the classification of differential structures on quantum 2 spheres exterior algebras of differential forms on quantum 2 spheress qc 2 q 1 1 0 c 0 c 0 forq 1 are classified in the definition of exterior algebras we assume the invariance w r t the action of the quantumsu 2 group and dimensionality conditions which imply that we deal with two dimensional manifolds the exterior algebras exist only forc 0 and are unique in that case the corresponding generalized directional derivatives are provided

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

quantum 2 spheres and big q jacobi polynomials orthogonal bases for the algebras of functions of podles quantum 2 spheres are explicitly determined in terms of bigq jacobi polynomials this gives a group theoretic interpretation of the symmetric bigq jacobi polynomials and the symmetricq hahn polynomials

existence and examples of quantum isometry groups for a class of compact metric spaces we formulate a definition of isometric action of a compact quantum group cqg on a compact metric space generalizing banica s definition for finite metric spaces for metric spaces x d which can be isometrically embedded in some euclidean space we prove the existence of a universal object in the category of the compact quantum groups acting isometrically on x d in fact our existence theorem applies to a larger class namely for any compact metric space x d which admits a one to one continuous map f x rn for some n such that d0 f x f y d x y where d0 is the euclidean metric for some homeomorphism of r as concrete examples we obtain wang s quantum permutation group sn and also the free wreath product of z2 by sn as the quantum isometry groups for certain compact connected metric spaces constructed by taking topological joins of intervals in 13

the spatial rokhlin property for actions of compact quantum groups abstract we introduce the spatial rokhlin property for actions of coexact compact quantum groups on c algebras generalizing the rokhlin property for both actions of classical compact groups and finite quantum groups two key ingredients in our approach are the concept of sequentially split homomorphisms and the use of braided tensor products instead of ordinary tensor products we show that various structure results carry over from the classical theory to this more general setting in particular we show that a number of c algebraic properties relevant to the classification program pass from the underlying c algebra of a rokhlin action to both the crossed product and the fixed point algebra towards establishing a classification theory we show that rokhlin actions exhibit a rigidity property with respect to approximate unitary equivalence regarding duality theory we introduce the notion of spatial approximate representability for actions of discrete quantum groups the spatial rokhlin property for actions of a coexact compact quantum group is shown to be dual to spatial approximate representability for actions of its dual discrete quantum group and vice versa

induced representations of the multiparameter hopf superalgebras uuq gl m n and uuq sl m n we construct the induced holomorphic representations of the multiparameter hopf superalgebras uuq gl m n and uuq sl m n the first superalgebra we constructed earlier as the dual of the multiparameter quantum deformation of the supergroup gl m n the second superalgebra is a hopf subalgebra of the first for a special choice of the parameters the representations are labelled by m n integer numbers respectively m n 1 complex numbers and act in the space of formal power series of m n m n 1 2 non commuting variables of which mn are odd and the rest are even these variables generate a q deformation of a flag supermanifold of the supergroup gl m n respectively sl m n

quantum isometry groups examples and computations in this follow up of 4 where the quantum isometry group of a noncommutative manifold has been defined we explicitly compute such quantum groups for a number of classical as well as noncommutative manifolds including the spheres and the tori it is also proved that the quantum isometry group of an isospectral deformation of a classical or noncommutative manifold is a suitable deformation of the quantum isometry group of the original undeformed manifold

compact quantum ergodic systems we develop theory of multiplicity maps for compact quantum groups as an application we obtain a complete classification of right coideal c algebras of c suq 2 for q 1 1 0 they are labeled with dynkin diagrams but classification results for positive and negative cases of q are different many of the coideals are quantum spheres or quotient spaces by quantum subgroups but we do have other ones in our classification list

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 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 andrews sellers family of partition congruences in 1994 james sellers conjectured an infinite family of ramanujan type congruences for 2 colored frobenius partitions introduced by george e andrews these congruences arise modulo powers of 5 in 2002 dennis eichhorn and sellers were able to settle the conjecture for powers up to 4 in this article we prove sellers conjecture for all powers of 5 in addition we discuss why the andrews sellers family is significantly different from classical congruences modulo powers of primes

quantum homogeneous spaces as quantum quotient spaces it is shown that certain embeddable homogeneous spaces of a quantum group that do not correspond to a quantum subgroup still have the structure of quantum quotient spaces a construction of quantum fibre bundles on such spaces is proposed the quantum plane and the general quantum two spheres are discussed in detail

induced representations of uqso 5 we study the reduction of the left regular representation of the quantum algebra uqso 5 this yields induced representations of uqso 5 on some quotient space of the matrix quantum group soq 5 to facilitate the construction of a suitable basis the gauss decomposition of soq 5 is worked out explicitly we then investigate the behavior of these representations for a generic q as well as when q is an nth root of unity leading to some finite dimensional irreducible representations of uqso 5 two basic intertwining operators are also constructed via the right action

a generalized fluctuation dissipation theorem for the one dimensional diffusion process the langevin equation describes the time evolution of a real stationary process witht positivity reflection positivity originating in the axiomatic quantum field theory for this langevin equation a generalized fluctuation dissipation theorem is proved we shall obtain as its application a generalized fluctuation dissipation theorem for the one dimensional non linear diffusion process which presents one solution of ryogo kubo s problem in physics

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

spectra for compact quantum group coactions and crossed products we present definitions of both connes spectrum and strong connes spectrum for actions of compact quantum groups on c algebras and obtain necessary and sufficient conditions for a crossed product to be a prime or a simple c algebra our results extend to the case of compact quantum ac tions the results in 8 which in turn generalize results by connes olesen and pedersen and kishimoto for abelian group actions we prove in addition that the connes spectra are closed under tensor products these results are new for compact nonabelian groups as well

a characterization of right coideals of quotient type and its application to classification of poisson boundaries let mathbb g be a co amenable compact quantum group we show that a right coideal of mathbb g is of quotient type if and only if it is the range of a conditional expectation preserving the haar state and is globally invariant under the left action of the dual discrete quantum group we apply this result to the theory of poisson boundaries introduced by izumi for discrete quantum groups and generalize a work of izumi neshveyev tuset on suq n for co amenable compact quantum groups with the commutative fusion rules more precisely we prove that the poisson integral is an isomorphism between the poisson boundary and the right coideal of quotient type by a maximal quantum subgroup of kac type in particular the poisson boundary and the quantum flag manifold are isomorphic for any q deformed classical compact lie group

on the theory of brownian motion with the alder wainwright effect the stokes boussinesq langevin equation which describes the time evolution of brownian motion with the alder wainwright effect can be treated in the framework of the theory of kmo langevin equations which describe the time evolution of a real stationary gaussian process witht positivity reflection positivity originating in axiomatic quantum field theory after proving the fluctuation dissipation theorems for kmo langevin equations we obtain an explicit formula for the deviation from the classical einstein relation that occurs in the stokes boussinesq langevin equation with a white noise as its random force we are interested in whether or not it can be measured experimentally

quantum automorphism group of the lexicographic product of finite regular graphs we study the quantum automorphism group of the lexicographic product of two finite regular graphs providing a quantum generalization of sabidussi s structure theorem on the automorphism group of such a graph

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

left regular representation of slq 3 reduction and intertwiners reduction of the left regular representation of the quantum algebra slq 3 is studied and q difference intertwining operators are constructed the irreducible representations correspond to a q deformation of the spaces of local sections of certain line bundles over the flag manifold

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

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

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

quantum differential operators on the quantum plane abstract the universal enveloping algebra u g of a lie algebra g acts on its representation ring r through d r the ring of differential operators on r a quantised universal enveloping algebra or quantum group is a deformation of a universal enveloping algebra and acts not through the differential operators of its representation ring but through the quantised differential operators of its representation ring we present this situation for the quantum group of sl2

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

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

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

exponential representation of jordanian matrix quantum group glh 2 an exponential representation of the jordanian matrix quantum group glh 2 is constructed in explicit form by the generators of the classical lie algebra sl 2 c

commutation relations among quantum minors in o q m n k in this paper we compute explicit formulas for the commutation relations between any two quantum minors in the quantum matrix bialgebra oq mn k the product of any two minors is expressed as linear combination of products of minors strictly less in certain orderings

multiparametric quantum deformation of the general linear supergroup in the work l d faddeev and his collaborators and subsequently v g drinfeld m jimbo s l woronowicz a new class of hopf algebras was constructed they can be considered as one parametric deformations of either group ring or the universal enveloping algebra of a simple algebraic group in this paper we define and investigate a multiparametric deformation of the general linear supergroup this is the simplest example of some general constructions described in 5 6

quantumsu 2 ande 2 groups contraction procedure in 3 it was shown in the framework of deformed enveloping algebras that quantumsu 2 ande 2 groups are related by the contraction procedure we consider the same problem on thec level as a result we find a number of formulae coupling the comultiplications in quantumsu 2 ande 2 in particular we show that the comultiplications in both groups are implemented by partial isometries an unexpected feature of quantume 2 is discovered and the corresponding strange behavior of quantumsu 2 is described

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

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

non existence of genuine compact quantum symmetries of compact connected smooth manifolds abstract suppose that a compact quantum group q acts faithfully on a smooth compact connected manifold m i e has a c co action on c m such that c m c m q and the linear span of c m 1 q is dense in c m q with respect to the frechet topology it was conjectured by the author quite a few years ago that q must be commutative as a c algebra i e q c g for some compact group g acting smoothly on m the goal of this paper is to prove the truth of this conjecture a remarkable aspect of the proof is the use of probabilistic techniques involving brownian stopping time

equivalent notions of normal quantum subgroups compact quantum groups with properties f and fd and other applications abstract the notion of normal quantum subgroup introduced in algebraic context by parshall and wang when applied to compact quantum groups is shown to be equivalent to the notion of normal quantum subgroup introduced by the author as applications a quantum analog of the third fundamental isomorphism theorem for groups is obtained which is used along with the equivalence theorem to obtain results on structure of quantum groups with property f and quantum groups with property fd other results on normal quantum subgroups for tensor products free products and crossed products are also proved

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

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

quantum isometries of the finite noncommutative geometry of the standard model we compute the quantum isometry group of the finite noncommutative geometry f describing the internal degrees of freedom in the standard model of particle physics we show that this provides genuine quantum symmetries of the spectral triple corresponding to m f where m is a compact spin manifold we also prove that the bosonic and fermionic part of the spectral action are preserved by these symmetries

the exceptional lie algebra e7 25 multiplets and invariant differential operators in the present paper we continue the project of systematic construction of invariant differential operators on the example of the non compact exceptional algebra e7 25 our choice of this particular algebra is motivated by the fact that it belongs to a narrow class of algebras which we call conformal lie algebras which have very similar properties to the conformal algebras of n dimensional minkowski spacetime this class of algebras is identified and summarized in a table another motivation is related to the ads cft correspondence we give the multiplets of indecomposable elementary representations including the necessary data for all relevant invariant differential operators

on fock space representations of quantized enveloping algebras related to noncommutative differential geometry in this paper explicitly natural from the geometrical point of view fock space representations contragradient verma modules of the quantized enveloping algebras are constructed in order to do so one starts from the gauss decomposition of the quantum group and introduces the differential operators on the corresponding q deformed flag manifold assumed as a left comodule for the quantum group by a projection to it of the right action of the quantized enveloping algebra on the quantum group finally the representatives of the elements of the quantized enveloping algebra corresponding to the left invariant vector fields on the quantum group are expressed as first order differential operators on the q deformed flag manifold

subsingular vectors and conditionally invariant q deformed equations we give a systematic discussion of the relation between subsingular vectors of verma modules over semisimple lie algebras g and differential equations which are conditionally g invariant this is extended to the drinfeld jimbo q deformation uq g of g we treat in detail the conformal algebra su 2 2 its complexification sl 4 and their q deformations the conditionally invariant equations are the d alembert equation and a new equation arising from a subsingular vector proposed by bernstein gel fand gel fand 1971 we also give the q difference analogues of these equations

the martin boundary of a discrete quantum group we consider the markov operator p on a discrete quantum group given by convolution with a q tracial state in the study of harmonic elements x p x x we define the martin boundary a it is a separable c algebra carrying canonical actions of the quantum group and its dual we establish a representation theorem to the effect that positive harmonic elements correspond to positive linear functionals on a the c algebraa has a natural time evolution and the unit can always be represented by a kms state any such state gives rise to a u c p map from the von neumann closure of a in its gns representation to the von neumann algebra of bounded harmonic elements which is an analogue of the poisson integral under additional assumptions this map is an isomorphism which respects the actions of the quantum group and its dual next we apply these results to identify the martin boundary of the dual of suq 2 with the quantum homogeneous sphere of podle this result extends and unifies previous results by ph biane and m izumi

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

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

critical scale invariance in a healthy human heart rate we demonstrate the robust scale invariance in the probability density function pdf of detrended healthy human heart rate increments which is preserved not only in a quiescent condition but also in a dynamic state where the mean level of the heart rate is dramatically changing this scale independent and fractal structure is markedly different from the scale dependent pdf evolution observed in a turbulentlike cascade heart rate model these results strongly support the view that a healthy human heart rate is controlled to converge continually to a critical state

traveling salesman problem and tsallis statistics a generalization of the stochastic method of simulated annealing algorithm based on tsallis statistics is proposed this algorithm is considerably faster than the traditional ones in solving the traveling salesman problem acceptable solutions are found in fewer steps and higher temperatures than both the classical and the fast simulated annealings recent developments in solving np complete problems can be incorporated and improve the performance even more

quantum symmetry groups of hilbert modules equipped with orthogonal filtrations abstract we define and show the existence of the quantum symmetry group of a hilbert module equipped with an orthogonal filtration our construction unifies the constructions of banica skalski s quantum symmetry group of a c algebra equipped with an orthogonal filtration and goswami s quantum isometry group of an admissible spectral triple

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

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

podleś spheres for the braided quantum su 2 abstract starting with the braided quantum group su q 2 for a complex deformation parameter q we perform the construction of the quotient su q 2 t which serves as a model of a quantum sphere then we follow the reasoning of podle who for real q classified quantum spaces with the action of su q 2 with appropriate spectral properties these properties can also be expressed in the context of the braided quantum su q 2 with complex q and we find that they lead to precisely the same family of quantum spaces as found by podle for the real parameter q

quantum heisenberg group and algebra contraction left and right regular representations it is shown that the quantum heisenberg group hq 1 can be obtained by means of contraction from the quantum suq 2 group its dual hopf algebra is the quantum heisenberg algebra uq h 1 left and right regular representations for uq h 1 as acting on its dual hq 1 are derived by imposing conditions on the right representation the left representation is reduced to an irreducible holomorphic representation with an associated quantum coherent state by duality left and right regular representations for quantum heisenberg group with the quantum heisenberg algebra as a representation module are also constructed as before the reduction of left representations leads to finite dimensional irreducible ones for which the intertwining operator is also investigated

compact quantum metric spaces and ergodic actions of compact quantum groups we show that for any co amenable compact quantum group a c g there exists a unique compact hausdorff topology on the set ea g of isomorphism classes of ergodic actions of g such that the following holds for any continuous field of ergodic actions of g over a locally compact hausdorff space t the map t ea g sending each t in t to the isomorphism class of the fibre at t is continuous if and only if the function counting the multiplicity of in each fibre is continuous over t for every equivalence class of irreducible unitary representations of g generalizations for arbitrary compact quantum groups are also obtained in the case g is a compact group the restriction of this topology on the subset of isomorphism classes of ergodic actions of full multiplicity coincides with the topology coming from the work of landstad and wassermann podle spheres are shown to be continuous in the natural parameter as ergodic actions of the quantum su 2 group we also introduce a notion of regularity for quantum metrics on g and show how to construct a quantum metric from any ergodic action of g starting from a regular quantum metric on g furthermore we introduce a quantum gromov hausdorff distance between ergodic actions of g when g is separable and show that it induces the above topology

quantum spheres and projective spaces as graph algebras the c algebras of continuous functions on quantum spheres quantum real projective spaces and quantum complex projective spaces are realized as cuntz krieger algebras corresponding to suitable directed graphs structural results about these quantum spaces especially about their ideals and k theory are then derived from the general theory of graph algebras it is shown that the quantum even and odd dimensional spheres are produced by repeated application of a quantum double suspension to two points and the circle respectively

the baum connes conjecture for free orthogonal quantum groups we prove an analogue of the baum connes conjecture for free orthogonal quantum groups more precisely we show that these quantum groups have a element and that 1 it follows that free orthogonal quantum groups are k amenable we compute explicitly their k theory and deduce in the unimodular case that the corresponding reduced c algebras do not contain nontrivial idempotents our approach is based on the reformulation of the baum connes conjecture by meyer and nest using the language of triangulated categories an important ingredient is the theory of monoidal equivalence of compact quantum groups developed by bichon de rijdt and vaes this allows us to study the problem in terms of the quantum group suq 2 the crucial part of the argument is a detailed analysis of the equivariant kasparov theory of the standard podle

an invariant for homogeneous spaces of compact quantum groups abstract the central notion in connes formulation of noncommutative geometry is that of a spectral triple given a homogeneous space of a compact quantum group restricting our attention to all spectral triples that are well behaved with respect to the group action we construct a certain dimensional invariant in particular taking the quantum group itself as the homogeneous space this gives an invariant for a compact quantum group computations of this invariant in several cases including all type a quantum groups are given

low dimensional chaos in a driven damped nonlinear schro dinger equation abstract we study bifurcations of attractors in a driven damped nonlinear schrodinger equation which models nonlinear responses of a plasma driven by a rf field and a sequence of period doubling bifurcations is shown to lead to a chaotic attractor in a low dimensional subspace spanned by a soliton and long wavelength radiation

a duality theorem for ergodic actions of compact quantum groups on c algebras the spectral functor of an ergodic action of a compact quantum group g on a unital c algebra is quasitensor in the sense that the tensor product of two spectral subspaces is isometrically contained in the spectral subspace of the tensor product representation and the inclusion maps satisfy natural properties we show that any quasitensor functor from rep g to the category of hilbert spaces is the spectral functor of an ergodic action of g on a unital c algebra

quantum flag manifolds quantum symmetric spaces and their associated universal k matrices abstract let u be a connected simply connected compact lie group with complexification g let u and g be the associated lie algebras let be the dynkin diagram of g with underlying set i and let u q u be the associated quantized universal enveloping algebra of u for some 0 q distinct from 1 let o q u be the coquasitriangular quantized function hopf algebra of u whose drinfeld double o q g r we view as the quantized function algebra of g considered as a real algebraic group we show how the datum of an involution of and a invariant function i r can be used to deform o q g r into a algebra o q id g r by a modification of the drinfeld double construction we then show how by a generalized theory of universal k matrices a specific subalgebra of o q id g r admits homomorphisms into both u q u and o q u the images being coideal subalgebras of respectively u q u and o q u we illustrate the theory by showing that two main classes of examples arise by such coideals namely quantum flag manifolds and quantum symmetric spaces except possibly for certain exceptional cases in the former case this connects to work of the first author and neshveyev while for the latter case we heavily rely on recent results of balagovic and kolb

q difference intertwining operators for a lorentz quantum algebra representations r r of a lorentz quantum algebra u are constructed they are labeled by two complex numbers r r and act in the space of formal power series of two noncommuting variables these variables are built from elements of the matrix lorentz quantum group l which is dual to u the conditions for reducibility of r r are given q difference intertwining operators in which realize partial equivalences of the representations r r are constructed explicitly the whole construction is a generalization of a known procedure for q 1 the case when q is a root of unity is also considered in detail

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

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

existence and rigidity of quantum isometry groups for compact metric spaces we prove the existence of a quantum isometry groups for new classes of metric spaces i geodesic metrics for compact connected riemannian manifolds possibly with boundary and ii metric spaces admitting a uniformly distributed probability measure in the former case it also follows from recent results of the second author that the quantum isometry group is classical i e the commutative c algebra of continuous functions on the riemannian isometry group

q difference intertwining operators for uq sl n general setting and the case n 3 we construct representations pi r of the quantum algebra uq sl n labelled by n 1 complex numbers ri and acting in the space of formal power series of n n 1 2 non commuting variables these variables generate a flag manifold of the matrix quantum group slq n which is dual to uq sl n the conditions for reducibility of pi r and the procedure for the construction of the q difference intertwining operators are given the representations and q difference intertwining operators are given in the most explicit form for n 3

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

induced representations of with subrepresentations of integer spin only we construct induced representations of on suitable cosets of the matrix quantum group from these we obtain canonically finite dimensional representations of only of odd dimension i e of integer spin the matrix elements of these finite dimensional representations are different from the standard ones which will be essential at least for the roots of unity case

on off intermittency power spectrum and fractal properties of time series abstract some dynamical systems possess invariant submanifolds such that the dynamics restricted to the invariant submanifold is chaotic this situation arises in systems with a spatial symmetry or in the synchronization of chaotic oscillators the invariant submanifold could become unstable to perturbations in the transverse directions when a parameter of the system is changed through a critical blow out value this could result in an extreme form of temporally intermittent bursting called on off intermittency we propose a model that incorporates the universal features of systems that display on off intermittency we study this model both with and without additive noise and we derive scaling results for the power spectral density of the on off intermittent process and for the box counting dimension for the set of time intervals when the process takes on values above a given threshold we then present numerical simulations realizing these results

maximum likelihood estimators for generalized cauchy processes maximum likelihood estimator mle for a generalized cauchy process gcp is studied with the aid of the method of information geometry in statistics our gcp is described by the langevin equation with multiplicative and additive noises the exact expressions of mles are given for the two cases that the two types of noises are uncorrelated and mutually correlated it is shown that the mles for these two gcps are free from divergence even in the parameter region wherein the ordinary moments diverge the mle relations can be regarded as a generalized fluctuation dissipation theorem for the present langevin equation availability of them and of some other higher order statistics is demonstrated theoretically and numerically

congruence properties for a certain kind of partition functions in light of the modular equations of fifth and seventh order we derive some congruence properties for a certain kind of partition functions a n which satisfy n 0 a n qn q q k modm where k is a positive integer with 1 k 24 and m 2 3 in view of these properties we obtain many infinite families of congruences for c k n the number of generalized frobenius partitions of n with k colors and c k n the number of generalized frobenius partitions of n with k colors whose order is k under cyclic permutation of the k colors meanwhile we also apply the main theorems to some other kinds of partition functions

quantum isometry groups of the podles spheres abstract for 0 1 c 0 we identify the quantum group so 3 as the universal object in the category of compact quantum groups acting by orientation and volume preserving isometries in the sense of bhowmick and goswami 2009 4 on the natural spectral triple on the podles sphere s c 2 constructed by dabrowski d andrea landi and wagner 2007 in 9