a singularity at y x 0 such integrals appear in galerkin discretization of integral operators in r d we construct a family of quadrature rules q n with n function evaluations for a class of integrands f which may have algebraic singularities at y x 0 and are email protected regular for y x 0 the main tool is an explicit regularizing coordinate transformation simultaneously simplifying the singular support and the domain of integration for the full tensor product variant of the suggested quadrature family we prove that q n achieves the exponential convergence rate o exp rn c with the exponent c 1 email protected 1 in the special case of a singularity of the form email protected a with real a we prove that the improved convergence rate of c 1 email protected is achieved if a certain modified one dimensional gauss jacobi quadrature rule is used in the singular direction we give numerical results for various types of the quadrature rules in particular based on tensor product rules standard smolyak optimized and adaptive sparse grid quadratures and sobol sequences

classifying spaces for quantum principal bundles a construction of a quantum analogue of principal bundles is discussed deformations of quantum groups in the sense of woronowicz allow to relax the condition of local triviality of a principal bundle the fibres need not be all identical any longer this leads to deformations of structure group and bundles there is still a classifying space in the sense that homotopy classes of bundles are classified by homotopy classes of maps from the base space into the classifying space

harmonic analysis on rational numbers abstract harmonic analysis on q is studied the fourier transforms of complex functions f y y q in the schwartz bruhat space s are functions f u where u a q q it is shown that the corresponding heisenberg weyl group is a locally compact topological group various time frequency analysis methods properties of displacement operators wigner and ambiguity functions etc are discussed in this context

irregular uq sl 3 representations at roots of unity via gel fand weyl zetlin basis the gel fand weyl zetlin gwz description of the uq sl 3 irregular irreps at roots of unity is explicitly given those are irreps fixed by the same parameters as the unitary irreducible representations uirs of su 3 yet having dimensions smaller than their classical counterparts the reason being that to obtain an irregular irrep one has to make factorization of an additional submodule this description is made geometrically transparent by an arrangement of the standard su 3 gwz basis in a hexagonal pyramid which is valid for any q and seems new even for q 1 the pyramid has as a base the standard hexagon which gives the weight space of the uirs of su 3 in the plane of third component of isospin iz and hypercharge y while third dimension of this pyramid is related to the isospin i algebraically this arrangement is related to a one to one correspondence between the abstract gwz states and monomials in the algebra of raising generators uq g however those monomials are not in the standard poi

replica symmetry breaking related to a general ultrametric space iii the case of general measure we study replica symmetry breaking related to a general ultrametric space in the case of general measure

quantization of lie group and algebra of g2 type in the faddeev reshetikhin takhtajan approach based on the quantized universal enveloping que algebras a quantization of the automorphism group of some nonassociative algebras is given in the formulation employing noncommuting matrix entries a quantum group of g2 type included in this scheme is studied in detail in the faddeev reshetikhin takhtajan frt approach also in the formulation employing noncommuting matrix entries the que algebra of g2 type is reconstructed through the pairing induced by the r matrix between the quantum group and the que algebra