cellular networks as models for planck scale physics starting from the working hypothesis that both physics and the corresponding mathematics have to be described by means of discrete concepts on the planck scale one of the many problems one has to face in this enterprise is to find the discrete protoforms of the building blocks of our ordinary continuum physics and mathematics we base our own approach on what we call cellular networks consisting of cells nodes interacting with each other via bonds figuring as elementary interactions according to a certain local law geometrically our dynamical networks are living on graphs hence a substantial amount of the investigation is devoted to the development of various versions of discrete functional analysis and geometry on such almost random webs another important topic we address is a suitable concept of intrinsic fractal dimension on erratic structures of this kind in the course of the investigation we make comments concerning both different and related approaches to quantum gravity as say the spin network framework it may perhaps be said that certain parts of our programme seem to be a realization of ideas sketched by smolin some time ago

exchange dynamical quantum groups for any simple lie algebra and any complex number q which is not zero or a nontrivial root of unity but may be equal to 1 we construct a dynamical quantum group hopf algebroid whose representation theory is essentially the same as the representation theory of the quantum group uq this dynamical quantum group is obtained from the fusion and exchange relations between intertwining operators in representation theory of uq and is an algebraic structure standing behind these relations

a geometric renormalization group in discrete quantum space time we model quantum space time on the planck scale as dynamical networks of elementary relations or time dependent random graphs the time dependence being an effect of the underlying dynamical network laws we formulate a kind of geometric renormalization group on these random networks leading to a hierarchy of increasingly coarse grained networks of overlapping lumps we provide arguments that this process may generate a fixed limit phase representing our continuous space time on a mesoscopic or macroscopic scale provided that the underlying discrete geometry is critical in a specific sense geometric long range order our point of view is corroborated by a series of analytic and numerical results which allow us to keep track of the geometric changes taking place on the various scales of the resolution of space time of particular conceptual importance are the notions of dimension of such random systems on the various scales and the notion of geometric criticality

double quantization on some orbits in the coadjoint representations of simple lie groups let be the function algebra on a semisimple orbit m in the coadjoint representation of a simple lie group g with the lie algebra we study one and two parameter quantizations h and t h of such that the multiplication on the quantized algebra is invariant under action of the drinfeld jimbo quantum group uh in particular the algebra t h specializes at h 0 to a u invariant g invariant quantization ascr t 0

dirac operators and the calculation of the connes metric on arbitrary infinite graphs as an outgrowth of our investigation of non regular spaces within the context of quantum gravity and non commutative geometry we develop a graph hilbert space framework on arbitrary infinite graphs and use it to study spectral properties of graph laplacians and graph dirac operators we define a spectral triplet sharing most of the properties of what connes calls a spectral triple with the help of this scheme we derive an explicit expression for the connes distance function on general directed or undirected graphs we derive a series of a priori estimates and calculate it for a variety of examples of graphs as a possibly interesting side we show that the natural setting for approaching such problems may be the framework of non linear programming or optimization we compare our results arrived at within our particular framework with those of other authors and show that the seeming differences depend on the use of different graph geometries and or dirac operators

triangular dynamical r matrices and quantization abstract we study some general aspects of triangular dynamical r matrices using poisson geometry we show that a triangular dynamical r matrix r h 2 g always gives rise to a regular poisson manifold using the fedosov method we prove that non degenerate triangular dynamical r matrices i e those such that the corresponding poisson manifolds are symplectic are quantizable and that the quantization is classified by the relative lie algebra cohomology h2 g h

on dynamical r matrices obtained from dirac reduction and their generalizations to affine lie algebras according to etingof and varchenko the classical dynamical yang baxter equation is a guarantee for the consistency of the poisson bracket on certain poisson lie groupoids here it is noticed that dirac reductions of these poisson manifolds give rise to a mapping from dynamical r matrices on a pair to those on another pair where is a chain of lie algebras for which admits a reductive decomposition as several known dynamical r matrices appear naturally in this setting and its application provides new r matrices too in particular we exhibit a family of r matrices for which the dynamical variable lies in the grade zero subalgebra of an extended affine lie algebra obtained from a twisted loop algebra based on an arbitrary finite dimensional self dual lie algebra

a kohno drinfeld theorem for the monodromy of cyclotomic kz connections we compute explicitly the monodromy representations of cyclotomic analogs of the knizhnik zamolodchikov differential system these are representations of the type b braid group b n 1 we show how the representations of the braid group b n obtained using quantum groups and universal r matrices may be enhanced to representations of b n 1 using dynamical twists then we show how these algebraic representations may be identified with the above analytic monodromy representations

constructive mathematics a foundation for computable analysis this paper introduces bishop s constructive mathematics which can be regarded as the constructive core of mathematics and whose theorems can be translated into many formal systems of computable analysis the real numbers are presented using a set of constructive axioms from which are derived some elementary properties of the real line r including its completeness

dynamical yang baxter equation and quantum vector bundles we develop a categorical approach to the dynamical yang baxter equation dybe for arbitrary hopf algebras in particular we introduce the notion of a dynamical extension of a monoidal category which provides a natural environment for quantum dynamical r matrices dynamical twists etc in this context we define dynamical associative algebras and show that such algebras give quantizations of vector bundles on coadjoint orbits we build a dynamical twist for any pair of a reductive lie algebra and its levi subalgebra using this twist we obtain an equivariant star product quantization of vector bundles on semisimple coadjoint orbits of reductive lie groups

quantization of formal classical dynamical r matrices the reductive case abstract in this paper we prove the existence of a formal dynamical twist quantization for any triangular and non modified formal classical dynamical r matrix in the reductive case the dynamical twist is constructed as the image of the dynamical r matrix by a l quasi isomorphism this quasi isomorphism also allows us to classify formal dynamical twist quantizations up to gauge equivalence

dimension theory of graphs and networks starting from the working hypothesis that both physics and the corresponding mathematics have to be described by means of discrete concepts on the planck scale one of the many problems one has to face in this enterprise is to find the discrete protoforms of the building blocks of continuum physics and mathematics a core concept is the notion of dimension in the following we develop such a notion for irregular structures such as large graphs and networks and derive a number of its properties among other things we show its stability under a wide class of perturbations which is important if one has dimensional phase transitions in mind furthermore we systematically construct graphs with almost arbitrary fractal dimension which may be of some use in the context of dimensional renormalization or statistical mechanics on irregular sets

quantization of symplectic dynamical r matrices and the quantum composition formula in this paper we quantize symplectic dynamical r matrices over a possibly nonabelian base the proof is based on the fact that the existence of a star product with a nice property called strong invariance is sufficient for the existence of a quantization we also classify such quantizations and prove a quantum analogue of the classical composition formula for coboundary dynamical r matrices

t g t a toy model for conformal field theory we study a chiral operator algebra of conformal field theory and quantum deformation of the finite dimensional lie group to obtain the definition of t g t and its representation the closeness of the kac moody algebras constituting the chiral operator algebra of a typical and generic conformal field theory model namely the wznw model and quantum deformation of corresponding finite dimensional lie groupg has become more and more evident in recent years 1 5 this in particular prompts further investigation of the differential geometry of such deformations the notion of tangent and cotangent bundles is basic in classical differential geometry it is only natural that the quantum deformations oftg andt g are to be introduced alongside those forg itself physical ideas could be useful for this goal indeed thet g can be interpreted as a phase space for a kind of a top generalizing the usual top associated withg so 3 the classical mechanics is a natural language to describe differential geometry whereas the usual quantization is nothing but the representation theory in this paper we put corresponding formulas in such a fashion that their deformation becomes almost evident given the experience in this domain as a result we get the definition of t g t and its representation t is the deformation parameter to make the exposition most simple and formulas transparent we shall work on an example ofg sl 2 and present results in such a way that the generalizations become evident we shall stick to generic complex versions real and especially compact forms requiring some additional consideration not all of which are self evident

quantum groupoids and dynamical categories abstract in this paper we realize the dynamical categories introduced in our previous paper as categories of modules over bialgebroids we study the bialgebroids arising in this way we define quasitriangular structure on bialgebroids and present examples of quasitriangular bialgebroids related to the dynamical categories we show that dynamical twists over an arbitrary base give rise to bialgebroid twists we prove that the classical dynamical r matrices over an arbitrary base manifold are in one to one correspondence with a special class of coboundary lie bialgebroids

explicit quantization of dynamical r matrices for finite dimensional semisimple lie algebras 1 1 classical r matrices in the early eighties belavin and drinfeld bd classified nonskewsymmetric classical r matrices for simple lie algebras it turned out that such r matrices up to isomorphism and twisting by elements from the exterior square of the cartan subalgebra are classified by combinatorial objects which are now called belavin drinfeld triples by definition a belavin drinfeld triple for a simple lie algebra g is a triple 1 2 t where 1 2 are subsets of the dynkin diagram of g and t 1 2 is an isomorphism which preserves the inner product and satisfies the nilpotency condition if 1 then there exists k such that t k 1 1 but t 1 the r matrix corresponding to such a triple is given by a certain explicit formula these results generalize in a straightforward way to semisimple lie algebras in s the third author generalized the work of belavin and drinfeld and classified classical nonskewsymmetric dynamical r matrices for simple lie algebras it turns out that they have an even simpler classification up to gauge transformations they are classified by generalized belavin drinfeld triples which are defined as the usual belavin drinfeld triples but without any nilpotency condition the dynamical rmatrix corresponding to such a triple is given by a certain explicit formula as before these results can be generalized to semisimple lie algebras

insight in discrete geometry and computational content of a discrete model of the continuum this article presents a synthetic and self contained presentation of the discrete model of the continuum introduced by harthong and reeb j harthong elements pour une theorie du continu asterisque 109 110 1983 235 244 1 j harthong une theorie du continu in h barreau j harthong eds la mathematiques non standard editions du cnrs 1989 pp 307 329 2 and the related arithmetization process which led reveilles j p reveilles geometrie discrete calcul en nombres entiers et algorithmique ph d thesis universite louis pasteur strasbourg france 1991 3 j p reveilles d richard back and forth between continuous and discrete for the working computer scientist annals of mathematics and artificial intelligence mathematics and informatic 16 1 4 1996 89 152 4 to the definition of a discrete analytic line we present then some basis on constructive mathematics e bishop d bridges constructive analysis springer berlin 1985 5 its link with programming p martin lof constructive mathematics and computer programming in logic methodology and philosophy of science vol vi 1980 pp 153 175 6 w a howard the formulae as types notion of construction to h b curry essays on combinatory logic lambda calculus and formalism 1980 pp 479 490 7 and we propose an analysis of the computational content of the so called harthong reeb line more precisely we show that a suitable version of this new model of the continuum partly fits with the constructive axiomatic of r proposed by bridges constructive mathematics a foundation for computable analysis theoretical computer science 219 1 2 1999 95 109 8 this is the first step of a more general program on a constructive approach of the scaling transformation from discrete to continuous space

quantum conjugacy classes of simple matrix groups let g be a simple complex classical group and mathfrak g its lie algebra let mathcal u hbar mathfrak g be the drinfeld jimbo quantization of the universal enveloping algebra mathcal u mathfrak g we construct an explicit mathcal u hbar mathfrak g equivariant quantization of conjugacy classes of g with levi subgroups as the stabilizers

on the exchange matrix for wznw model the connection between the exchange algebra in thesu 2 wess zumino novikov witten model and the quantum groupsu 2 is discussed it is shown that on the quasiclassical level this connection has the simple interpretation in terms of the lie poisson action ofsu 2 on the chiral components of the fields in the wznw model

the non commutative weil algebra for any compact lie group g together with an invariant inner product on its lie algebra g we define the non commutative well algebra w g as a tensor product of the universal enveloping algebra u g and the clifford algebra cl 8 just like the usual we

a certified corecursive implementation of exact real numbers we implement exact real numbers in the logical framework coq using streams i e infinite sequences of digits and characterize constructive real numbers through a minimal axiomatization we prove that our construction inhabits the axiomatization working formally with coinductive types and corecursive proofs thus we obtain reliable corecursive algorithms for computing on real numbers

a constructive theory of point set nearness an axiomatic constructive development of the theory of nearness and apartness of a point and a set is introduced as a setting for topology

unique existence approximate solutions and countable choice an existential statement seems to admit of a constructive proof without countable choice only if the object to be constructed is uniquely determined or is intended as an approximate solution of the problem in question this conjecture is substantiated by re examining some basic tools of mathematical analysis from a choice free constructive point of view starting from dedekind cuts as an appropriate notion of real numbers by establishing a fairly general version of the approximate intermediate value theorem we also indicate that strong continuity is a practicable substitute for uniform continuity where a mapping between metric spaces is called strongly continuous if two subsets of its domain are bounded away from each other whenever so are their images

geometry and classificatin of solutions of the classical dynamical yang baxter equation the classical yang baxter equation cybe is an algebraic equation central in the theory of integrable systems its nondegenerate solutions were classified by belavin and drinfeld quantization of cybe led to the theory of quantum groups a geometric interpretation of cybe was given by drinfeld and gave rise to the theory of poisson lie groups

dynamical weyl groups and applications abstract following a preceding paper of tarasov and the second author we define and study a new structure which may be regarded as the dynamical analog of the weyl group for lie algebras and of the quantum weyl group for quantized enveloping algebras we give some applications of this new structure

quantization of classical dynamical r matrices with nonabelian base we construct some classes of dynamical r matrices over a nonabelian base and quantize some of them by constructing dynamical pseudo twists in the sense of xu this way we obtain quantizations of r matrices obtained in earlier work of the second author with schiffmann and varchenko a part of our construction may be viewed as a generalization of the donin mudrov nonabelian fusion construction we apply these results to the construction of equivariant star products on poisson homogeneous spaces which include some homogeneous spaces introduced by de concini