quantum symmetric kac moody pairs the present paper develops a general theory of quantum group analogs of symmetric pairs for involutive automorphism of the second kind of symmetrizable kac moody algebras the resulting quantum symmetric pairs are right coideal subalgebras of quantized enveloping algebras they give rise to triangular decompositions including a quantum analog of the iwasawa decomposition and they can be written explicitly in terms of generators and relations moreover their centers and their specializations are determined the constructions follow g letzter s theory of quantum symmetric pairs for semisimple lie algebras the main additional ingredient is the classification of involutive automorphisms of the second kind of symmetrizable kac moody algebras due to kac and wang the resulting theory comprises various classes of examples which have previously appeared in the literature such as q onsager algebras and the twisted q yangians introduced by molev ragoucy and sorba

equivariant quantizations of symmetric algebras abstract let g be a lie bialgebra and let v be a finite dimensional g module we study deformations of the symmetric algebra of v which are equivariant with respect to an action of the quantized enveloping algebra u h g resp u q g we investigate in particular such quantizations obtained from the quantization of certain lie bialgebra structures on the semidirect product of g and v we classify these structure in the important special case when g is complex simple with quasitriangular lie bialgebra structure and v is a simple g module we then introduce a more general notion co poisson module algebras and their quantizations to further address the problem and show that many known examples of quantized symmetric algebras can be described in this language

on the minimum ropelength of knots and links the ropelength of a knot is the quotient of its length by its thickness the radius of the largest embedded normal tube around the knot we prove existence and regularity for ropelength minimizers in any knot or link type these are c 1 1 curves but need not be smoother we improve the lower bound for the ropelength of a nontrivial knot and establish new ropelength bounds for small knots and links including some which are sharp

primitive ideals of cq sl n abstract the primitive ideals of the quantum group c q sl n are classified in the case where q is a non zero complex number which is not a root of unity it is shown that the orbits in prim c q sl n under the action of the character group h c n 1 are parameterized naturally by w w where w is the associated weyl group it is shown that there is a natural one to one correspondence between primitive ideals of c q sl n and symplectic leaves of the associated poisson algebraic group sl n c

harish chandra isomorphisms for quantum algebras the center of the quantum algebra is studied especially an analogue of the harish chandra isomorphism is established

singular localization for quantum groups at generic q abstract we quantize parabolic flag manifolds and describe categories of quantum d modules on them at a singular central character we compute global sections of generators for these categories for any q c for generic q we prove a singular version of beilinson bernstein localization for a quantized enveloping algebra

the locally finite part of the dual coalgebra of quantized irreducible flag manifolds for quantized irreducible flag manifolds the locally finite part of the dual coalgebra is shown to coincide with a natural quotient coalgebra overline u of u q mathfrak g on the way the coradical filtration of overline u is determined a graded version of the duality between overline u and the quantized coordinate ring is established this leads to a natural construction of several examples of quantized vector spaces as an application covariant first order differential calculi on quantized irreducible flag manifolds are classified

localization for quantum groups at a root of unity let c be the field of complex numbers and fix q g c let g be a semi simple lie algebra over c and let g be the corresponding simply connected algebraic group let uq be the quantized enveloping algebra of q let oq be the algebra of quantized functions on g let oq b be the quotient hopf algebra of oq corresponding to a borel subgroup b of g in the paper bk we defined categories of equivariant quantum o modules and pg modules on the quantum flag variety of g we proved that the beilinson bernstein localization theorem holds at a generic q namely the global section functor gives an equivalence between categories of modules and vq iaod es on the quantum flag variety thus one can translate questions about the represen tation theory of quantum groups to the study of the noncommutative geometry of the quantum flag variety in this paper we prove that a derived version of this theorem holds at the root of unity case using this equivalence we get that one can understand the representation theory of quantum groups at roots of unity through the now commutative geometry of the springer fibers we now recall the main results in bk the constructions given there are crucial for the present paper and a fairly detailed survey of the material there is given in the next section we defined an equivariant sheaf of quasi coherent modules over the quantum flag variety to be a left oq mod e equipped with a right q b comod ile structure satisfying certain compatibility conditions such objects form a category denoted ai q gq it contains certain line bundles oq for in the weight lattice we proved that oq x is ample for 0 holds for every q this implies that the category m bq gq is a proj category in the sense of serre

lower upper symmetric gauss seidel method for the euler and navier stokes equations on developpe un schema de relaxation multigrille application a des ecoulements transsoniques

area inequalities for embedded disks spanning unknotted curves author s hass joel lagarias jeffrey c thurston william p abstract we show that a smooth unknotted curve in r 3 satisfies an isoperimetric inequality that bounds the area of an embedded disk spanning the curve in terms of two parameters the length l of the curve and the thickness r maximal radius of an embedded tubular neighborhood of the curve for fixed length the expression giving the upper bound on the area grows exponentially in 1 r 2 in the direction of lower bounds we give a sequence of length one curves with r approaching 0 for which the area of any spanning disk is bounded from below by a function that grows exponentially with 1 r in particular given any constant a there is a smooth unknotted length one curve for which the area of a smallest embedded spanning disk is greater than a

classification of bicovariant differential calculi on quantum groups a representation theoretic approach the restricted dual of a quantized enveloping algebra can be viewed as the algebra of functions on a quantum group according to woronowicz there is a general notion of bicovariant differential calculus on such an algebra we give a classification theorem of these calculi the proof uses the notion due to reshetikhin and semenov tian shansky of a factorizable quasi triangular hopf algebra and relies on results of joseph and letzter on the way we also give a new formula for rosso s bilinear form

quantum flag varieties equivariant quantum d modules and localization of quantum groups abstract let o q g be the algebra of quantized functions on an algebraic group g and o q b its quotient algebra corresponding to a borel subgroup b of g we define the category of sheaves on the quantum flag variety of g to be the o q b equivariant o q g modules and prove that this is a proj category we construct a category of equivariant quantum d modules on this quantized flag variety and prove the beilinson bernstein s localization theorem for this category in the case when q is transcendental

cuts for 3 d magnetic scalar potentials visualizing unintuitive surfaces arising from trivial knots abstract a wealth of literature exists on computing and visualizing cuts for the magnetic scalar potential of a current carrying conductor via finite element methods fem and harmonic maps to the circle by a cut we refer to an orientable surface bounded by a given current carrying path such that the flux through it may be computed that restricts contour integrals on a curl zero vector field to those that do not link the current carrying path analogous to branch cuts of complex analysis this work is concerned with a study of a peculiar contour that illustrates topologically unintuitive aspects of cuts obtained from a trivial loop and raises questions about the notion of an optimal cut specifically an unknotted curve that bounds only high genus surfaces in its convex hull is analyzed the current work considers the geometric realization as a current carrying wire in order to construct a magnetic scalar potential moreover we consider the problem of choosing an energy functional on the space of maps suggesting an algorithm for computing cuts via minimizing a conformally invariant functional utilizing newton iteration

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

quantum zonal spherical functions and macdonald polynomials abstract a unified theory of quantum symmetric pairs is applied to q special functions previous work characterized certain left coideal subalgebras in the quantized enveloping algebra and established an appropriate framework for quantum zonal spherical functions here a distinguished family of such functions invariant under the weyl group associated to the restricted roots is shown to be a family of macdonald polynomials as conjectured by koornwinder and macdonald our results place earlier work for lie algebras of classical type in a general context and extend to the exceptional cases

quantification de la localisation de dixmier de u sln 1 c resume in this article we extend to the quantum case the classical result of dixmier according to which up to localization the enveloping algebra of sl n 1 c is a central polynomial extension of the enveloping algebra of some parabolic subalgebra in fact the latter has a semiinvariant element d by which it is sufficient to localize to obtain the result the methods we use here are quite different from those used by dixmier indeed in the quantum case the powers of the indeterminate q would make calculations similar to those of dixmier much too complicated on the other hand when we use the rosso form which allows us to work in the hopf dual of q sl n 1 c the calculations become much easier than they would be in q sl n 1 c

convex partitions of polyhedra a lower bound and worst case optimal algorithm the problem of partitioning a polyhedron into a minimum number of convex pieces is known to be np hard we establish here a quadratic lower bound on the complexity of this problem and we describe an algorithm that produces a number of convex parts within a constant factor of optimal in the worst case the algorithm is linear in the size of the polyhedron and cubic in the number of reflex angles since in most applications areas the former quantity greatly exceeds the latter the algorithm is viable in practice

the number of reidemeister moves needed for unknotting there is a positive constant c1 such that for any diagram d representing the unknot there is a sequence of at most 2 c1n reidemeister moves that will convert it to a trivial knot diagram where n is the number of crossings in d a similar result holds for elementary moves on a polygonal knot k embedded in the 1 skeleton of the interior of a compact orientable triangulated p l 3 manifold m there is a positive constant c2 such that for each t 1 if m consists of t tetrahedra and k is unknotted then there is a sequence of at most 2 c 2t elementary moves in m which transforms k to a triangle contained inside one tetrahedron of m we obtain explicit values for c1 and c2

symmetric pairs for quantized enveloping algebras abstract let be an involution of a semisimple lie algebra g let g denote the fixed lie subalgebra and assume the cartan subalgebra of g has been chosen in a suitable way we construct a quantum analog of u g which can be characterized as the unique subalgebra of the quantized enveloping algebra of g which is a maximal right coideal that specializes to u g

quantized multiplicative quiver varieties abstract beginning with the data of a quiver q and its dimension vector d we construct an algebra d q d q mat d q which is a flat q deformation of the algebra of differential operators on the affine space mat d q the algebra d q is equivariant for an action by a product of quantum general linear groups acting by conjugation at each vertex we construct a quantum moment map for this action and subsequently define the hamiltonian reduction a d q of d q with moment parameter we show that a d q is a flat formal deformation of lusztig s quiver varieties and their multiplicative counterparts for all dimension vectors satisfying a flatness condition of crawley boevey indeed the product on a d q yields a fedosov quantization the of symplectic structure on multiplicative quiver varieties as an application we give a description of the category of representations of the spherical double affine hecke algebra of type a n 1 and its generalization constructed by etingof oblomkov and rains in terms of a quotient of the category of equivariant d q modules by a serre subcategory of aspherical modules

on modified block ssor iteration methods for linear systems from steady incompressible viscous flow problems abstract in order to solve the large sparse systems of linear equations arising from numerical solutions of two dimensional steady incompressible viscous flow problems in primitive variable formulation we present block ssor and modified block ssor iteration methods based on the special structures of the coefficient matrices in each step of the block ssor iteration we employ the block lu factorization to solve the sub systems of linear equations we show that the block lu factorization is existent and stable when the coefficient matrices are block diagonally dominant of type ii by columns under suitable conditions we establish convergence theorems for both block ssor and modified block ssor iteration methods in addition the block ssor iteration and af adi method are considered as preconditioners for the nonsymmetric systems of linear equations numerical experiments show that both block ssor and modified block ssor iterations are feasible iterative solvers and they are also effective for preconditioning krylov subspace methods such as gmres and bicgstab when used to solve this class of systems of linear equations

sharp error bounds of some krylov subspace methods for non hermitian linear systems1project 19601036 supported by the national natural science foundation of china 1 the convergence of the krylov subspace methods e g full orthogonal method fom and generalized minimal residual method gmres etc for solving large non hermitian linear systems is studied in a unified and detailed way when the coefficient matrix is defective in particular when its spectrum lies in the open right left half plane or is on the real axis related theoretical error bounds are established which reveal some intrinsic relationships between the convergence properties and the eigen characteristics of the coefficient matrix these results not only generalize all the known ones for the diagonalizable matrices in the literature but also sharp the corresponding estimates in jia acta mathematica sinica new series 14 1998 507 518

structured preconditioners for nonsingular matrices of block two by two structures for the large sparse block two by two real nonsingular matrices we establish a general framework of practical and efficient structured preconditioners through matrix transformation and matrix approximations for the specific versions such as modified block jacobi type modified block gauss seidel type and modified block unsymmetric symmetric gauss seidel type preconditioners we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices also we show that when these structured preconditioners are employed to precondition the krylov subspace methods such as gmres and restarted gmres fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two by two coefficient matrices in particular these structured preconditioners can lead to efficient and high quality preconditioning matrices for some typical matrices from the real world applications

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

quantum enveloping algebras with von neumann regular cartan like generators and the pierce decomposition quantum bialgebras derivable from uq sl2 which contain idempotents and von neumann regular cartan like generators are introduced and investigated various types of antipodes invertible and von neumann regular on these bialgebras are constructed which leads to a hopf algebra structure and a von neumann hopf algebra structure respectively for them explicit forms of some particular r matrices also invertible and von neumann regular are presented and the latter respects the pierce decomposition

the center of the reflection equation algebra via quantum minors abstract we give simple formulas for the elements c k appearing in a quantum cayley hamilton formula for the reflection equation algebra re algebra associated to the quantum group u q gl n answering a question of kolb and stokman the c k s are certain canonical generators of the center of the re algebra and hence of u q gl n itself they have been described by reshetikhin using graphical calculus by nazarov tarasov using quantum yangians and by gurevich pyatov and saponov using quantum schur functions however no explicit formulas for these elements were previously known as byproducts we prove a quantum girard newton identity relating the c k s to the so called quantum power traces and we give a new presentation for the quantum group u q gl n as a localization of the re algebra along certain principal minors

representations of quantum groups at roots of unity whittaker vectors and q w algebras abstract let u e g be the standard simply connected version of the drinfeld jumbo quantum group at an odd m th root of unity e the center of u e g contains a huge commutative subalgebra isomorphic to the algebra z g of regular functions on a finite covering of a big cell in a complex connected simply connected algebraic group g with lie algebra g let v be a finite dimensional representation of u e g on which z g acts according to a non trivial character g given by evaluation of regular functions at g g then v is a representation of the finite dimensional algebra u g u e g u e g ker g we show that in this case under certain restrictions on m u g contains a subalgebra u g m of dimension m 1 2 dim o where o is the conjugacy class of g and u g m has a one dimensional representation c g we also prove that if v is not trivial then the space of whittaker vectors hom u g m c g v is not trivial and the algebra w g end u g u g u g m c g naturally acts on it which gives rise to a schur type duality between representations of the algebra u g and of the algebra w g called a q w algebra

primitive ideals of c q sl 3 the primitive ideals of the hopf algebrac q sl 3 are classified in particular it is shown that the orbits in primc q sl 3 under the action of the representation grouph c c are parameterized naturally byw w wherew is the associated weyl group it is shown that there is a natural one to one correspondence between primitive ideals ofc q sl 3 and symplectic leaves of the associated poisson algebraic groupsl 3 c

the computational complexity of knot and link problems we consider the problem of deciding whether a polygonal knot in 3 dimensional euclidean space is unknotted ie capable of being continuously deformed without self intersection so that it lies in a plane we show that this problem unknotting problem is in np we also consider the problem splitting problem of determining whether two or more such polygons can be split or continuously deformed without self intersection so that they occupy both sides of a plane without intersecting it we show that it also is in np finally we show that the problem of determining the genus of a polygonal knot a generalization of the problem of determining whether it is unknotted is in pspace we also give exponential worst case running time bounds for deterministic algorithms to solve each of these problems these algorithms are based on the use of normal surfaces and decision procedures due to w haken with recent extensions by w jaco and j l tollefson

on preconditioned iterative methods for unsteady incompressible navier stokes equations when the implicit finite difference scheme and linearization technique in conjunction with the artificial compressibility method are employed to discretize the unsteady incompressible navier stokes equations in each pseudo time step a structured system of linear equations is obtained and needs to be solved numerically in this paper preconditioning techniques are applied to solve such linear systems numerical example is given to illustrate the effectiveness of the proposed method we also find that a combination of the linearization technique with the preconditioned gmres method is quite efficient for the implicit finite difference discretization of the incompressible navier stokes equations

localization of quantum biequivariant ɗ modules and q w algebras the author is grateful to y kremnizer for useful discussions and to the referee for careful reading of the manuscript