quantization of poisson lie groups and applications letg be a connected poisson lie group we discuss aspects of the question of drinfel d can g be quantized and give some answers wheng is semisimple a case where the answer isyes we introduce quantizable poisson subalgebras ofc g related to harmonic analysis ong they are a generalization of f r t models of quantum groups and provide new examples of quantized poisson algebras

quantum algebras as quantizations of dual poisson lie groups a systematic computational approach for the explicit construction of any quantum hopf algebra uz g z starting from the lie bialgebra g that gives the first order deformation of the coproduct map z is presented the procedure is based on the well known quantum duality principle namely the fact that any quantum algebra can be viewed as the quantization of the unique poisson lie structure g g on the dual group g which is obtained by exponentiating the lie algebra g defined by the dual map from this perspective the coproduct for uz g is just the pull back of the group law for g and the poisson analogues of the quantum commutation rules for uz g are given by the unique poisson lie structure g on g whose linearization is the poisson analogue of the initial lie algebra g this approach is shown to be a very useful technical tool in order to solve the lie bialgebra quantization problem explicitly since once a lie bialgebra g is given the full dual poisson lie group g can be obtained either by applying standard poisson lie group techniques or by implementing the algorithm presented here with the aid of symbolic manipulation programs as a consequence the quantization of g will give rise to the full uz g quantum algebra provided that ordering problems are appropriately fixed through the choice of certain local coordinates on g whose coproduct fulfils a precise quantum symmetry property the applicability of this approach is explicitly demonstrated by reviewing the construction of several instances of quantum deformations of physically relevant lie algebras such as the 2 1 anti de sitter algebra so 2 2 and the poincare algebra in 3 1 dimensions

deformations of lie group and lie algebra representations a complete study of deformations of lie group and lie algebra representations including differentiability and integrability results is given adapted results are given in the semisimple case a notion of induced deformations is introduced various examples are given including deformations of indecomposable representations

basic twist quantization of the exceptional lie algebra g2 we present the formulas for twist quantization of g2 corresponding to the solution of a classical yb equation with support in the eight dimensional borel subalgebra of g2 the considered chain of twists consists of the four factors describing the four steps of quantization jordanian twist the two twist factors extending jordanian twist and the deformed jordanian or in a second variant additional abelian twist the first two steps describe as well the sl 3 quantization the coproducts are calculated for each step in explicit form and for that purpose we present new formulas for the calculation of similarity transformations on the tensor product we introduce new basic generators in universal enveloping algebra u g2 that provide nonlinearities in the algebraic sector maximally simplifying the deformed coproducts

peripheric extended twists the properties of the set of extended jordanian twists are studied it is shown that the boundaries of contain twists whose characteristics differ considerably from those of internal points the extension multipliers of these peripheric twists are factorizable this leads to simplifications in the twisted algebra relations and helps to find the explicit form for coproducts the peripheric twisted algebra u sl 4 is obtained to illustrate the construction it is shown that the corresponding deformation up sl 4 cannot be connected with the drinfeld jimbo one by a smooth limit procedure all the carrier algebras for the extended and the peripheric extended twists are proved to be frobenius

deformations and contractions of lie algebras in this contributed presentation we discuss and compare the mutually opposite procedures of deformations and contractions of lie algebras we suggest that with approp riate combinations of both procedures one may construct new lie algebras we first discuss low dimensional lie algebras and illustrate thereby that whereas for every contraction there exists a reverse deformation the converse is not true in general also we note that some lie algebras belonging to parameterized families are singled out by the irreversibility of defor mations and contractions after reminding that global deformations of the witt virasoro and affine kac moody algebras allow one to retrieve lie algebras of krichever novikov type we contract the latter to find new infinite dimensional lie algebras

chains of frobenius subalgebras of so m and the corresponding twists chains of extended jordanian twists are studied for the universal enveloping algebras u so m the carrier subalgebra of a canonical chain fb0 pmax cannot cover the maximal nilpotent subalgebra n so m we demonstrate that there exist other types of frobenius subalgebras in so m that can be large enough to include n so m the problem is that the canonical chains fb0 p do not preserve the primitivity on these new carrier spaces we show that this difficulty can be overcome and the primitivity can be restored if one changes the basis and passes to the deformed carrier spaces finally the twisting elements for the new frobenius subalgebras are explicitly constructed this gives rise to a new family of universal r matrices for orthogonal algebras for a special case of g so 5 and its defining representation we present the corresponding matrix solution of the yang baxter equation

random rates in anisotropic regression in the context of minimax theory we propose a new kind of risk normalized by a random variable measurable with respect to the data we present a notion of optimality and a method to construct optimal procedures accordingly we apply this general setup to the problem of selecting significant variables in gaussian white noise in particular we show that our method essentially improves the accuracy of estimation in the sense of giving explicit improved confidence sets in l2 norm links to adaptive estimation are discussed 1 introduction searching for significant variables is certainly one of the oldest and most popular problems in statistics one of the simplest models where the issue of selecting significant variables was first stated mathematically is linear regression a vast literature has been devoted to this topic since and different approaches have been proposed over the last forty years both for estimation and for hypothesis testing among many authors we refer to akaike 1 breiman and freedman 3 chernoff 5 csiszar and korner 6 dychakov 10 patel 42 renyi 46 freidlina 13 meshalkin 35 malyutov and tsitovich 34 schwarz 47 and stone 48 in classical parametric regression if we consider a linear model we first have to measure the possible gain of searching for a limited number of significant variables if the model comes from a specific field of application then only an adequate description together with its solution is relevant however from a mathematical point of view a theory of selecting significant variables does not lead at least asymptotically to a substantial improvement of the accuracy of estimation in a regular parametric model the classical n rate of convergence is not affected by the number of significant variables however even in this setup let us emphasize that asymptotically has to be understood as up to a constant and that the correct choice of significant variables may possibly improve this constant if instead of a linear model we consider a nonparametric regression model the search for significant variables becomes crucial for estimating the regression function the rate of convergence explicitly depends on the set of significant variables let us develop this statement with the following example of multivariate regression suppose we observe z n xi y i i 1 n in the model

strongly pseudoconvex cr structures over small balls part i an a priori estimate the purpose of the present series of papers is to develop the theory of harmonic integrals on strongly pseudoconvex cr structures over small balls along the line developed by d c spencer c b morrey j j kohn and l nirenberg 5 6 7 8 for strongly pseudoconvex complex manifolds we consider a strongly pseudoconvex cr structure on a manifold of real dimension 2n 1 in part i we establish the a priori estimate for the neumann boundary value problem on the complex associated with the structure in the case the structure is induced by an embedding in c and restricted to a small ball of special type provided i c q c n 3 where q is the degree of differential forms we consider in part ii we derive the regularity theorem of solutions of the neumann boundary value problem based on the a priori estimate of part i as an application we prove in part il that when n 5 the structure is realized on a neighborhood of a reference point by an embedding in cn we note here that l nirenberg gave an example of a strongly pseudoconvex cr structure which cannot be induced by such an embedding when n 2 9 l boutet de monvel proved that in the case of such structures over a compact manifold we can find such an embedding when n 3 1 there are two major differences between the cases of strongly pseudoconvex integrable almost complex structures and cr structures firstly the principal parts of the associated laplacians in cr structures are no longer elliptic secondly we have characteristic points on the boundary as a result when we follow over the predecessor s calculation in our case we end up with terms which may not be absorbed in the main non negative terms fortunately these leftover terms depend mostly on the levi form of the cr structure and the hessian form with respect to the cr structure of the equation of the boundary of our

uqosp 2 2 lattice models in this paper i construct lattice models with an underlying uqosp 2 2 superalgebra symmetry i find new solutions to the graded yang baxter equation these trigonometric r matrices depend on three continuous parameters the spectral parameter the deformation parameter q and the u 1 parameter b of the superalgebra it must be emphasized that the parameter q is generic and the parameter b does not correspond to the nilpotency parameter of the rational limits are given they also depend on the u 1 parameter and this dependence cannot be rescaled away i give the bethe ansatz solution of the lattice models built from some of these r matrices while for other matrices due to the particular nature of the representation theory of osp 2 2 i conjecture the result the parameter b appears as a continuous generalized spin finally i briefly discuss the problem of finding the ground state of these models

general twisting of algebras abstract we introduce the concept of pseudotwistor with particular cases called twistor and braided twistor for an algebra a u in a monoidal category as a morphism t a a a a satisfying a list of axioms ensuring that a t u is also an algebra in the category this concept provides a unifying framework for various deformed or twisted algebras from the literature such as twisted tensor products of algebras twisted bialgebras and algebras endowed with fedosov products pseudotwistors appear also in other topics from the literature e g durdevich s braided quantum groups and ribbon algebras we also focus on the effect of twistors on the universal first order differential calculus as well as on lifting twistors to braided twistors on the algebra of universal differential forms

the quantum group structure of 2d gravity and minimal models i on the unit circle an infinite family of chiral operators is constructed whose exchange algebra is given by the universalr matrix of the quantum groupsl 2 q this establishes the precise connection between the chiral algebra of two dimensional gravity or minimal models and this quantum group the method is to relate the monodromy properties of the operator differential equations satisfied by the generalized vertex operators with the exchange algebra ofsl 2 q the formulae so derived which generalize an earlier particular case worked out by babelon are remarkably compact and may be entirely written in terms of q deformed factorials and binomial coefficients

pseudo kahler quantization on flag manifolds a unified approach to geometric symbol and deformation quantizations on a generalized flag manifold endowed with an invariant pseudo kahler structure is proposed in particular cases we arrive at berezin s quantization via covariant and contravariant symbols

quantum deformation of lorentz group a one parameter quantum deformations l 2 ofsl 2 is introduced and investigated an analog of the iwasawa decomposition is proved the compact part of this decomposition coincides withs u 2 whereas the solvable part is identified as a pontryagin dual ofs u 2 it shows thats l 2 is the result of the dual version of drinfeld s double group construction applied tos u 2 the same construction applied to any compact quantum groupgc is discussed in detail in particular the explicit formulae for the haar measures on the pontryagin dualgd ofgc and on the double groupg are given we show that there exists remarkable 1 1 correspondence between representations ofg and bicovariant bimodules tensor bundles overgc the theory of smooth representations ofs l 2 is the same as that ofsl 2 clebsh gordon coefficients are however modified the corresponding tame bicovariant bimodules ons u 2 are classified an application to 4d differential calculus is presented the nonsmooth case is also discussed

non realizable cr structures the examples given by nirenberg in 5 and 6 show that not all three dimensional cr manifolds may be realized as real hypersurfaces in 1172 these examples are perturbations of the cr structure on the sphere in this paper we extend his construction and show that any strictly pseudo convex realizable cr structure of dimension 3 may be perturbed so as to obtain a non realizable one this is done in sect 1 in sect 2 we show non realizability for higher dimensional structures provided the levi form is non degenerate and has only one positive eigenvalue in our last section we summarize cartan s approach to cr geometry from this viewpoint it was important to decide if nirenberg s example was really limited to perturbations of the sphere we have made this paper self contained and have provided proofs of several known results for instance lemma 1 l and the first part of lemma 1 5 were proved by lewy 2 and lemma 1 2 is the well known observation that a strictly pseudo convex hypersurface is convex in some coordinate system

non standard quantum 1 1 poincare group a t matrix approach the hopf algebra dual form for the non standard uniparametric deformation of the 1 1 poincare algebra iso 1 1 is deduced in this framework the quantum coordinates that generate funw iso 1 1 define an infinite dimensional lie algebra the t matrix formalism is used to derive a universal r matrix for both uwiso 1 1 and funw iso 1 1 it is also shown how these results can be generalized for the triangular deformations of 1 1 poincare and galilei algebras that include a spacetime dilation generator

classification of quantum groups and lie bialgebra structures on sl n f relations with brauer group given an arbitrary field f of characteristic 0 we study lie bialgebra structures on sl n f based on the description of the corresponding classical double for any lie bialgebra structure 5 the classical double d sl n f delta is isomorphic to sl n f circle times f a where a is either f epsilon with epsilon 2 0 or f circle plus f or a quadratic field extension of f in the first case the classification leads to quasi frobenius lie subalgebras of sl n f in the second and third cases a belavin drinfeld cohomology can be introduced which enables one to classify lie bialgebras on sl n f up to gauge equivalence the belavin drinfeld untwisted and twisted cohomology sets associated to an r matrix are computed for the cremmer gervais r matrix in sl 3 we also construct a natural map of sets between the total belavin drinfeld twisted cohomology set and the brauer group of the field f

the maschke theorem for l r smash products in this paper we give a sufficient and necessary condition for generalized l r smash products or generalized diagonal crossed products to be bialgebras hopf algebras and mainly give a maschke theorem for l r smash products and hence obtain maschke theorems for twisted smash products and diagonal crossed products

hom l r smash products hom diagonal crossed products and the drinfeld double of a hom hopf algebra abstract we introduce the hom analogue of the l r smash product and use it to define the hom analogue of the diagonal crossed product when h is a finite dimensional hom hopf algebra with bijective antipode and bijective structure map we define the drinfeld double of h its algebra structure is a hom diagonal crossed product and it has all expected properties namely it is quasitriangular and modules over it coincide with left right yetter drinfeld modules over h

extended jordanian twists for lie algebras jordanian quantizations of lie algebras are studied using the factorizable twists for a restricted borel subalgebra b of sl n the explicit expressions are obtained for the twist element f universal r matrix and the corresponding canonical element t it is shown that the twisted hopf algebra uf b is self dual the cohomological properties of the involved lie bialgebras are studied to justify the existence of a contraction from the dinfeld jimbo quantization to the jordanian one the construction of the twist is generalized to a certain type of inhomogenious lie algebras

diagonal crossed products and l r smash products over weak hopf algebras the notions of the diagonal crossed product and the l r smash product over weak hopf algebras which can be seen as generalizations of smash products over weak hopf algebras are introduced and studied in particular the weak hopf algebraic structure of the drinfeld double of a finite dimensional weak hopf algebra which is a special case of diagonal crossed products is given moreover two canonical examples the kadison products and the connes moscovici products over weak hopf algebras are introduced and investigated

deformation of orthosymplectic lie superalgebra osp 1 4 triangular deformation of the orthosymplectic lie superalgebra osp 14 is defined by chains of twists the corresponding classical r matrix is obtained by a contraction procedure from the trigonometric r matrix the carrier space of the constant r matrix and the twist element is the borel subalgebra the known super jordanian twist of osp 12 is generalized as an extended twist to the orthosymplectic lie superalgebras of higher rank the dimension of the cobracket kernel is related to the number of generators with primitive twisted coproduct of the deformed algebra

berezin s quantization on flag manifolds and spherical modules we show that the theory of spherical harish chandra modules naturally gives rise to berezin s symbol quantization on generalized flag manifolds it provides constructions of symbol algebras and of their representations for covariant and contravariant symbols and also for symbols which so far have no explicit definition for all these symbol algebras we give a general proof of the correspondence principle introduction berezin s quantization on a symplectic manifold q see 1 is givexl by a family of associative algebras h whose elements are smooth functions on q together with their representations in hilbert spaces hh where the parameter h plays the role of the planck constant the rnultiplication h in ah must satisfy the so called correspondence principle as h o a formal definition is given in section 1 representation of the algebra ah in hh maps a function h c ah to the operator a in hh in that case the function h is called a symbol of the operator f the concrete examples of quantizatiorls are usually based on explicit collstructions of symbols see 1 2 in this paper we show how the theory of spherical harish chazdra modules naturally provides the algebras of covariant arld corltravariant syrrlbols on generalized flag manifolds moreover it also gives rise to the algebras of symbols which so far have no general construction or definition as in the case of covariant and contravariant symbols for all these symbol algebras we give a general proof of the correspondence principle we also prove the conjecture from 3 on rational dependence of the product c h g on h on an arbitrary generalized flag manifold 1 teie definition of qijantization we will give now a formal definition of quantization on a syrrlplectic manifold q let f be a set of positive numbers with a limit point 0 for each h c f let ah be an algebra which is a linear subspace of c q with a multiplication h and a given representation in a hilbert space hh define a uh ah and let a be dense in c q e these data d termine a quantization on q if for f g c and h o the correspondence principle holds l s h g fg h f h g g h f tf 9 qreceived by the editors december 14 1994 and in revised form june 14 1996 199l mathematics subject classification primary 58f06 secondary 22e47

the range of the tangential cauchy riemann operator over a small ball in this paper we study the closed range property of tangential cauchy riemann equations of an abstract cr structure over a small ball let d be the set x e r 1 1x1 4 kuranishi 141 and akahori 11 have studied the l theory

reconstruction of universal drinfeld twists from representations universal drinfeld twists are inner automorphisms which relate the coproduct of a quantum enveloping algebra to the coproduct of the undeformed enveloping algebra even though they govern the deformation theory of classical symmetries and have appeared in numerous applications no twist for a semisimple quantum enveloping algebra has ever been computed it is argued that universal twists can be reconstructed from their well known representations a method to reconstruct an arbitrary element of the enveloping algebra from its irreducible representations is developed for the twist this yields an algebra valued generating function to all orders in the deformation parameter expressed by a combination of basic and ordinary hypergeometric functions it is shown how the generating function can be expanded to the formal power series of the twist an explicit expression for the universal twist of su 2 is given up to third order

star exponentials for any ordering of the elements of the inhomogeneous symplectic lie algebra we compute for any ordering the star exponentials of all polynomials of degree not greater than two on the 2l dimensional phase space of a quantum system with l degrees of freedom and we show in the particular case of the moyal star product that the weyl transform of the moyal star exponential of the one dimensional harmonic oscillator hamiltonian is the evolution operator of this quantum system

rtt relations a modified braid equation and noncommutative planes with the known group relations for the elements a b c d of a quantum matrix t as input a general solution of the rtt relations is sought without imposing the yang baxter yb constraint for r or the braid equation for r pr for three biparametric deformatios gl p q 2 gl g h 2 and gl q h 1 1 the standard the nonstandard and the hybrid one respectively r or r is found to depend apart from the two parameters defining the deformation in question on an extra free parameter k such that r 12 r 23 r 12 r 23 r 12 r 23 k k1 1 k k2 1 r 23 r 12 with k1 k2 1 p q 1 1 and 1 1 q respectively only for k k1 or k k2 one has the braid equation arbitray k corresponds to a class conserving the group relations independent of k of the mqybe or modified quantum yb equations studied by gerstenhaber giaquinto and schack various properties of the triparametric r k p q r k g h and r k q h are studied in the larger space of the modified braid equation mbe even r k p

classical flag varieties for quantum groups the standard quantum sl n c abstract we suggest a possible programme to associate geometric flag like data to an arbitrary simple quantum group in the spirit of the noncommutative al gebraic geometry developed by artin tate and van den bergh we then carry out this programme for the standard quantum sl n of drinfel d and jimbo where the varieties involved are certain t stable subvarieties of the ordinary flag variety

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

minimax number of strata for online stratified sampling the case of noisy samples we consider online stratied sampling for monte carlo estimation of the integral of a function given a nite budget n of noisy evaluations to the function in this paper we address the problem of choosing the best number k of strata as a function of n a large k provides a high quality stratication where an accurate estimate of the integral of f could be computed by an optimal oracle allocation if the variances within each stratum were known however the performance of an adaptive allocation which does not know the variance within the strata compared to the oracle one deteriorates with k this denes a trade o between the stratication quality and the pseudo regret of an adaptive strategy

jordanian twists on deformed carrier subspaces the nontrivial subspaces with primitive coproducts are found in the deformed universal enveloping algebras they can form carrier subspaces for additional jordanian twists the latter can be used to construct sequences of twists for algebras whose root systems contain long series of roots the corresponding twist for the so 5 algebra is given explicitly

extended and reshetikhin twists for sl 3 the properties of the set of extended jordanian twists for algebra sl 3 are studied starting from the simplest algebraic construction the peripheric hopf algebra we explicitly construct the complete family of extended twisted algebras sl 3 corresponding to the set of four dimensional frobenius subalgebras l in sl 3 it is proved that the extended twisted algebras with different values of the parameter are connected by a special kind of reshetikhin twist we study the relations between the family sl 3 and the one dimensional set sl 3 produced by the standard reshetikhin twist from the drinfeld jimbo quantization sl 3 these sets of deformations are in one to one correspondence each element of sl 3 can be obtained by a limiting procedure from the unique point in the set sl 3

on some lie bialgebra structures on polynomial algebras and their quantization we study classical twists of lie bialgebra structures on the polynomial current algebra mathfrak g u where mathfrak g is a simple complex finite dimensional lie algebra we focus on the structures induced by the so called quasi trigonometric solutions of the classical yang baxter equation it turns out that quasi trigonometric r matrices fall into classes labelled by the vertices of the extended dynkin diagram of mathfrak g we give the complete classification of quasi trigonometric r matrices belonging to multiplicity free simple roots which have coefficient 1 in the decomposition of the maximal root we quantize solutions corresponding to the first root of mathfrak sl n

cyclic homology of brzezinski s crossed products and of braided hopf crossed products abstract let k be a field a a unitary associative k algebra and v a k vector space endowed with a distinguished element 1 v we obtain a mixed complex simpler than the canonical one that gives the hochschild cyclic negative and periodic homologies of a crossed product e a f v in the sense of brzezinski we actually work in the more general context of relative cyclic homology specifically we consider a subalgebra k of a that satisfies suitable hypothesis and we find a mixed complex computing the hochschild cyclic negative and periodic homologies of e relative to k then when e is a cleft braided hopf crossed product we obtain a simpler mixed complex that also gives the hochschild cyclic negative and periodic homologies of e

the quantum group structure associated with non linearly extended virasoro algebras recently an infinite family of chiral virasoro vertex operators whose exchange algebra is given by the universalr matrix ofsl 2 q has been constructed in the present paper the case of non linearly w extended virasoro symmetries related to the algebrasan n 1 is considered along the same line contrary to the previous case a1 the standardr matrix ofsl n 1 q does not come out and a different solution of yang and baxter s equations is derived the associated quantum group structure is displayed

classical a n w geometry by analyzing theextrinsic geometry of two dimensional surfaces chirally embedded incp n thecp n w surface 1 we give exact treatments in various aspects of the classical w geometry in the conformal gauge first the basis of tangent and normal vectors are defined at regular points of the surface such that their infinitesimal displacements are given by connections which coincide with the vector potentials of the conformal a n toda lax pair since the latter is known to be intrinsically related with the w symmetries this gives the geometrical meaning of thea n w algebra second w surfaces are put in one to one correspondence with solutions of the conformally reduced wznw model which is such that the toda fields give the cartan part in the gauss decomposition of its solutions third the additional variables of the toda hierarchy are used as coordinates ofcp n this allows us to show that w transformations may be extended as particular diffeomorphisms of this target space higher dimensional generalizations of the wznw equations are derived and related with the zakharov shabat equations of the toda hierarchy fourth singular points are studied from a global viewpoint using our earlier observation 1 that w surfaces may be regarded as instantons the global indices of the w geometry which are written in terms of the toda fields are shown to be the instanton numbers for associated mappings of w surfaces into the grassmannians the relation with the singularities of w surface is derived by combining the toda equations with the gauss bonnet theorem

a cohomological construction of quantization functors of lie bialgebras abstract we propose a variant to the etingof kazhdan construction of quantization functors we construct the twistor j associated to an associator using cohomological techniques we then introduce a criterion ensuring that the left hopf algebra of a quasitriangular que algebra is flat we prove that this criterion is satisfied at the universal level this gives a construction of quantization functors equivalent to the etingof kazhdan construction

noncommutative manifolds the instanton algebra and isospectral deformations we give new examples of noncommutative manifolds that are less standard than the nc torus or moyal deformations of n they arise naturally from basic considerations of noncommutative differential topology and have non trivial global features

confidence bands in density estimation given a sample from some unknown continuous density f we construct adaptive confidence bands that are honest for all densities in a generic subset of the union of t holder balls 0 t r where r is a fixed but arbitrary integer the exceptional nongeneric set of densities for which our results do not hold is shown to be nowhere dense in the relevant holder norm topologies in the course of the proofs we also obtain limit theorems for maxima of linear wavelet and kernel density estimators which are of independent interest

quantization of kahler manifolds ii we use berezin s dequantization procedure to define a formal product on a dense subalgebra of the algebra ofsmooth functions on a compact homogeneous kahler manifold m we prove that this formal product isconvergent when m isa hermitian symmetric space 1993 american mathematical society

symbolic calculus on the time frequency half plane this study concerns a special symbolic calculus of interest for signal analysis this calculus associates functions on the time frequency half plane f 0 with linear operators defined on the positive frequency signals full attention is given to its construction which is entirely based on the study of the affine group in a simple and direct way the correspondence rule is detailed and the associated wigner function is given formulas expressing the basic operation star bracket of the lie algebra of symbols which is isomorphic to that of the operators are obtained in addition it is shown that the resulting calculus is covariant under a three parameter group which contains the affine group as subgroup this observation is the starting point of an investigation leading to a whole class of symbolic calculi which can be considered as modifications of the original one

deformation quantization and nambu mechanics starting from deformation quantization star products the quantization problem of nambu mechanics is investigated after considering some impossibilities and pushing some analogies with field quantization a solution to the quantization problem is presented in the novel approach of zariski quantization of fields observables functions in this case polynomials this quantization is based on the factorization over of polynomials in several real variables we quantize the infinite dimensional algebra of fields generated by the polynomials by defining a deformation of this algebra which is abelian associative and distributive this procedure is then adapted to derivatives needed for the nambu brackets which ensures the validity of the fundamental identity of nambu mechanics also at the quantum level our construction is in fact more general than the particular case considered here it can be utilized for quite general defining identities and for much more general star products

local existence theorems with estimates for b on weakly pseudo convex cr manifolds in particular we prove the solvability for 0 1 on co for 1 q n 2 when m is a pseudo convex cr hypersurface of finite type in the sense defined in d angelo 11 and c co is the intersection of m with a flat hypersurface m0 see theorem 1 in sect i near a point zo of finite type we also show that there exists a neighborhood basis co of z0 such that 0 1 is solvable in co for any ct satisfying 0 2 see corol lary 1 1 we also obtain the interior regularity of the solutions for 0 1 in the sobolev spaces and show that the range of b is closed in the frechet norm l p q co loc see theorem 2 and corol lary 2 1 in sect 1 since the existence of the solution u of 0 1 was proved on co for any c satisfying 0 2 in co theorems 1 and 2 are semi global results when we assume that mo is only levi flat and with a stein ne ighborhood basis then we prove that for any relatively compact subset co c co there exists a u satisfying 0 2 in co see theorem 3 we note that the local solvability for 0 1 was not known previously even on a small subset in co on general weakly pseudo convex cr manifolds in this paper we shall prove the

chains of twists for classical lie algebras for chains of regular injections of hopf algebras the sets of maximal extended jordanian twists k are considered we prove that under certain conditions there exists for 0 the twist composed of the factors k the general construction of a chain of twists is applied to the universal envelopings u of classical lie algebras we study the chains for the infinite series an bn and dn the properties of the deformation produced by a chain u are explicitly demonstrated for the case of so 9

bialgebra cohomology deformations and quantum groups abstract we introduce cohomology and deformation theories for a bialgebra a over a commutative unital ring k such that the second cohomology group is the space of infinitesimal deformations our theory gives a natural identification between the underlying k modules of the original and the deformed bialgebra certain explicit deformation formulas are given for the construction of quantum groups i e hopf algebras that are neither commutative nor cocommutative whether or not they arise from quantum yang baxter operators these formulas yield in particular all glq n and slq n as deformations of gl n and sl n using a hodge decomposition of the underlying cochain complex we compute our cohomology for gl n with this we show that every deformation of gl n is equivalent to one in which the comultiplication is unchanged not merely on elements of degree one but on all elements settling in the strongest way a decade old conjecture and in which the quantum determinant as an element of the underlying k module is identical with the usual one

quantum weyl algebras abstract let r be a hecke symmetry there is then a natural quantization a n r of the n th weyl algebra a n based on r the aim of this paper is to study some general ring theoretical aspects of a n r and its relation to formal deformations of a n we also obtain further information on those quantizations obtained from some well known hecke symmetries

on two dimensional supergravity and the quantum super mobius group we describe the quantum group structure of two dimensionaln 1 supergravity in the conformal gauge the operator algebra of this super liouville theory is shown to correspond to a quantum deformation of the super mobius group which provides a new solution of yang and baxter s equation this structure is used to investigate the strong coupling regime of the theory 1 d 9 ford 5 a unitary truncation theorem similar to the one of the bosonic case is derived for the fundamental representation

the hidden group structure of quantum groups strong duality rigidity and preferred deformations a notion of well behaved hopf algebra is introduced reflexivity for strong duality between hopf algebras of drinfeld type and their duals algebras of coefficients of compact semi simple groups is proved a hidden classical group structure is clearly indicated for all generic models of quantum groups moyal product like deformations are naturally found for all frt models on coefficients andc functions strong rigidity hbi2 0 under deformations in the category of bialgebras is proved and consequences are deduced

rational top and its classical r matrix we construct a rational integrable system the rational top on a co adjoint orbit of sl n lie group it is described by the lax operator with spectral parameter and classical non dynamical skew symmetric r matrix in the case of the orbit of minimal dimension the model is gauge equivalent to the rational calogero moser cm system to obtain the results we represent the lax operator of the cm model in two different factorized forms without spectral parameter related to the spinless case and another one with the spectral parameter the latter gives rise to the rational top while the first one is related to generalized cremmer gervais r matrices the gauge transformation relating the rational top and cm model provides the classical rational version of the irf vertex correspondence from the geometrical point of view it describes the modification of bundles over degenerated elliptic curve in view of the symplectic hecke correspondence the rational top is related to the rational spin cm model possible applications and generalizations of the suggested construction are discussed in particular the obtained r matrix defines a class of kzb equations

an infinite product formula for uq sl 2 dynamical coboundary element we give a short summary of results and conjectures in the theory of dynamical quantum group related to the dynamical coboundary equation also known as irf vertex transform babelon has shown that the dynamical twist f x of uq sl 2 is a dynamical coboundary m x i e f x m 1 xq h 2 m 2 x a m x we give a new formula for this element m x as an infinite product and give a new proof of the coboundary relation our proof involves the quantum weyl group element giving a possible hint for the generalization to higher rank case

generalization of the h deformation to higher dimensions we construct glh 3 from glq 3 by a singular map we show that there exist two singular maps which map glq 3 to new quantum groups we also construct their r matrices and will show that although the maps are singular their r matrices are not then we generalize these singular maps to the case gl n and for cn series

the gervais neveu felder equation for the jordanian quasi hopf uh y sl 2 algebra using a contraction procedure we construct a twist operator that satisfies a shifted cocycle condition and leads to the jordanian quasi hopf uh y sl 2 algebra the corresponding universal h y matrix obeys a gervais neveu felder equation associated with the uh y sl 2 algebra for a class of representations the dynamical yang baxter equation may be expressed as a compatibility condition for the algebra of the lax operators

a noncommutative hopf structure on c sl 2 c as a quantum lorentz group we introduce a classical frechet hopf algebra a containing the quantum deformation of the enveloping lorentz algebra of ogievestky and wess et al as a dense hopf subalgebra we also show that a is isomorphic to a quasitriangular twisted topological tensor product hopf algebra by duality we recover on the topological dual space of a the non commutative algebra structure introduced by podles and woronowicz finally this non commutative product can be extended to c sl 2 c which then becomes a topological hopf algebra for its natural frechet topology

twisted conformal algebra so 4 2 a new twisted deformation uz so 4 2 of the conformal algebra of the 3 1 dimensional minkowskian spacetime is presented this construction is provided by a classical r matrix spanned by ten weyl poincare generators which generalizes non standard quantum deformations previously obtained for so 2 2 and so 3 2 however by introducing a conformal null plane basis it is found that the twist can indeed be supported by an eight dimensional carrier subalgebra by construction the weyl poincare subalgebra remains as a hopf subalgebra after deformation non relativistic limits of uz so 4 2 are shown to be well defined and they give rise to new twisted conformal algebras of galilean and carroll spacetimes furthermore a difference differential massless klein gordon or wave equation with twisted conformal symmetry is constructed through deformed momenta and position operators the deformation parameter is interpreted as the lattice step on a uniform minkowskian spacetime lattice discretized along two basic null plane directions

representations des groupes exponentiels we give a complete and explicit realization of the unitary irreducible representations of any exponential group g by deformation of the associative and poisson algebra of functions on the dual g of the lie algebra of g we define an adapted fourier transform which is a deformation of the usual one and which gives a natural description of the harmonic analysis of g on the restrictions to the coadjoint orbits of the considered functions

jordanian quantum algebra h sl n via contraction method and mapping using a contraction procedure developed earlier we construct in the first part of the present paper the jordanian quantum hopf algebra h sl 3 which has a remarkably simple coalgebraic structure and contains the jordanian algebra h sl 2 obtained by ohn as a hopf subalgebra a nonlinear map between the quantum h sl 3 and the classical sl 3 algebras is then established and the universal h matrix of the h sl 3 algebra is given in the second part we give the higher dimensional jordanian algebras h sl n for all n the universal h matrix of the h sl n algebra is also given

least uncertainty principle in deformation quantization deformation quantization generally produces families of cohomologically equivalent quantizations of a single physical system we conjecture that the physically meaningful ones i allow enough observable energy distributions i e ones for which no pure quantum state has negative probability and ii reduce the uncertainty in the probability distribution of the resulting quantum states for the simple harmonic oscillator this principle selects the classic groenewold moyal or weyl product on phase space while for the free particle in which there is no real quantization all cohomologically equivalent quantizations are equally good

quantization of lie poisson structures by peripheric chains the quantization properties of composite peripheric twists are studied peripheric chains of extended twists are constructed for u sl n in order to obtain composite twists with sufficiently large carrier subalgebras it is proved that the peripheric chains can be enlarged with additional reshetikhin and jordanian factors this provides the possibility to construct new solutions to drinfeld equations and thus to quantize new sets of lie poisson structures when the jordanian additional factors are used the carrier algebras of the enlarged peripheric chains are transformed into algebras of motion of the form the factor algebra is a direct sum of borel and contracted borel subalgebras of lower dimensions the corresponding form is a coboundary the enlarged peripheric chains represent the twists that contain operators external with respect to the lie poisson structure the properties of new twists are illustrated by quantizing r matrices for the algebras u sl 3 u sl 4 and u sl 7

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

l r smash product for quasi hopf algebras we introduce a more general version of the so called l r smash product and study its relations with other kinds of crossed products two sided smash and crossed product and diagonal crossed product we also give an interpretation of the l r smash product in terms of an l r twisting datum

quantum groups and deformation quantization explicit approaches and implicit aspects deformation quantization which gives a development of quantum mechanics independent of the operator algebra formulation and quantum groups which arose from the inverse scattering method and a study of yang baxter equations share a common idea abstracted earlier in algebraic deformation theory that algebraic objects have infinitesimal deformations which may point in the direction of certain continuous global deformations i e quantizations in deformation quantization the algebraic object is the algebra of observables functions on symplectic phase space whose infinitesimal deformation is the poisson bracket and global deformation a star product in quantum groups it is a hopf algebra generally either of functions on a lie group or often its dual in the topological vector space sense as we briefly explain a completed universal enveloping algebra of a lie algebra with for infinitesimal a matrix satisfying the modified classical yang baxter equation mcybe frequently existence proofs are known but explicit formulas useful for physical applications have been difficult to extract one success here comes from universal deformation formulas udfs expressions built from a lie algebra which deform any algebra on which the lie algebra operates as derivations the most famous of these is the moyal product a special case of a class in which the lie algebra is abelian another comes from recognition that the belavin drinfel d solutions to the mcybe are in fact infinitesimal deformations for which in the case of the special linear groups it is possible to give explicit formulas for the corresponding quantum yang baxter equations this review paper discusses necessarily in brief these and related topics including twisting as a form of udf and finding formulas for preferred deformations of hopf algebras in which the multiplication or comultiplication is rigid and must be preserved in the course of deformation

physics for algebraists non commutative and non cocommutative hopf algebras by a bicrossproduct construction abstract the initial part of this paper presents physics for algebraists in the context of quantum mechanics combined with gravity such physical notions as the yang baxter equations position observables momentum space momentum and position quantization etc are described many readers may wish to just read this initial part of the paper the physics leads to the search for self dual algebraic structures and finally to non commutative and non cocommutative hopf algebras by a bicrossproduct construction the entire paper contains numerous examples the non commutative and non cocommutative hopf algebras are obtained as a simultaneous smash product and smash coproduct and denoted h1 h2 among the examples is one obtained by modifying the weyl algebra we also give the context in which the compatibility requirements on the structure maps reduce to the classical yang baxter equations and an example related to drinfel d s double hopf algebra d h

quantum jordanian twist the quantum deformation of the jordanian twist q for the standard quantum borel algebra uq b is constructed it gives the family uq b of quantum algebras depending on parameters and h in a generic point these algebras represent the hybrid standard nonstandard quantization the quantum jordanian twist can be applied to the standard quantization of any kac moody algebra the corresponding classical r matrix is a linear combination of the drinfeld jimbo and the jordanian ones the two parametric families of hopf algebras obtained here are smooth and for the limit values of the parameters the standard and nonstandard quantizations are recovered the twisting element q also has correlated limits in particular when q tends to unity it acquires the canonical form of the jordanian twist to illustrate the properties of the quantum jordanian twist we construct the hybrid quantizations for u sl 2 and for the corresponding affine algebra the universal quantum matrix and its defining representation are presented