groupoid equivariant prequantization in their 2005 paper c laurent gengoux and p xu define prequantization for pre hamiltonian actions of quasi presymplectic lie groupoids in terms of s1 central extensions of lie groupoids the definition requires that the quasi presymplectic structure be exact i e the closed 3 form on the unit space of the lie groupoid must be exact in the present paper we define prequantization for pre hamiltonian actions of not necessarily exact quasi presymplectic lie groupoids in terms of dixmier douady bundles the definition is a natural adaptation of e meinrenken s treatment of prequantization for quasi hamiltonian lie group actions with group valued moment map the definition given in this paper is shown to be compatible with the definition of laurent gengoux and xu when the underlying quasi preysmplectic structure is exact properties related to morita invariance and symplectic reduction are established

pseudodifferential symbols on riemann surfaces and krichever novikov algebras we define the krichever novikov type lie algebras of differential operators and pseudodifferential symbols on riemann surfaces along with their outer derivations and central extensions we show that the corresponding algebras of meromorphic operators and symbols have many invariant traces and central extensions given by the logarithms of meromorphic vector fields very few of these extensions survive after passing to the algebras of operators and symbols holomorphic away from several fixed points we also describe the associated manin triples and kdv type hierarchies emphasizing the similarities and differences with the case of smooth symbols on the circle

a new integral formula for heckman opdam hypergeometric functions abstract we provide harish chandra type formulas for the multivariate bessel functions and heckman opdam hypergeometric functions as representation valued integrals over dressing orbits our expression is the quasi classical limit of the realization of macdonald polynomials as traces of intertwiners of quantum groups given by etingof and kirillov jr in 10 integration over the liouville tori of the gelfand tsetlin integrable system and adjunction for higher calogero moser hamiltonians recovers and gives a new proof of the integral realization over gelfand tsetlin polytopes which appeared in the recent work 4 of borodin and gorin on the jacobi corners ensemble

classical solutions of the quantum yang baxter equation the classical analogue is developed here for part of the construction in which knot and link invariants are produced from representations of quantum groups whereas previous work begins with a quantum group obtained by deforming the multiplication of functions on a poisson lie group we work directly with a poisson lie groupg and its associated symplectic groupoid the classical analog of the quantumr matrix is a lagrangian submanifold open image in new window in the cartesian square of the symplectic groupoid for any symplectic leafs ing open image in new window induces a symplectic automorphism ofs s which satisfies the set theoretic yang baxter equation when combined with the flip map exchanging components and suitably implanted in each cartesian powers n generates a symplectic action of the braid groupb n ons n application of a symplectic trace formula to the fixed point set of the action of braids should lead to link invariants but work on this last step is still in progress

lie theory of multiplicative tensors we study tensors on lie groupoids suitably compatible with the groupoid structure called multiplicative our main result gives a complete description of these objects only in terms of infinitesimal data special cases include the infinitesimal counterparts of multiplicative forms multivector fields and holomorphic structures obtained through a unifying and conceptual method we also give a full treatment of multiplicative vector valued forms particularly nijenhuis operators and related structures

further facts about input to state stabilization previously published results about input to state stabilizability are shown to hold even for systems which are not linear in controls provided that a more general type of feedback is allowed applications to certain stabilization problems and coprime factorizations as well as comparisons to other results on input to state stability are briefly discussed

unified products for leibniz algebras applications abstract let g be a leibniz algebra and e a vector space containing g as a subspace all leibniz algebra structures on e containing g as a subalgebra are explicitly described and classified by two non abelian cohomological type objects hl g 2 v g provides the classification up to an isomorphism that stabilizes g and hl 2 v g will classify all such structures from the view point of the extension problem here v is a complement of g in e a general product called the unified product is introduced as a tool for our approach the crossed resp bicrossed products between two leibniz algebras are introduced as special cases of the unified product the first one is responsible for the extension problem while the bicrossed product is responsible for the factorization problem the description and the classification of all complements of a given extension g e of leibniz algebras are given as a converse of the factorization problem they are classified by another cohomological object denoted by ha 2 h g where is the canonical matched pair associated to a given complement h several examples are worked out in details

poisson lie group of pseudodifferential symbols we introduce a lie bialgebra structure on the central extension of the lie algebra of differential operators with scalar or matrix coefficients on the line and on the circle this defines a poisson lie structure on the dual group of pseudodifferential symbols of an arbitrary real or complex order we show that the usual second benney gl n kdv or gl n adler gelfand dickey and kp poisson structures are naturally realized as restrictions of this poisson structure to submanifolds of this universal poisson lie group moreover the reduced sl n versions of these manifolds orw n algebras in physical terminology can be viewed as certain subspaces of the quotient of this poisson lie group by the dressing action of the group of functions on the circle or as a result of a poisson reduction finally we define an infinite set of commuting functions on the poisson lie group that give the standard families of hamiltonians when restricted to the submanifolds mentioned above the poisson structure and hamiltonians on the whole group interpolate between the poisson structures and hamiltonians of benney kp and kdv flows we also discuss the geometrical meaning ofw as a limit of poisson algebraswe as e 0

quantization of poisson manifolds from the integrability of the modular function we discuss a framework for quantizing a poisson manifold via the quantization of its symplectic groupoid combining the tools of geometric quantization with the results of renault s theory of groupoid c algebras this setting allows very singular polarizations in particular we consider the case when the modular function is multiplicatively integrable i e when the space of leaves of the polarization inherits a groupoid structure if suitable regularity conditions are satisfied then one can define the quantum algebra as the convolution algebra of the subgroupoid of leaves satisfying the bohr sommerfeld conditions

algebraic structure of multiparameter quantum groups let g be a connected semi simple complex lie group we define and study the multi parameter quantum group cq p g in the case where q is a complex parameter that is not a root of unity using a method of twisting bigraded hopf algebras by a cocycle 2 we develop a unified approach to the construction of cq p g and of the multi parameter drinfeld double dq p using a general method of deforming bigraded pairs of hopf algebras we construct a hopf pairing between these algebras from which we deduce a peterweyl type theorem for cq p g we then describe the prime and primitive spectra of cq p g generalizing a result of joseph in the one parameter case this description was conjectured and established in the sl n case by the first and second authors in 15 16 it was proved in the general case by joseph in 18 19 in particular the orbits in prim cq p g under the natural action of the maximal torus h are indexed as in the one parameter case by the elements of the double weyl group w w unlike the one parameter case there is not in general a bijection between symp g and prim cq p g however in the case when

double products and hypersymplectic structures on ℝ4n in this paper we give a procedure to construct hypersymplectic structures on 4n beginning with affine symplectic data on 2n these structures are shown to be invariant by a 3 step nilpotent double lie group and the resulting metrics are complete and not necessarily flat explicit examples of this construction are exhibited

leaf preserving quantizations of poisson su 2 are not coalgebra homomorphisms although it has been found that some deformation quantizations of the poissonsu 2 preserve symplectic leaves and some preserve the group i e coalgebra operation this paper shows that a quantization ofsu 2 cannot be both leaf preserving and group preserving

finding community structure in networks using the eigenvectors of matrices we consider the problem of detecting communities or modules in networks groups of vertices with a higher than average density of edges connecting them previous work indicates that a robust approach to this problem is the maximization of the benefit function known as modularity over possible divisions of a network here we show that this maximization process can be written in terms of the eigenspectrum of a matrix we call the modularity matrix which plays a role in community detection similar to that played by the graph laplacian in graph partitioning calculations this result leads us to a number of possible algorithms for detecting community structure as well as several other results including a spectral measure of bipartite structure in networks and a centrality measure that identifies vertices that occupy central positions within the communities to which they belong the algorithms and measures proposed are illustrated with applications to a variety of real world complex networks

prime spectra of quantum semisimple groups we study the prime ideal spaces of the quantized function algebras rq g for g a semisimple lie group and q an indeterminate our method is to examine the structure of algebras satisfying a set of seven hypotheses and then to demonstrate using work of joseph hodges and levasseur that the algebras rq g satisfy this list of assumptions rings satisfying the assumptions are shown to satisfy normal separation and therefore jategaonkar s strong second layer condition for such rings much representation theoretic information is carried by the graph of links of the prime spectrum and so we proceed to a detailed study of the prime links of algebras satisfying the list of assumptions homogeneity is a key feature it is proved that the clique of any prime ideal coincides with its orbit under a finite rank free abelian group of automorphisms bounds on the ranks of these groups are obtained in the case of rq g in the final section the results are specialized to the case g sln c where detailed calculations can be used to illustrate the general results as a preliminary set of examples we show also that the multiparameter quantum coordinate rings of affine n space satisfy our axiom scheme when the group generated by the parameters is torsionfree 0 introduction and background 0 1 let p and q be prime ideals of a noetherian ring r we say that p is linked to q and write p q if there is an ideal a of r with pq c a c p n q such that p n q a is a nonzero torsionfree left r p module and a torsionfree right r q module this condition can be simplified in case r satisfies the second layer condition see below in that case p q if and only if p n q pq is faithful as a left r p module and as a right r q module 39 10 1 4 the graph of links of r is the directed graph whose vertices are the points of spec r the set of prime ideals of r with a directed edge from p to q if and only if p q the primes in the connected component of the graph of links containing p constitute the clique of p denoted clique p there is a close connection between the graph of links of r and the representation theory of r cf 20 4 this connection is particularly strong when r satisfies the strong second layer condition which requires that whenever u is a cyclic uniform r module with the annihilator received by the editors november 4 1994 and in revised form september 5 1995 1991 mathematics subject classification primary 16d30 16d60 16p40 17b37 the research of the second author was partially supported by a grant from the national science foundation usa part of the work was carried out while he visited the university of glasgow mathematics department during october 1993 supported by the edinburgh and london mathematical societies work on a revised version of the paper was completed in summer 1995 during a visit by both authors to the department of mathematics of the university of washington whom both thank for its hospitality the travel costs of the first author were in part covered by a grant from the carnegie trust for the universities of scotland 1996 american mathematical society

on poisson actions of compact lie groups on symplectic manifolds let g p be a compact simple poisson lie group equipped with a poisson structure p and m omega be a symplectic manifold assume that m carries a poisson action of g p and there is an equivariant moment map in the sense of lu and weinstein which maps

a class of quadratic matrix algebras arising from the quantized enveloping algebra uq a2n 1 a natural family of quantized matrix algebras is introduced it includes the two best studied such located inside uq a2n 1 it consists of quadratic algebras with the same hilbert series as polynomials in n2 variables we discuss their general properties and investigate some members of the family in great detail with respect to associated varieties degrees centers and symplectic leaves finally the space of rank r matrices becomes a poisson submanifold and there is an associated tensor category of rank r matrices

quantum grassmann manifolds orbits of the quantum dressing transformation forsu q n acting on its solvable dual are introduced the case is considered when the corresponding classical orbits coincide with grassmann manifolds quantization of the poisson bracket on a zariski open subset of the grassmann manifold yields a algebra generated by the quantum coordinate functions the commutation relations are written in a compact form with the help of ther matrix finite dimensional irreducible representations ofu h mathfrak s mathfrak l n mathbb c are derived from the algebra structure

characterization of all nonlinear stabilizing controllers via observer based kernel representations this paper deals with the characterization of all nonlinear stabilizing controllers observer based kernel representations are introduced to avoid the difficulty which occurs in constructing the state space realizations of the parametrization of input output operators the parametrization of all stabilizing controllers for a given plant and a class of stabilizing plant controller pairs are derived by employing observer based kernel representations furthermore the relationship between the results via kernel representation approach and the existing results via state space approach is clarified

quantum and classical pseudogroups part ii differential and symplectic pseudogroups the category of sympletic pseudospaces analogical to the category of pseudospaces in the sense of 2 is introduced and used to define symplectic pseudogroups structures analogical to pseudogroups 3 or quantum groups 4 it is shown that symplectic pseudogroups are in one to one correspondence with manin groups also introduced in this paper the set theoretical part of these structures has been described in i

formal symplectic groupoid the multiplicative structure of the trivial symplectic groupoid over d associated to the zero poisson structure can be expressed in terms of a generating function we address the problem of deforming such a generating function in the direction of a non trivial poisson structure so that the multiplication remains associative we prove that such a deformation is unique under some reasonable conditions and we give the explicit formula for it this formula turns out to be the semi classical approximation of kontsevich s deformation formula for the case of a linear poisson structure the deformed generating function reduces exactly to the cbh formula of the associated lie algebra the methods used to prove existence are interesting in their own right as they come from an at first sight unrelated domain of mathematics the runge kutta theory of the numeric integration of ode s

free motion on the poisson su n group is the phase space of the poisson su n we explicitly calculate the symplectic structure of define an analogue of the hamiltonian of the free motion on su n and solve the corresponding equations of motion velocity is related to the momentum by a nonlinear legendre transformation

quantum lobachevsky planes we classify all sl 2 r covariant poisson structures on the lobachevsky plane with respect to all multiplicative poisson structures on sl 2 r and describe quantizations for all these poisson structures

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

noncommutative instantons on the 4 sphere from quantum groups we describe an approach to the noncommutative instantons on the 4 sphere based on quantum group theory we quantize the hopf bundle 7 4 making use of the concept of quantum coisotropic subgroups the analysis of the semiclassical poisson lie structure of u 4 shows that the diagonal su 2 must be conjugated to be properly quantized the quantum coisotropic subgroup we obtain is the standard su q 2 it determines a new deformation of the 4 sphere 4 q as the algebra of coinvariants in q 7 we show that the quantum vector bundle associated to the fundamental corepresentation of su q 2 is finitely generated and projective and we compute the explicit projector we give the unitary representations of 4 q we define two 0 summable fredholm modules and we compute the chern connes pairing between the projector and their characters it comes out that even the zero class in cyclic homology is non trivial

compact quantum groups and groupoidc algebras abstract we study simple compact matrix quantum groups in the context of groupoid c algebras and obtain explicit descriptions of the structure of the c algebra of quantum su n and a concrete formula for its haar functional

hochschild and cyclic homology of quantum groups for an arbitrary complex linear semisimple lie groupg we consider hopf algebras of the deformations of the formal and algebraic functions ong the hochschild and cyclic homology of these hopf algebras are computed when the value of the deformation parameter is generic

representations of uh su n derived from quantum flag manifolds a relationship between quantum flag and grassmann manifolds is revealed this enables a formal diagonalization of quantum positive matrices the requirement that this diagonalization defines a homomorphism leads to a left uh su n module structure on the algebra generated by quantum antiholomorphic coordinate functions living on the flag manifold the module is defined by prescribing the action on the unit and then extending it to all polynomials using a quantum version of the leibniz rule the leibniz rule is shown to be induced by the dressing transformation for discrete values of parameters occurring in the diagonalization one can extract finite dimensional irreducible representations of uh su n as cyclic submodules

quantization of the poisson su 2 and its poisson homogeneous space the 2 sphere we show that deformation quantizations of the poisson structures on the poisson lie groupsu 2 and its homogeneous space the 2 sphere are compatible with woronowicz s deformation quantization ofsu 2 s group structure and podles deformation quantization of 2 sphere s homogeneous structure respectively so in a certain sense the multiplicativity of the lie poisson structure onsu 2 at the classical level is preserved under quantization

integrable systems on semidirect product lie groups we study integrable systems on the semidirect product of a lie group and its lie algebra as the representation space of the adjoint action regarding the tangent bundle of a lie group as phase space endowed with this semidirect product lie group structure we construct a class of symplectic submanifolds equipped with a dirac bracket on which integrable systems in the adler kostant symes sense are naturally built through collective dynamics in doing so we address other issues such as factorization poisson lie structures and dressing actions we show that the procedure becomes recursive for some particular hamilton functions giving rise to a tower of nested integrable systems

input output to state stability this work explores lyapunov characterizations of the input output to state stability ioss property for nonlinear systems the notion of ioss is a natural generalization of the standard zero detectability property used in the linear case the main contribution of this work is to establish a complete equivalence between the ioss property and the existence of a certain type of smooth lyapunov function as corollaries one shows the existence of norm estimators and obtains characterizations of nonlinear detectability in terms of relative stability and of finite energy estimates

generalized quantization scheme for central extensions of lie algebras we present the method for finding nonlinear poisson lie group structures on the vector spaces and for their quantization explicit quantization formulas are proposed for arbitrary central extension of a lie algebra

on the youla kucera parametrization for nonlinear systems abstract we consider the problem of stabilizing a nonlinear plant through the use of a possibly unstable pre compensator and a stable feedback compensator and parameterizing a class of such stabilizers in terms of a bibo stable map q in the linear case by means of superposition the pre and feedback compensator can be combined in such a way that the bibo map q occurs in just one feedback loop within the controller feeding back output residuals prediction errors to the control inputs the main contribution of this paper is to develop a nonlinear generalization of this property building on earlier work in forming the youla kucera parametrization for nonlinear systems we show the equivalence of the class of all bounded input stabilizing nonlinear pre and feedback compensators to a class of possibly unstable feedback controllers in which a map q s is present in only the one feedback loop we then show that necessary and sufficient condition to achieve stability of the system is that q s be bibo stable one advantage of the new formulation is that differential boundedness assumptions do not involve the parametrization q s in any way just as the linear versions of our results have applications in the areas of optimal control and adaptive control of linear systems it is conjectured that the present results will underlie more general results for adaptive control and nonlinear systems

hopf von neumann algebra bicrossproducts kac algebra bicrossproducts and the classical yang baxter equations two groups g1 g2 are said to be a matched pair if each acts on the space of the other and these actions say obey a certain compatibility condition for every matched pair of locally compact groups g1 g2 we construct an associated coinvolutive hopf von neumann algebra m g1 l g2 by simultaneous cross product and cross coproduct for non trivial these bicrossproduct hopf von neumann algebras are non commutative and non cocommutative if the modules for the actions are also matched then these bicrossproducts are kac algebras in this case we show that the dual kac algebra is of the same form with the roles of g1 g2 and of interchanged examples exist with g1 a simply connected lie group and choices of g2 determined by suitable solutions of the classical yang baxter equations on the complexification of the lie algebra of g1

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

the krichever map vector bundles over algebraic curves and heisenberg algebras we study the grassmanniangr x n consisting of equivalence classes of rankn algebraic vector bundles over a riemann surfacex with an holomorphic trivialization at a fixed pointp commutative subalgebras ofgl n h h being the ring of functions holomorphic on a punctured disc aboutp define flows on the grassmannian giving rise to classes of solutions to multi component kp hierarchies these commutative subalgebras correspond to heisenberg algebras in the kac moody algebra associated togl n h one can obtain by the krichever map points ofgr x n and solutions of mckp from coveringsf y x and other geometric data conversely for every point ofgr x n and for every choice of heisenberg algebra we construct using the cotangent bundle ofgr x n an algebraic curve coveringx and other data thus inverting the krichever map we show the explicit relation between the choice of heisenberg algebra and the geometry of the covering space

phase spaces related to standard classical r matrices fundamental representations of real simple poisson lie groups are poisson actions with a suitable choice of the poisson structure on the underlying real vector space we study these mostly quadratic poisson structures and corresponding phase spaces symplectic groupoids

dirac method and symplectic submanifolds in the cotangent bundle of a factorizable lie group we study some symplectic submanifolds in the cotangent bundle of a factorizable lie group defined by second class constraints by applying the dirac method we study many issues of these spaces as fundamental dirac brackets symmetries and collective dynamics as the main application we study integrable systems on these submanifolds as inherited from a system on the whole cotangent bundle meeting in a natural way with the adler kostant symes theory of integrability

one shot detection with laplacian object and fast matrix cosine similarity one shot generic object detection involves searching for a single query object in a larger target image relevant approaches have benefited from features that typically model the local similarity patterns in this paper we combine local similarity encoded by local descriptors with a global context i e a graph structure of pairwise affinities among the local descriptors embedding the query descriptors into a low dimensional but discriminatory subspace unlike principal components that preserve global structure of feature space we actually seek a linear approximation to the laplacian eigenmap that permits us a locality preserving embedding of high dimensional region descriptors our second contribution is an accelerated but exact computation of matrix cosine similarity as the decision rule for detection obviating the computationally expensive sliding window search we leverage the power of fourier transform combined with integral image to achieve superior runtime efficiency that allows us to test multiple hypotheses for pose estimation within a reasonably short time our approach to one shot detection is training free and experiments on the standard data sets confirm the efficacy of our model besides low computation cost of the proposed codebook free object detector facilitates rather straightforward query detection in large data sets including movie videos

the poisson geometry of su 1 1 we study the natural poisson structure on the group su 1 1 and related questions in particular we give an explicit description of the ginzburg weinstein isomorphism for the sets of admissible elements we also establish an analog of thompson s conjecture for this group

a characterization of integral input to state stability the notion of input to state stability iss is now recognized as a central concept in nonlinear systems analysis it provides a nonlinear generalization of finite gains with respect to supremum norms and also of finite l sup 2 gains it plays a central role in recursive design coprime factorizations controllers for nonminimum phase systems and many other areas in this paper a newer notion that of integral input to state stability iiss is studied the notion of iiss generalizes the concept of finite gain when using an integral norm on inputs but supremum norms of states in that sense generalizing the linear h sup 2 theory it allows one to quantify sensitivity even in the presence of certain forms of nonlinear resonance we obtain several necessary and sufficient characterizations of the iiss property expressed in terms of dissipation inequalities and other alternative and nontrivial characterizations

semi supervised eigenvectors for large scale locally biased learning in many applications one has side information e g labels that are provided in a semi supervised manner about a specific target region of a large data set and one wants to perform machine learning and data analysis tasks nearby that prespecified target region for example one might be interested in the clustering structure of a data graph near a prespecified seed set of nodes or one might be interested in finding partitions in an image that are near a prespecified ground truth set of pixels locally biased problems of this sort are particularly challenging for popular eigenvector based machine learning and data analysis tools at root the reason is that eigenvectors are inherently global quantities thus limiting the applicability of eigenvector based methods in situations where one is interested in very local properties of the data in this paper we address this issue by providing a methodology to construct semisupervised eigenvectors of a graph laplacian and we illustrate how these locally biased eigenvectors can be used to perform locally biased machine learning these semi supervised eigenvectors capture successively orthogonalized directions of maximum variance conditioned on being well correlated with an input seed set of nodes that is assumed to be provided in a semi supervised manner we show that these semi supervised eigenvectors can be computed quickly as the solution to a system of linear equations and we also describe several variants of our basic method that have improved scaling properties we provide several empirical examples demonstrating how these semi supervised eigenvectors can be used to perform locally biased learning and we discuss the relationship between our results and recent machine learning algorithms that use global eigenvectors of the graph laplacian

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

classical intertwiner space and quantization given two symplectic realizations a symplectic manifold called the classical intertwiner space is introduced as a classical analogue of an intertwiner space of representations of an associative algebra we describe explicitly how a quantum data on realizations induces a quantum data on their classical intertwiner space

on tangential star products for the coadjoint poisson structure we derive necessary conditions on a lie algebra from the existence of a star product on a neighbourhood of the origin in the dual of the lie algebra for the coadjoint poisson structure which is both differential and tangential to all the coadjoint orbits in particular we show that when the lie algebra is semisimple there are no differential and tangential star products on any neighbourhood of the origin in the dual of its lie algebra

quantum line bundles on s2 and method of orbits for suq 2 the quantum sphere is covered by two quantum complex planes quantum line bundle on s2 is introduced by specifying a transition function the structure of the quantum complex planes enables one to construct representations of suq 2 acting in spaces of holomorphic sections in the quantum case dressing transformation replaces coadjoint action

classification of real three dimensional poisson lie groups all real three dimensional poisson lie pl groups are explicitly constructed and fully classified under group automorphisms by making use of their one to one correspondence with the complete classification of real three dimensional lie bialgebras given in gomez 2000 j math phys 41 4939 many of these 3d pl groups are non coboundary structures whose poisson brackets are given here for the first time casimir functions for all three dimensional pl groups are given and some features of several pl structures are commented

variational perspective on local graph clustering modern graph clustering applications require the analysis of large graphs and this can be computationally expensive in this regard local spectral graph clustering methods aim to identify well connected clusters around a given seed set of reference nodes without accessing the entire graph the celebrated approximate personalized pagerank appr algorithm in the seminal paper by andersen et al in focs 06 proceedings of the 47th annual ieee symposium on foundations of computer science pp 475 486 2006 is one such method appr was introduced and motivated purely from an algorithmic perspective in other words there is no a priori notion of objective function optimality conditions that characterizes the steps taken by appr here we derive a novel variational formulation which makes explicit the actual optimization problem solved by appr in doing so we draw connections between the local spectral algorithm of andersen et al 2006 and an iterative shrinkage thresholding algorithm ista in particular we show that appropriately initialized ista applied to our variational formulation can recover the sought after local cluster in a time that only depends on the number of non zeros of the optimal solution instead of the entire graph in the process we show that an optimization algorithm which apparently requires accessing the entire graph can be made to behave in a completely local manner by accessing only a small number of nodes this viewpoint builds a bridge across two seemingly disjoint fields of graph processing and numerical optimization and it allows one to leverage well studied numerically robust and efficient optimization algorithms for processing today s large graphs

double quantum groups and iwasawa decomposition abstract the double quantum groups c q d g c q g c q g are the hopf algebras underlying the complex quantum groups of which the simplest example is the quantum lorentz group they are nonstandard quantizations of the double groupg g we construct a corresponding quantized universal enveloping algebrauq d g and prove that the pairing between c q d g anduq d g is nondegenerate we analyze the representation theory of these c q d g give a detailed version of the iwasawa decomposition proved by podles and woronowicz for the quantum lorentz group and show that c q d g is noetherian finally we outline how to construct more general nonstandard quantum groups using quantum double groups and their generalizations

the hyperbolic moduli space of flat connections and the isomorphism of symplectic multiplicity spaces the hyperbolic moduli space of flat connections and the isomorphism of symplectic multiplicity spaces

nonlinear unity feedback systems and q parametrization this paper concerns nonlinear systems defines a new concept of stability and extends to unity feedback systems the technique of q parametrization introduced by zames and developed by desoer chen and gustafson we specify 1 a global parametrization of all controllers that s stabilize a given s stable plant 2 a parametrization of a class of controllers that stabilize an unstable plant 3 necessary and sufficient conditions for a nonlinear controller to simultaneously stabilize two nonlinear plants

decomposable representations and lagrangian submanifolds of moduli spaces associated to surface groups the importance of explicit examples of lagrangian submanifolds of moduli spaces is revealed by papers such as dostoglou and salamon ann of math 2 139 3 581 640 1994 and salamon proceedings of the international congress of mathematicians vol 1 2 zurich 1994 pp 526 536 birkhauser basel 1995 given a 3 manifold m with boundary m dostoglou and salamon use such examples to obtain a proof of the atiyah floer conjecture relating the symplectic floer homology of the representation space hom 1 m u u associated to an explicit pair of lagrangian submanifolds of this representation space and the instanton homology of the 3 manifold m in the present paper we construct a lagrangian submanifold of the space of representations mathcal m g l hom mathcal c pi g l u u of the fundamental group g lof a punctured riemann surface g l into an arbitrary compact connected lie group u this lagrangian submanifold is obtained as the fixed point set of an anti symplectic involution hat beta defined on mathcal m g l we show that the involution hat beta is induced by a form reversing involution defined on the quasi hamiltonian space u times u g times mathcal c 1 times cdots times mathcal c l the fact that hat beta has a non empty fixed point set is a consequence of the real convexity theorem for group valued momentum maps proved in schaffhauser a real convexity theorem for quasi hamiltonian actions submitted 25 p 2007 http arxiv org abs math 0705 0858 the notion of decomposable representation provides a geometric interpretation of the lagrangian submanifold thus obtained

symplectic structures associated to lie poisson groups the lie poisson analogues of the cotangent bundle and coadjoint orbits of a lie group are considered for the natural poisson brackets the symplectic leaves in these manifolds are classified and the corresponding symplectic forms are described thus the construction of the kirillov symplectic form is generalized for lie poisson groups

variational calculus with constraints on general algebroids variational calculus on a vector bundle e equipped with a structure of a general algebroid is developed together with the corresponding analogs of euler lagrange equations constrained systems are introduced in the variational and geometrical settings the constrained euler lagrange equations are derived for analogs of holonomic vakonomic and nonholonomic constraints this general model covers the majority of first order lagrangian systems which are present in the literature and reduces to the standard variational calculus and the euler lagrange equations in classical mechanics for e tm

stokes phenomenon gelfand zeitlin systems and relative ginzburg weinstein linearization abstract in 2007 alekseev meinrenken proved that there exists a ginzburg weinstein diffeomorphism from the dual lie algebra u n to the dual poisson lie group u n compatible with the gelfand zeitlin integrable systems in this paper we explicitly construct such diffeomorphisms via stokes phenomenon and boalch s dual exponential maps then we introduce a relative version of the ginzburg weinstein linearization motivated by irregular riemann hilbert correspondence and generalize the results of enriquez etingof marshall to this relative setting in particular we prove the connection matrix for a certain irregular riemann hilbert problem satisfies a relative gauge transformation equation of the alekseev meinrenken dynamical r matrices this gauge equation is then derived as the semiclassical limit of the relative drinfeld twist equation

output to state stability and detectability of nonlinear systems the notion of input to state stability iss has proved to be useful in nonlinear systems analysis this paper discusses a dual notion output to state stability oss a characterization is provided in terms of a dissipation inequality involving storage lyapunov functions combining iss and oss there results the notion of input output to state stability ioss which is also studied and related to the notion of detectability the existence of observers and output injection

formal symplectic groupoid of a deformation quantization we give a self contained algebraic description of a formal symplectic groupoid over a poisson manifold m to each natural star product on m we then associate a canonical formal symplectic groupoid over m finally we construct a unique formal symplectic groupoid with separation of variables over an arbitrary kahler poisson manifold

a state space approach to parameterization of stabilizing controllers for nonlinear systems a state space approach to the youla parameterization of stabilizing controllers for linear and nonlinear systems is suggested the stabilizing controllers or a class of stabilizing controllers for nonlinear systems are characterized as fractional transformations of stable parameters the main idea behind this approach is to decompose the output feedback stabilization problem into state feedback and state estimation problems the parameterized output feedback controllers have separation structures this machinery allows the parameterization of stabilizing controllers to be conducted directly in state space without using coprime factorization

quantized flag manifolds and irreducible representations we study irreducible representations of a certain quantization of the algebra of polynomial functions on a generalized flag manifold regarded as a real manifold all irreducible representations are classified for a subclass of flag manifolds containing in particular the irreducible compact hermitian symmetric spaces for this subclass it is shown that the irreducible representations are parametrized by the symplectic leaves of the underlying poisson bracket we also discuss the relation between the quantized flag manifolds studied in this paper and the quantum flag manifolds studied by soibel man lakshimibai reshetikhin juraco s ovicek and korogodsky

double bracket equations and geodesic flows on symmetric spaces in this paper we consider the geometry of hamiltonian flows on the cotangent bundle of coadjoint orbits of compact lie groups and on symmetric spaces a key idea here is the use of the normal metric to define the kinetic energy this leads to hamiltonian flows of the double bracket type we analyze the integrability of geodesic flows according to the method of thimm we obtain via the double bracket formalism a quite explicit form of the relevant commuting flows and a correspondingly transparent proof of involutivity we demonstrate for example integrability of the geodesic flow on the real and complex grassmannians we also consider right invariant systems and the generalized rigid body equations in this setting

symplectic double groupoids over poisson ax b groups first we classify all the multiplicative poisson structures on the ax b group and determine their dual poisson lie groups next we show the existence of symplectic groupoid over the poisson ax b group finally by the hamilton jacobi method we construct nontrivial symplectic double groupoids and conclude that for each pair of nondegenerate multiplicative poisson structures of the ax b group there exists a symplectic double groupoid

quantum dressing orbits on compact groups the quantum double is shown to imply the dressing transformation on quantum compact groups and the quantum iwasawa decompositon in the general case quantum dressing orbits are described explicitly as algebras the dual coalgebras consisting of differential operators are related to the quantum weyl elements besides the differential geometry on a quantum leaf allows a remarkably simple construction of irreducible representations of the algebras of quantum functions representation spaces then consist of analytic functions on classical phase spaces these representations are also interpreted in the framework of quantization in the spirit of berezin applied to symplectic leaves on classical compact groups convenient coherent states are introduced and a correspondence between classical and quantum observables is given

lie poisson structure on some poisson lie groups poisson lie groups appeared in the work of drinfel d see e g drl dr2 as classical objects corresponding to quantum groups going in the other direction we may say that a poisson lie group is a group of symmetries of a phase space that are allowed to twist in a certain sense the symplectic or poisson structure the poisson structure on the group controls this twisting in a precise way quantizing both the phase space and the symmetries one may obtain a quantum group acting on a quantum phase space in recent work lu and ratiu lr used so called standard poisson structures on a compact semisimple lie group k and on its poisson dual k in order to give a new proof of the nonlinear convexity theorem of kostant ko their method is analogous to the famous symplectic proof of the linear convexity theorem given by atiyah a and guillemin and stemnberg gs the nonlinear convexity theorem like the linear one follows from a very general result on convexity of the image of the momentum map a gs however in lr it is applied not to a coadjoint orbit in t but to a symplectic leaf in k the main result of this paper is that the standard poisson structure on the poisson dual k to a compact semisimple poisson lie group k is actually isomorphic to the linear one on t this theorem seems to be related to some facts in the theory of quantum groups namely for generic q the universal enveloping algebra u t and its quantum deformation uq t are isomorphic as algebras though not as coalgebras of course since the quantum version is not cocommutative in particular there is a bijective correspondence between their representations a direct connection between our work and its quantum analogues though is still to be found the present work supplies a positive answer to question 5 1 in lr and strongly depends on that paper to simplify reading we keep the notations of lr wherever possible on one hand but give all necessary definitions on the other our work is also related to du in which the nonlinear convexity theorem is reduced to the linear one by a deformation argument not unlike the one that we use in 5 the paper is organized as follows in 2 we define poisson lie groups discuss

quantized algebras of functions on homogeneous spaces with poisson stabilizers let g be a simply connected semisimple compact lie group with standard poisson structure k a closed poisson lie subgroup 0 q 1 we study a quantization c gq kq of the algebra of continuous functions on g k using results of soibelman and dijkhuizen stokman we classify the irreducible representations of c gq kq and obtain a composition series for c gq kq we describe closures of the symplectic leaves of g k refining the well known description in the case of flag manifolds in terms of the bruhat order we then show that the same rules describe the topology on the spectrum of c gq kq next we show that the family of c algebras c gq kq 0 q 1 has a canonical structure of a continuous field of c algebras and provides a strict deformation quantization of the poisson algebra mathbb c g k finally extending a result of nagy we show that c gq kq is canonically kk equivalent to c g k

local network community detection with continuous optimization of conductance and weighted kernel k means local network community detection is the task of finding a single community of nodes concentrated around few given seed nodes in a localized way conductance is a popular objective function used in many algorithms for local community detection this paper studies a continuous relaxation of conductance we show that continuous optimization of this objective still leads to discrete communities we investigate the relation of conductance with weighted kernel k means for a single community which leads to the introduction of a new objective function conductance conductance is obtained by setting to 0 two algorithms emc and pgdc are proposed to locally optimize conductance and automatically tune the parameter they are based on expectation maximization and projected gradient descent respectively we prove locality and give performance guarantees for emc and pgdc for a class of dense and well separated communities centered around the seeds experiments are conducted on networks with ground truth communities comparing to state of the art graph diffusion algorithms for conductance optimization on large graphs results indicate that emc and pgdc stay localized and produce communities most similar to the ground while graph diffusion algorithms generate large communities of lower quality

a class of globally stabilizing controllers for nonlinear systems abstract a class of globally stabilizing controllers for nonlinear systems are parametrized as fractional transformations on some input to state stable parameters

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 and classical pseudogroups part i union pseudogroups and their quantization union pseudogroups structures analogical to pseudogroups in the sense of 1 are defined using the category dual to the category of groupoids instead of the category of pseudospaces in the sense of 2 it is shown that these structures are equivalent to double groups in the sense of 3 moreover it is shown that a quantization procedure associates with each finite union pseudogroup a quantum pseudogroup therefore for each finite double group there is a finite pseudogroup

a local spectral method for graphs with applications to improving graph partitions and exploring data graphs locally the second eigenvalue of the laplacian matrix and its associated eigenvector are fundamental features of an undirected graph and as such they have found widespread use in scientific computing machine learning and data analysis in many applications however graphs that arise have several local regions of interest and the second eigenvector will typically fail to provide information fine tuned to each local region in this paper we introduce a locally biased analogue of the second eigenvector and we demonstrate its usefulness at highlighting local properties of data graphs in a semi supervised manner to do so we first view the second eigenvector as the solution to a constrained optimization problem and we incorporate the local information as an additional constraint we then characterize the optimal solution to this new problem and show that it can be interpreted as a generalization of a personalized pagerank vector and finally as a consequence we show that the solution can be computed in nearly linear time in addition we show that this locally biased vector can be used to compute an approximation to the best partition near an input seed set in a manner analogous to the way in which the second eigenvector of the laplacian can be used to obtain an approximation to the best partition in the entire input graph such a primitive is useful for identifying and refining clusters locally as it allows us to focus on a local region of interest in a semi supervised manner finally we provide a detailed empirical evaluation of our method by showing how it can applied to finding locally biased sparse cuts around an input vertex seed set in social and information networks

deformation quantization of heisenberg manifolds form a smooth manifold equipped with a poisson bracket we formulate ac algebra framework for deformation quantization including the possibility of invariance under a lie group of diffeomorphisms preserving the poisson bracket we then show that the much studied non commutative tori give examples of such deformation quantizations invariant under the usual action of ordinary tori going beyond this the main results of the paper provide a construction of invariant deformation quantizations for those poisson brackets on heisenberg manifolds which are invariant under the action of the heisenberg lie group and for various generalizations suggested by this class of examples interesting examples are obtained of simplec algebras on which the heisenberg group acts ergodically

sontag s input to state stability condition and global stabilization using state detection this paper deals with the global stabilization problem for nonlinear system using state detection sufficient algebraic conditions for stabilization and detectability concerning particular classes of systems are included these conditions are based to a version of a lyapunov like condition proposed by sontag which guarantees input to state stability

ginzburg weinstein via gelfand zeitlin let u n be the unitary group and u n the dual of its lie al gebra equipped with the kirillov poisson structure in their 1983 paper guillemin sternberg introduced a densely defined hamil tonian action of a torus of dimension n 1 n 2 on u n with mo ment map given by the gelfand zeitlin coordinates a few years later flaschka ratiu described a similar multiplicative gelfand zeitlin system for the poisson lie group u n by the ginzburg weinstein theorem u n is isomorphic to u n as a poisson manifold flaschka ratiu conjectured that one can choose the ginzburg weinstein diffeomorphism in such a way that it intertwines the linear and nonlinear gelfand zeitlin systems our main result gives a proof of this conjecture and produces a canonical ginzburg weinstein diffeomorphism

a fast parametric maximum flow algorithm and applications the classical maximum flow problem sometimes occurs in settings in which the arc capacities are not fixed but are functions of a single parameter and the goal is to find the value of the parameter such that the corresponding maximum flow or minimum cut satisfies some side condition finding the desired parameter value requires solving a sequence of related maximum flow problems in this paper it is shown that the recent maximum flow algorithm of goldberg and tarjan can be extended to solve an important class of such parametric maximum flow problems at the cost of only a constant factor in its worst case time bound faster algorithms for a variety of combinatorial optimization problems follow from the result

a family of poisson structures on hermitian symmetric spaces we investigate the compatibility of symplectic kirillov kostant souriau structure and poisson lie structure on coadjoint orbits of semisimple lie group we prove that they are compatible for an orbit compact lie group iff the orbit is hermitian symmetric space we prove also the compatibility statement for non compact hermitian symmetric space as an example we describe a structure of symplectic leaves oncp n for this family these leaves may be considered as a perturbation of schubert cells possible applications to infinite dimensional examples are discussed