dirac geometry quasi poisson actions and d g valued moment maps we study dirac structures associated with manin pairs d g and give a dirac geometric approach to hamiltonian spaces with d g valued moment maps originally introduced by alekseev and kosmann schwarzbach 3 in terms of quasi poisson structures we explain how these two distinct frameworks are related to each other proving that they lead to isomorphic categories of hamiltonian spaces we stress the connection between the viewpoint of dirac geometry and equivariant differential forms the paper discusses various examples including q hamiltonian spaces and poisson lie group actions explaining how presymplectic groupoids are related to the notion of double in each context

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

lie bialgebroids and poisson groupoids lie bialgebras arise as infinitesimal invariants of poisson lie groups a lie bialgebra is a lie algebra g with a lie algebra structure on the dual g which is compatible with the lie algebra g in a certain sense for a poisson group g the multiplicative poisson structure induces a lie algebra structure on the lie algebra dual g which makes g g into a lie bialgebra in fact there is a one one correspondence between poisson lie groups and lie bialgebras if the lie groups are assumed to be simply connected 7 16 19 the importance of poisson lie groups themselves arises in part from their role as classical limits of quantum groups 8 and in part because they provide a class of poisson structures for which the realization problem is tractable 15 poisson groupoids were introduced by weinstein 24 as a generalization of both poisson lie groups and the symplectic groupoids which arise in the integration of arbitrary poisson manifolds 4 11 he noted that the lie algebroid dual a g of a poisson groupoid g itself has a lie algebroid structure but did not develop the infinitesimal structure further in this paper we introduce and study a natural infinitesimal invariant for poisson groupoids the lie bialgebroids of the title our ultimate purpose is to develop a lie theory for poisson groupoids parallel to that for poisson groups in this paper we are primarily concerned with the first half of this process that is with the construction of the lie bialgebroid of a poisson groupoid after the preliminary 2 in which we describe the generalization to arbitrary lie algebroids of the exterior calculus and schouten calculus in 3 we define a lie bialgebroid to be a lie algebroid a whose dual a is also equipped with a lie algebroid structure such that the coboundary operator d a a associated to a satisfies a cocycle equation with respect to a the lie algebra of sections of a this is clearly a straightforward extension of the concept of a lie bialgebra 16 but cannot be formalized in lie algebroid cohomological terms since there is no satisfactory adjoint representation for a general lie algebroid most of 3 is devoted to proving that this definition is self dual if a a is a lie bialgebroid then a a is also in 4 we briefly consider the special case of lie bialgebroids satisfying a triangularity condition which include some important examples such as the usual triangular lie bialgebras and lie bialgebroids associated to poisson manifolds the techniques used in 2 4 are similar to those known for lie bialgebras it would be possible by suitably generalizing the proof for poisson groups to prove

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

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

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

artin level modules we introduce level modules and show that these form a natural class of modules over a polynomial ring we prove that there exist compressed level modules i e level modules with the expected maximal hilbert function given socle type and the number of generators we also show how to use the theory of level modules to compute minimal free resolutions of the coordinate ring of points from the back which reveals new examples where random sets of points fail to satisfy the minimal resolution conjecture

on the cohen macaulay type of s lines in an 1 abstract let a be the coordinate ring of s lines through the origin in a n 1 k we discuss what it means for these lines to be in sufficiently general position and then with this restriction on the lines we attempt to verify our belief that the cohen macaulay type of a depends only on s and n we characterize those s and n for which a is a gorenstein ring i e of c m type 1 and explicitly calculate the cohen macaulay type in several other cases we then extend some of our results to the case of an arbitrary reduced curve whose tangent cone consists of lines in sufficiently general position finally we calculate the hubert samuel polynomials of the curves we have been considering

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

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

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

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

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

poly poisson sigma models and their relational poly symplectic groupoids the main idea of this paper is to describe the integration procedure for poly poisson structures that is to find a poly symplectic groupoid integrating a poly poisson structure in terms of topological field theories namely via the path space construction this will be given in terms of the poly poisson sigma model and we prove that every poly poisson structure has a natural integration via relational poly symplectic groupoids extending known results for poisson structures we provide familiar examples trivial linear constant and symplectic within this formulation and give some applications of this construction regarding the classification of poly symplectic integrations as well as morita equivalence of poly poisson manifolds the main idea of this paper is to describe the integration procedure for poly poisson structures that is to find a poly symplectic groupoid integrating a poly poisson structure in terms of topological field theories namely via the path space construction this will be given in terms of the poly poisson sigma model and we prove that every poly poisson structure has a natural integration via relational poly symplectic groupoids extending known results for poisson structures we provide familiar examples trivial linear constant and symplectic within this formulation and give some applications of this construction regarding the classification of poly symplectic integrations as well as morita equivalence of poly poisson manifolds

multiplicative forms at the infinitesimal level we describe arbitrary multiplicative differential forms on lie groupoids infinitesimally i e in terms of lie algebroid data this description is based on the study of linear differential forms on lie algebroids and encompasses many known integration results related to poisson geometry we also revisit multiplicative multivector fields and their infinitesimal counterparts drawing a parallel between the two theories

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

integrability of lie brackets in this paper we present the solution to a longstanding problem of differential geometry lie s third theorem for lie algebroids we show that the integrability problem is controlled by two computable obstructions as applications we derive explain and improve the known integrability results we establish integrability by local lie groupoids we clarify the smoothness of the poisson sigma model for poisson manifolds and we describe other geometrical applications

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

on meromorphic vector fields on projective spaces fix integers n and r with n 2 and r 0 a section s e ho pn tpn r was called gk a meromorphic vector field of degree r on pn let z s 0 be its zero locus s was called nondegenerate if dim z 0 and z is reduced i e if s has only isolated zeroes and each zero has multiplicity one it was proved in gk that a nondegenerate meromorphic vector field is uniquely determined by its zero locus and that its zero locus is stable in the sense of mu set x n r cn tpn r r 1 n l 1 r the last equality following from the euler s sequence it is a natural problem to know if for a given integer y x n r and a given set z of y points of pn there is a nondegenerate meromorphic vector field of degree n vanishing on z we can ask also for the dimension of such vector fields we work over an algebraically closed field k but of course the interesting case is when k is the complex number field for a real number u let u be its integral part set w n r h0 pn tpn r 1 n note that by the euler s sequence we have ho pn tpn r n 1 n r 1 n r 1 n r n r here are the main results of this paper

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

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

tangent lifts of poisson and related structures the derivation dt on the exterior algebra of forms on a manifold m with values in the exterior algebra of forms on the tangent bundle tm is extended to multivector fields these tangent lifts are studied with application to the theory of poisson structures their symplectic foliations canonical vector fields and poisson lie groups

reduction of polysymplectic manifolds the aim of this paper is to generalize the classical marsden weinstein reduction procedure for symplectic manifolds to polysymplectic manifolds in order to obtain quotient manifolds which inherit the polysymplectic structure this generalization allows us to reduce polysymplectic hamiltonian systems with symmetries such as those appearing in certain kinds of classical field theories as an application of this technique an analogue to the kirillov kostant souriau theorem for polysymplectic manifolds is obtained and some other mathematical examples are also analyzed our procedure corrects some mistakes and inaccuracies in previous papers gunther 1987 j differ geom 25 23 53 munteanu et al 2004 j math phys 45 1730 51 on this subject

on the resolution of certain graded algebras let a r i be a graded algebra over the polynomial ring r k x 0 x n some properties of the numerical invariants in a minimal free resolution of a are discussed in the case a is a short graded algebra when a is the homogeneous coordinate ring of a set of points in generic position in the projective space several result are obtained on the line traced by some conjectures proposed by green and lazarsfeld in gl and lorenzini in l1

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

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

lie algebroid structures on double vector bundles and representation theory of lie algebroids abstract a vb algebroid is essentially defined as a lie algebroid object in the category of vector bundles there is a one to one correspondence between vb algebroids and certain flat lie algebroid superconnections up to a natural notion of equivalence in this setting we are able to construct characteristic classes which in special cases reproduce characteristic classes constructed by crainic and fernandes we give a complete classification of regular vb algebroids and in the process we obtain another characteristic class of lie algebroids that does not appear in the ordinary representation theory of lie algebroids

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

riemannian metrics on lie groupoids we introduce a notion of metric on a lie groupoid compatible with multipli cation and we study its properties we show that many families of lie groupoids admit such metrics including the important class of proper lie groupoids the exponential map of these metrics allows us to establish a linearization theorem for riemannian groupoids obtaining both a simpler proof and a stronger version of the weinstein zung linearization theorem for proper lie groupoids this new notion of metric has a simplicial nature which will be explored in future papers of this series

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

low regularity global well posedness for the zakharov and klein gordon schrodinger systems we prove low regularity global well posedness for the id zakharov system and the 3d klein gordon schrodinger system which are systems in two variables u r d x x r t c and n r d x x r t r the zakharov system is known to be locally well posed in u n l 2 x h and the klein gordon schrodinger system is known to be locally well posed in u n l 2 x l 2 here we show that the zakharov and klein gordon schrodinger systems are globally well posed in these spaces respectively by using an available conservation law for the l 2 norm of u and controlling the growth of n via the estimates in the local theory

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

on the nonlinear schrodinger limit of the magnetic zakharov system abstract in this paper we study the limit behavior of a smooth solution for the magnetic type zakharov system as the parameters tend to infinity we prove that solutions of the magnetic zakharov system converge to the solution of the nonlinear schrodinger equation moreover the detailed convergence rate is also investigated

decay estimates for a class of wave equations abstract in this paper we use a unified way studying the decay estimate for a class of dispersive semigroup given by e i t where r r is smooth away from the origin especially the decay estimates for the solutions of the klein gordon equation and the beam equation are simplified and slightly improved

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

on the cauchy problem for the zakharov system abstract we study the local cauchy problem in time for the zakharov system 1 1 and 1 2 governing langmuir turbulence with initial data u 0 n 0 t n 0 h k h lscr h l 1 in arbitrary space dimension we define a natural notion of criticality according to which the critical values of k l are 2 3 2 2 2 using a method recently developed by bourgain we prove that the zakharov system is locally well posed for a variety of values of k l the results cover the whole subcritical range for 4 for 3 they cover only part of it and the lowest admissible values are k l 1 2 0 for 2 3 and k l 0 1 2 for 1 as a by product of the one dimensional result we prove well posedness of the benney system 1 14 and 1 15 governing the interaction of short and long waves for the same values of k l

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

holomorphic jacobi manifolds and holomorphic contact groupoids this paper belongs to a series of works aiming at exploring generalized complex geometry in odd dimensions holomorphic jacobi manifolds were introduced and studied by the authors in a separate paper as special cases of generalized contact bundles in fact generalized contact bundles are nothing but odd dimensional analogues of generalized complex manifolds in the present paper we solve the integration problem for holomorphic jacobi manifolds by proving that they integrate to holomorphic contact groupoids a crucial tool in our proof is what we call the homogenization scheme which allows us to identify holomorphic jacobi manifolds with homogeneous holomorphic poisson manifolds and holomorphic contact groupoids with homogeneous complex symplectic groupoids

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

integration of twisted dirac brackets given a lie groupoid g over a manifold m we show that multiplicative 2 forms on g relatively closed with respect to a closed 3 form phi on m correspond to maps from the lie algebroid of g into t m satisfying an algebraic condition and a differential condition with respect to the phi twisted courant bracket this correspondence describes as a special case the global objects associated to phi twisted dirac structures as applications we relate our results to equivariant cohomology and foliation theory and we give a new description of quasi hamiltonian spaces and group valued momentum maps

vector bundles over lie groupoids and algebroids abstract we study vb groupoids and vb algebroids which are vector bundles in the realm of lie groupoids and lie algebroids through a suitable reformulation of their definitions we elucidate the lie theory relating these objects i e their relation via differentiation and integration we also show how to extend our techniques to describe the more general lie theory underlying double lie algebroids and la groupoids

equations of langmuir turbulence and nonlinear schrodinger equation smoothness and approximation we consider the following family of systems parametrized by e 0 which describe langmuir s turbulence in the case where initial data are sufficiently small we study the asymptotic behaviour of the solutions ee ne when e goes to zero for d 1 2 3 namely we state convergence results of ee ne to the couple e e 2 where e is the solution of the nonlinear schrodinger equation moreover in three dimensions we show that the smooth solution ee ne is defined on 0 tmax e with tmax e going to infinity as e goes to 0

energy convergence for singular limits of zakharov type systems we prove existence and uniqueness of solutions to the klein gordon zakharov system in the energy space h1 l2 on some time interval which is uniform with respect to two large parameters c and these two parameters correspond to the plasma frequency and the sound speed in the simultaneous high frequency and subsonic limit we recover the nonlinear schrodinger system at the limit we are also able to say more when we take the limits separately

multiplicative forms and spencer operators motivated by our attempt to recast cartan s work on lie pseudogroups in a more global and modern language we are brought back to the question of understanding the linearization of multiplicative forms on groupoids and the corresponding integrability problem from this point of view the novelty of this paper is that we study forms with coefficients however the main contribution of this paper is conceptual the discovery of the relationship between multiplicative forms and cartan s work is explored here to provide a completely new approach to integrability theorems for multiplicative forms the multiplicative point of view shows that modulo lie s functor the cartan pfaffian system itself a multiplicative form with coefficients is the same thing as the classical spencer operator

symplectic groupoids and poisson manifolds on definit la notion de groupoide symplectique on donne la variete de poisson de ce groupoide symplectique on presente une construction

well posedness of the initial value problem for the korteweg de vries equation 1 1 itu axu u1xu o x t e r u x 0 uo x the kdv equation which was first derived as a model for unidirectional propagation of nonlinear dispersive long waves 21 has been considered in different contexts namely in its relation with the inverse scattering method in plasma physics and in algebraic geometry see 24 and references therein our purpose is to study local and global well posedness of the ivp 1 1 in classical sobolev spaces hs r we shall say that the ivp 1 1 is locally resp globally well posed in the function space x if it induces a dynamical system on x by generating a continuous local resp global flow it was established in the works of bona and smith 3 bona and scott 2 saut and temam 30 and kato 1 5 that the ivp 1 1 is locally resp globally well posed in hs with s 3 2 resp s 2 roughly speaking global well posedness in hs depends on the available local theory and on the conservation laws satisfied by solutions of 1 1 namely

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

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 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