fusion categories arising from semisimple lie algebras using tilting modules we equip certain semisimple categories with a reduced tensor product structure the fusion rules for this tensor product are determined via known character formulas for the involved modules

invariants of 3 manifolds via link polynomials and quantum groups the aim of this paper is to construct new topological invariants of compact oriented 3 manifolds and of framed links in such manifolds our invariant of a link in a closed oriented 3 manifold is a sequence of complex numbers parametrized by complex roots of 1 for a framed link in s 3 the terms of the sequence are equale to the values of the suitably parametrized jones polynomial of the link in the corresponding roots of 1 in the case of manifolds with boundary our invariant is a sequence of finite dimensional complex linear operators this produces from each root of unity q a 3 dimensional topological quantum field theory

multi step methods for random odes driven by ito diffusions linear multi step methods are derived for random ordinary differential equations rodes driven by the solutions of ito stochastic differential equations sodes via strong ito taylor schemes for sodes due to the special structure of the rode sode pair it is not necessary to restrict the intensity of the noise pathwise convergence is established as well as the b stability of implicit multi step methods numerical comparisons are provided for explicit schemes applied to a low dimensional rode and implicit schemes applied to a high dimensional rode obtained with the method of lines by spatially discretizing a random partial differential equation with finite difference quotients

a spin decomposition of the verlinde formulas for type a modular categories a modular category is a braided category with some additional algebraic features the interest of this concept is that it provides a topological quantum field theory in dimension 3 the verlinde formulas associated with a modular category are the dimensions of the tqft modules we discuss reductions and refinements of these formulas for modular categories related with su n our main result is a splitting of the verlinde formula corresponding to a brick decomposition of the tqft modules whose summands are indexed by spin structures modulo an even integer we introduce here the notion of a spin modular category and give the proof of the decomposition theorem in this general context

topological gauge theories and group cohomology we show that three dimensional chern simons gauge theories with a compact gauge groupg not necessarily connected or simply connected can be classified by the integer cohomology grouph4 bg z in a similar way possible wess zumino interactions of such a groupg are classified byh3 g z the relation between three dimensional chern simons gauge theory and two dimensional sigma models involves a certain natural map fromh4 bg z toh3 g z we generalize this correspondence to topological spin theories which are defined on three manifolds with spin structure and are related to what might be calledz2 graded chiral algebras or chiral superalgebras in two dimensions finally we discuss in some detail the formulation of these topological gauge theories for the special case of a finite group establishing links with two dimensional holomorphic orbifold models

chern simons perturbation theory ii we study the perturbation theory for three dimensional chern simons quantum field theory on a general compact three manifold without boundary we show that after a simple change of variables the action obtained by brs gauge fixing in the lorentz gauge has a superspace formulation the basic properties of the propagator and the feynman rules are written in a precise manner in the language of differential forms using the explicit description of the propagator singularities we prove that the theory is finite finally the anomalous metric dependence of the 2 loop partition function on the riemannian metric which was introduced to define the gauge fixing can be cancelled by a local counterterm as in the 1 loop case in fact the counterterm is equal to the chern simons action of the metric connection normalized precisely as one would expect based on the framing dependence of witten s exact solution invited talk at the xxth conference on differential geometric methods in physics new york june 1991

spectra of wess zumino witten models with arbitrary simple groups we consider the wess zumino witten two dimensional sigma models with fields taking values in an arbitrary connected but not necessarily simply connected simple lie groupg the quantum states of the theory are realized geometrically as sections of a line bundle over the loop grouplg the action of the current algebra of the theory is decomposed into highest weight representations by explicit construction of the highest weight states this solves for the spectrum of the models as a by product we obtain modular invariant partition functions of the theory on tori the present paper extends the results of 7 where the casesg su 2 andso 3 were treated

jones witten invariants for nonsimply connected lie groups and the geometry of the weyl alcove abstract the quotient process of muger and bruguieres is used to construct modular categories and tqfts out of closed subsets of the weyl alcove of a simple lie algebra in particular it is determined at which levels closed subsets associated to nonsimply connected groups lead to tqfts many of these tqfts are shown to decompose into a tensor product of tqfts coming from smaller subsets the prime subsets among these are classified and apart from some giving tqfts depending on homology as described by murakami ohtsuki and okada they are shown to be in one to one correspondence with the tqfts predicted by dijkgraaf and witten to be associated to chern simons theory with a nonsimply connected lie group thus in particular a rigorous construction of the dijkgraaf witten tqfts is given as a byproduct a purely quantum groups proof of the modularity of the full weyl alcove for arbitrary quantum groups at arbitrary levels is given

on minimum average stretch spanning trees in polygonal 2 trees a spanning tree of an unweighted graph is a minimum average stretch spanning tree if it minimizes the ratio of sum of the distances in the tree between the end vertices of the graph edges and the number of graph edges for a polygonal 2 tree on n vertices we present an algorithm to compute a minimum average stretch spanning tree in o n log n time this algorithm also finds a minimum fundamental cycle basis in polygonal 2 trees we show that there is a unique minimum cycle basis in a polygonal 2 tree and it can be computed in linear time

simple wzw currents a complete classification of simple currents of wzw theories is obtained the proof is based on an analysis of the quantum dimensions of the primary fields simple currents are precisely the primaries with unit quantum dimension for wzw theories explicit formulae for the quantum dimensions can be derived so that an identification of the fields with unit quantum dimension is possible

categories premodulaires modularisations et invariants des varietes de dimension 3 an abelian k linear semisimple category having a finite number of simple objects and endowed with a ribbon structure is called premodular it is modular in the sense of turaev if the so called s matrix is invertible a modular category defines invariants of 3 manifolds and a tqft t when is it possible to construct a modularisation of a given premodular category i e a functor to a modular category preserving the structures and dominant in a certain sense it turns out 2 3 that this amounts essentially to making transparent objects trivial we give a full answer to this problem in the case when k is a field of char 0 as well as partial answers in char p under a few obvious hypotheses a premodular category admits a modularisation which is unique th 3 1 and cor 3 5 in char 0 the proof relies on two main ingredients a new and very simple criterion for the s matrix to be invertible 1 1 and deligne s internal characterization of tannakian categories in char 0 d when simple transparent objects are invertible the criterion is simpler 4 2 and the modularisation can be described more explicitly prop 4 4 we conclude with two examples the premodular categories associated with quantum hbox sf sl n and hbox sf pgl n at roots of unity in the first case we obtain modular categories which were built independently by c blanchet b in the second case we obtain modularizations in all the cases where y yokota y found reshetikhin turaev invariants of 3 manifolds thereby improving as well as explaining yokota s results

invariants of spin three manifolds from chern simons theory and finite dimensional hopf algebras abstract a version of kirby calculus for spin and framed three manifolds is given and is used to construct invariants of spin and framed three manifolds in two situations the first is ribbon categories which possess odd degenerate objects this case includes the quantum group situations corresponding to the half integer level chern simons theories conjectured to give spin tqfts by dijkgraaf and witten 1990 commun math phys 129 393 429 in particular the spin invariants constructed by kirby and melvin 1991 invent math 105 473 545 are shown to be identical to the invariants associated to so 3 second an invariant of spin manifolds analogous to the hennings invariant is constructed beginning with an arbitrary factorizable unimodular quasitriangular hopf algebra in particular a framed manifold invariant is associated to every finite dimensional hopf algebra via its quantum double and is conjectured to be identical to kuperberg s noninvolutory invariant of framed manifolds associated to that hopf algebra

on a family of non unitarizable ribbon categories we consider several families of categories the first are quotients of h andersen s tilting module categories for quantum groups of lie type b at odd roots of unity the second consists of categories of type bc constructed from idempotents in bmw algebras our main result is to show that these families coincide as braided tensor categories using a recent theorem of tuba and wenzl by appealing to similar results of blanchet and beliakova we obtain another interesting equivalence with these two families of categories and the quantum group categories of lie type c at odd roots of unity the morphism spaces in these categories can be equipped with a hermitian form and we are able to show that these categories are never unitary and no braided tensor category sharing the grothendieck semiring common to these families is unitarizable

integrality and symmetry of quantum link invariants 0 introduction quantum invariants of framed links whose components are colored by modules of a simple lie algebra g are laurent polynomials in v1 d with integer coefficients where v is the quantum parameter and d an integer depending on g we show that quantum invariants with a suitable normalization are laurent polynomials in v2 we also establish two symmetry properties of quantum link invariants at roots of unity the first asserts that quantum link invariants at rth roots of unity are invariant under the action of the affine weyl group wr which acts on the weight lattice a fundamental domain of wr is the fundamental alcove cr a simplex let g be the center of the corresponding simply connected complex lie group there is a natural action of g on cr the second symmetry property in its simplest form asserts that quantum link invariants are invariant under the action ofg if the link has zero linking matrix the second symmetry property generalizes symmetry principles of kirby and melvin the sl2 case and kohno and takata the sln case to arbitrary simple lie algebra

a graph theoretic game and its application to the k server problem this paper investigates a zero sum game played on a weighted connected graph g between two players the tree player and the edge player at each play the tree player chooses a spanning tree t and the edge player chooses an edge e the payoff to the edge player is cost t e defined as follows if e lies in the tree t then cost t e 0 if e does not lie in the tree then cost t e cycle t e w e where w e is the weight of edge e and cycle t e is the weight of the unique cycle formed when edge e is added to the tree t the main result is that the value of the game on any n vertex graph is bounded above by exp o sqrt log n log log n it is conjectured that the value of the game is o log n the game arises in connection with the k server problem on a road network i e a metric space that can be represented as a multigraph g in which each edge e represents a road of length w e it is shown that if the value of the game on g is val g w then there is a randomized strategy that achieves a competitive ratio of k 1 val g w against any oblivious adversary thus on any n vertex road network there is a randomized algorithm for the k server problem that is k cdot exp o sqrt log n log log n competitive against oblivious adversaries at the heart of the analysis of the game is an algorithm that provides an approximate solution for the simple network design problem specifically for any n vertex weighted connected multigraph the algorithm constructs a spanning tree t such that the average over all edges e of cost t e is less than or equal to exp o sqrt log n log log n this result has potential application to the design of communication networks it also improves substantially known estimates concerning the existence of a sparse basis for the cycle space of a graph

galois theory for braided tensor categories and the modular closure abstract given a braided tensor category c with conjugate dual objects and irreducible unit together with a full symmetric subcategory s we define a crossed product c s this construction yields a tensor category with conjugates and an irreducible unit a category is a category enriched over vect c with positive operation a galois correspondence is established between intermediate categories sitting between c and c s and closed subgroups of the galois group gal c s c aut c c s of c the latter being isomorphic to the compact group associated with s by the duality theorem of doplicher and roberts denoting by d c the full subcategory of degenerate objects i e objects which have trivial monodromy with all objects of c the braiding of c extends to a braiding of c s iff s d under this condition c s has no non trivial degenerate objects iff s d if the original category c is rational i e has only finitely many isomorphism classes of irreducible objects then the same holds for the new one the category c c d is called the modular closure of c since in the rational case it is modular i e gives rise to a unitary representation of the modular group sl 2 z if all simple objects of s have dimension one the structure of the category c s can be clarified quite explicitly in terms of group cohomology

from simplicial chern simons theory to the shadow invariant ii this is the second of a series of papers in which we introduce and study a rigorous simplicial realization of the non abelian chern simons path integral for manifolds m of the form m s1 and arbitrary simply connected compact structure groups g more precisely we introduce for general links l in m a rigorous simplicial version wlorig l of the corresponding wilson loop observable wlo l in the so called torus gauge by blau and thompson nucl phys b 408 2 345 390 1993 for a simple class of links l we then evaluate wlorig l explicitly in a non perturbative way finding agreement with turaev s shadow invariant l

the trivial connection contribution to witten s invariant and finite type invariants of rational homology spheres we derive an analog of the melvin morton bound on the power series expansion of the colored jones polynomial of algebraically split links and boundary links this allows us to produce a simple formula for the trivial connection contribution to witten s invariant of rational homology spheres we show that then th term in the 1 k expansion of the logarithm of this contribution is a finite type invariant of ohtsuki order 3n and of at most garoufalidis ordern