from geometric quantization to moyal quantization it is shown how the moyal product of phase space functions and the weyl correspondence between symbols and operator kernels may be obtained directly using the procedures of geometric quantization applied to the symplectic groupoid constructed by doubling the phase space

semi classical properties of berezin toeplitz operators with c k symbol we obtain the semi classical expansion of the kernels and traces of toeplitz operators with ck symbol on a symplectic manifold we also give a semi classical estimate of the distance of a toeplitz operator to the space of self adjoint and multiplication operators we obtain the semi classical expansion of the kernels and traces of toeplitz operators with ck symbol on a symplectic manifold we also give a semi classical estimate of the distance of a toeplitz operator to the space of self adjoint and multiplication operators

on the noncyclic property of sylvester hadamard matrices in this paper we are concerned with hadamard matrices with a certain noncyclic property first we show that when the first column of a sylvester hadamard matrix of order 2m m 2 a positive integer is removed the number of shift distinct row vectors in the matrix is given by 2m m then for m 4 we construct an infinite family of hadamard matrices with a property that when the first column of the hadamard matrix is removed all the row vectors of the matrix are shift distinct these hadamard matrices are useful in constructing low correlation zone sequences

semiclassical spectral analysis of toeplitz operators on symplectic manifolds the case of discrete wells we consider toeplitz operators associated with the renormalized bochner laplacian on high tensor powers of a positive line bundle on a compact symplectic manifold we study the asymptotic behavior in the semiclassical limit of low lying eigenvalues and the corresponding eigensections of a self adjoint toeplitz operator under assumption that its principal symbol has a non degenerate minimum with discrete wells as an application we prove upper bounds for low lying eigenvalues of the bochner laplacian in the semiclassical limit

a class of pseudonoise sequences over gf p with low correlation zone a new class of pseudonoise sequences over gf p based on gordon mills welch 1962 sequences is constructed the sequences have the property that in a specified zone the out of phase autocorrelation and cross correlation values are all equal to 1 such sequences with low correlation zone lcz are suitable for approximately synchronized code division multiple access cdma system

a formal model of berezin toeplitz quantization we give a new construction of symbols of the differential operators on the sections of a quantum line bundle l over a kahler manifold m using the natural contravariant connection on l these symbols are the functions on the tangent bundle tm polynomial on fibres for high tensor powers of l the asymptotics of the composition of these symbols leads to the star product of a deformation quantization with separation of variables on tm corresponding to some pseudo kahler structure on tm surprisingly this star product is intimately related to the formal symplectic groupoid with separation of variables over m we extend the star product on tm to generalized functions supported on the zero section of tm the resulting algebra of generalized functions contains an idempotent element which can be thought of as a natural counterpart of the bergman projection operator using this idempotent we define an algebra of toeplitz elements and show that it is naturally isomorphic to the algebra of berezin toeplitz deformation quantization on m

deformed noncommutative tori we recall a construction of non commutative algebras related to a one parameter family of deformed spheres and tori and show that in the case of tori the algebras can be completed into c algebras isomorphic to the standard non commutative torus as the former was constructed in the context of matrix or fuzzy geometries it provides an important link to the framework of non commutative geometry and opens up for a concrete way to study deformations of non commutative tori furthermore we show how the well known fuzzy sphere and fuzzy torus can be obtained as formal scaling limits of finite dimensional representations of the deformed algebras and their projective modules are described together with connections of constant curvature

on the composition of berezin toeplitz operators on symplectic manifolds we compute the second coefficient of the composition of two berezin toeplitz operators associated with the text spin c dirac operator on a symplectic manifold making use of the full off diagonal expansion of the bergman kernel

quantized nambu poisson manifolds and n lie algebras we investigate the geometric interpretation of quantized nambu poisson structures in terms of noncommutative geometries we describe an extension of the usual axioms of quantization in which classical nambu poisson structures are translated to n lie algebras at quantum level we demonstrate that this generalized procedure matches an extension of berezin toeplitz quantization yielding quantized spheres hyperboloids and superspheres the extended berezin quantization of spheres is closely related to a deformation quantization of n lie algebras as well as the approach based on harmonic analysis we find an interpretation of nambu heisenberg n lie algebras in terms of foliations of rn by fuzzy spheres fuzzy hyperboloids and noncommutative hyperplanes some applications to the quantum geometry of branes in m theory are also briefly discussed

some topics on glq n abstract the quantum groups gl q n and sl q n are defined as pairs of hopf algebras and it is shown they are absolutely connected when q is a root of 1 some quantum analogue of the frobenius map and some related finite quantum groups are introduced and their simplicity is studied

gl and geometric quantization an axiomatic approach to the approximation of infinite dimensional algebras is presented examples illustrating the need for a rigorous treatment of this subject geometric quantization is employed to construct systematicallysu n approximations of diffeomorphism algebras which first appeared in the theory of relativistic membranes

the manifold of compatible almost complex structures and geometric quantization let m be an integral symplectic manifold we study a family of hermitian vector bundles on the space mathcal j of almost complex structures on m compatible with whose fibers consist of nearly holomorphic sections of powers of a prequantum line bundle we obtain asymptotics of the curvature of a natural connection in these bundles these results together with toeplitz operator theory provide another proof of donaldson s result that the action of ham m on mathcal j is hamiltonian with moment map the scalar curvature we also give an example involving teichmuller space and discuss a relationship between parallel transport and the schrodinger equation

deformation quantization of almost kahler models and lagrange finsler spaces finsler and lagrange spaces can be equivalently represented as almost kahler manifolds endowed with a metric compatible canonical distinguished connection structure generalizing the levi civita connection the goal of this paper is to perform a natural fedosov type deformation quantization of such geometries all constructions are canonically derived for regular lagrangians and or fundamental finsler functions on tangent bundles

some interrelations between hopf algebras and their duals abstract in this paper we study certain interrelations between subsets of hopf algebras h and their duals which stem from various morphisms t defined on h we define and study t cocommutative elements cocommutative elements generators of h as an h module and invariants under various actions and coactions are all examples an important role is played by nakayama automorphisms these ideas are applied to co quasitriangular hopf algebras in particular when they are factorizable then the drinfeld element gives rise to a nakayama automorphism and to a special element in the dual explicit calculations are carried out for the drinfeld double

hermite polynomials and quasi classical asymptotics we study an unorthodox variant of the berezin toeplitz type of quantization scheme on a reproducing kernel hilbert space generated by the real hermite polynomials and work out the associated quasi classical asymptotics

quantization and the hessian of mabuchi energy let l x be an ample bundle over a compact complex manifold fix a hermitian metric in l whose curvature defines a kahler metric on x the hessian of mabuchi energy is a fourth order elliptic operator d d on functions which arises in the study of scalar curvature we quantize d d by the hessian p k pk of balancing energy a function appearing in the study of balanced embeddings p k pk is defined on the space of hermitian endomorphisms of h0 x lk endowed with the l2 inner product we first prove that the leading order term in the asymptotic expansion of p k pk is d d we next show that if aut x l is discrete then the eigenvalues and eigenspaces of p k pk converge to those of d d we also prove convergence of the hessians in the case of a sequence of balanced embeddings tending to a constant scalar curvature kahler metric as consequences of our results we prove that an estimate of phong and sturm is sharp and give a negative answer to a question posed by donaldson we also discuss some possible applications to the study of calabi flow 2012

the semiclassical structure of low energy states in the presence of a magnetic field we consider a compact riemannian manifold with a hermitian line bundle whose curvature is non degenerate the laplacian acting on high tensor powers the semiclassical regime of the bundle exhibits a cluster of low energy states we demonstrate that the orthogonal projectors onto these states are the fourier components of an operator with the structure of the szego projector i e a fourier integral operator of hermite type this result yields semiclassical asymptotics for the low energy eigenstates

quantum riemann surfaces i the unit disc we construct a non commutative algebra c mu bar u which is a quantum deformation of the algebra of continuous functions on the closed unit disc bar u c mu bar u is generated by the toeplitz operators on a suitable hilbert space of holomorphic functions onu

generalized bergman kernels and geometric quantization in geometric quantization it is well known that if f is an observable and f a polarization on a symplectic manifold m then the condition xf leaves f invariant where xf denotes the hamiltonian vector field associated to f is sufficient to guarantee that one does not have to compute the bks kernel explicitly in order to know the corresponding quantum operator it is shown in this paper that this condition on f can be weakened to xf leaves f f invariant and the corresponding quantum operator is then given implicitly by formula 4 8 in particular when f is a positive kahler polarization all observables can be quantized directly and moreover an explicit formula for the corresponding quantum operator is derived theorem 5 8 applying this to the phase space r2n one obtains a quantization prescription which ressembles the normal ordering of operators in quantum field theory when we translate this prescription to the usual position representation of quantum mechanics the result i

the deformation quantizations of the hyperbolic plane we describe the space of all invariant both formal and non formal deformation quantizations on the hyperbolic plane mathbb d as solutions of the evolution of a second order hyperbolic differential operator the construction is entirely explicit and relies on non commutative harmonic analytical techniques on symplectic symmetric spaces the present work presents a unified method producing every quantization of mathbb d and provides in the 2 dimensional context an exact solution to weinstein s wkb quantization program within geometric terms the construction reveals the existence of a metric of lorentz signature canonically attached or dual to the geometry of the hyperbolic plane through the quantization process

an obstruction to quantization of the sphere in the standard example of strict deformation quantization of the symplectic sphere s2 the set of allowed values of the quantization parameter is not connected indeed it is almost discrete li recently constructed a class of examples including s2 in which can take any value in an interval but these examples are badly behaved here i identify a natural additional axiom for strict deformation quantization and prove that it implies that the parameter set for quantizing s2 is never connected

convergence of the wick star product we construct a frechet space as a subspace of c omega mathbb c n where the wick star product converges and is continuous the resulting frechet algebra mathcal a is studied in detail including a representation of mathcal a in the bargmann fock space and a discussion of star exponentials and coherent states

new constructions of quaternary low correlation zone sequences in this paper given a composite integer n we propose a method of constructing quaternary low correlation zone lcz sequences of period 2 sup n 1 from binary sequences of the same length with ideal autocorrelation these new sequences are optimal with respect to the bound by tang fan and matsufuji the correlation distributions of these new quaternary lcz sequences constructed from m sequences and gordon mills welch gmw sequences are derived

berezin transforms on pluriharmonic bergman spaces we show that perhaps surprisingly in several aspects the behaviour of the reproducing kernels of toeplitz operators and of the berezin transform on some weighted pluriharmonic bergman spaces is the same as in the holomorphic case

berezin toeplitz quantization on kaehler manifolds we study the berezin toeplitz quantization on kaehler manifolds we explain first how to compute various associated asymptotic expansions then we compute explicitly the first terms of the expansion of the kernel of the berezin toeplitz operators and of the composition of two berezin toeplitz operators as application we estimate the norm of donaldson s q operator

toeplitz quantization of kahler manifolds ang gl n n limits for general compact kahler manifolds it is shown that both toeplitz quantization and geometric quantization lead to a well defined by operator norm estimates classical limit this generalizes earlier results of teh authors and klimek and lesniewski obtained for the tours and higher genus riemann surfaces respectively we thereby arrive at an approximation of the poisson algebra by a sequence of finitedimensional matrix algebrasgl n n

quantum riemann surfaces for arbitrary planck s constant we continue our study of quantum riemann surfaces initiated in refs s klimek and a lesniewski commun math phys 146 103 122 1992 lett math phys 24 125 139 1992 32 45 61 1994 we construct a one parameter family of deformations of compact riemann surfaces of genus g 2 our construction does not require any discreteness condition on the value of planck s constant it coincides with the construction of ref lett math phys 24 125 139 1992 in the case when planck s constant assumes the discrete set of values dictated by geometric quantization

quantization of multiply connected manifolds the standard berezin toeplitz geometric quantization of a compact kahler manifold is restricted by integrality conditions these restrictions can be circumvented by passing to the universal covering space provided that the lift of the symplectic form is exact i relate this construction to the baum connes assembly map and prove that it gives a strict quantization of the original manifold i also propose a further generalization classify the required structure and provide a means of computing the resulting algebras these constructions involve twisted group c algebras of the fundamental group which are determined by a group cocycle constructed from the cohomology class of the symplectic form this provides an algebraic counterpart to the morita equivalence of a symplectic manifold with its fundamental group

categorical perspective on quantization of poisson algebra we propose a generalization of quantization using a categorical approach for a fixed poisson algebra quantization categories are defined as subcategories of the r module category equipped with the structure of classical limits we then construct the generalized quantization categories including matrix regularization strict deformation quantization prequantization and poisson enveloping algebra it is shown that the categories of strict deformation quantization prequantization and matrix regularization with certain conditions are equivalent categories on the other hand the categories of poisson enveloping algebra are not equivalent to the other categories

berezin transform on the harmonic fock space abstract we show that the berezin transform associated to the harmonic fock segal bargmann space on c n has an asymptotic expansion analogously as in the holomorphic case the proof involves a computation of the reproducing kernel which turns out to be given by one of horn s hypergeometric functions of two variables and an ad hoc determination of the asymptotic behaviour of the resulting integrals to which the ordinary stationary phase method is not directly applicable

geometric quantization of vector bundles and the correspondence with deformation quantization i repeat my definition for quantization of a vector bundle for the cases of the toeplitz and geometric quantizations of a compact kahler manifold i give a construction for quantizing any smooth vector bundle which depends functorially on a choice of connection on the bundle

toeplitz quantization on fock space for toeplitz operators t f t acting on the weighted fock space h t 2 we consider the semi commutator t f t t g t t fg t where t 0 is a certain weight parameter that may be interpreted as planck s constant hbar in rieffel s deformation quantization in particular we are interested in the semi classical limit begin equation tag lim limits t to 0 t f t t g t t fg t t end equation it is well known that t f t t g t t fg t t tends to 0 under certain smoothness assumptions imposed on f and g this result was recently extended to f g in textup buc c n by bauer and coburn we now further generalize to not necessarily bounded uniformly continuous functions and symbols in the algebra textup vmo cap l infty of bounded functions having vanishing mean oscillation on mathbb c n our approach is based on the algebraic identity t f t t g t t fg t h bar f t h g t where h g t denotes the hankel operator corresponding to the symbol g and norm estimates in terms of the weighted heat transform as a consequence only f or likewise only g has to be contained in one of the above classes for to vanish for g we only have to impose limsup t to 0 h g t t infty e g g in l infty c n we prove that the set of all symbols f in l infty mathbb c n with the property that lim t rightarrow 0 t t ft t g t t fg t lim t rightarrow 0 t g t t f t t gf t t 0 for all g in l infty mathbb c n coincides with textup vmo cap l infty additionally we show that lim limits t to 0 t f t t f infty holds for all f in l infty c n finally we present new examples including bounded smooth functions where does not vanish

quantization towards a comparison between methods in this paper it is shown that the procedure of geometric quantiztion applied to kahler manifolds gives the following result the hilbert space h consists roughly speaking of holomorphic functions on the phase space m and to each classical observable f i e a real function on m is associated an operator f on h as follows first multiply by f 1 4 drf dr being the laplace de rham operator on the kahler manifold m and then take the holomorphic part see g m tuynman j math phys 27 573 1987 this result is correct on compact kahler manifolds and correct modulo a boundary term md on noncompact kahler manifolds in this way these results can be compared with the quantization procedure of berezin math ussr izv 8 1109 1974 9 341 1975 commun math phys 40 153 1975 which is strongly related to quantization by products e g see c moreno and p ortega navarro amn inst h poincare sec a 38 215 1983 lett math phys 7 181 1983 c moreno lett math phys 11 3

weighted bergman kernels and quantization let be a bounded pseudoconvex domain in c n two positive functions on such that log log are plurisubharmonic and z a point at which log is smooth and strictly plurisubharmonic we show that as k the bergman kernels with respect to the weights k have an asymptotic expansion for x y near z where x y is an almost analytic extension of phi x x x and similarly for further if in addition is of finite type behave reasonably at the boundary and log log are strictly plurisubharmonic on we obtain also an analogous asymptotic expansion for the berezin transform and give applications to the berezin quantization finally for smoothly bounded and strictly pseudoconvex and a smooth strictly plurisubharmonic defining function for we also obtain results on the berezin toeplitz quantization

berezin transform of two arguments abstract we describe the asymptotic behavior of the berezin transform of two arguments which generalizes the standard notion of the berezin transform the asymptotic analysis we give is in the setting of the holomorphic and the harmonic fock spaces respectively

new sets of optimal p ary low correlation zone sequences in this correspondence three methods of constructing low correlation zone lcz sequences are proposed in the first method we constructed binary lcz sequence sets of period 2n 1 using the legendre sequences of period 2m 1 as a column sequence when m n in the second method we devise a column sequence set of length 2m 1 1 from a binary sequence of period 2m 1 having ideal autocorrelation property and this column sequence set is used to construct binary lcz sequence sets of period 2 n 1 when m 1 n in the third method p ary lcz sequence sets are constructed by adopting p ary sequence of period pm 1 with ideal autocorrelation for integers n and m such that m n as a column sequence the second and third methods give us the optimal sets with respect to the bound by tang fan and matsufuji finally a construction method of pntimespn p ary hadamard matrices from optimal lcz sequence sets is proposed

a note on low correlation zone signal sets in this correspondence we present a connection between designing low correlation zone lcz sequences and the results of correlation of sequences with subfield decompositions presented in a recent book by the first two authors this results in lcz signal sets with huge sizes over three different alphabetic sets finite field of size integer residue ring modulo and the subset in the complex field which consists of powers of a primitive th root of unity we show a connection between these sequence designs and ldquocompletely noncyclicrdquo hadamard matrices and a construction for those sequences we also provide some open problems along this direction

a class of drinfeld doubles that are ribbon algebras andruskiewitsch and schneider classify a large class of pointed hopf algebras with abelian coradical the drinfeld double of each such hopf algebra is investigated the drinfeld doubles of a family of hopf algebras from the above classification are ribbon hopf algebras

circular symmetry and the trace formula let m be a compact n dimensional manifold without boundary and let p be a positive first order elliptic self adjoint classical pseudodifferential operator on lz m the hilbert space of half densities on m the spectrum of p is discrete and accumulates only at we denote the eigenvalues with multiplicities by 2o 21 2 recall that the singularities of the inverse fourier transform of the spectral measure