drinfeld sokolov reduction for difference operators and deformations of w algebras ii the general semisimple case the paper is the sequel to 9 we extend the drinfeld sokolov reduction procedure to q difference operators associated with arbitrary semisimple lie algebras this leads to a new elliptic deformation of the lie bialgebra structure on the associated loop algebra the related classical r matrix is explicitly described in terms of the coxeter transformation we also present a cross section theorem for q gauge transformations which generalizes a theorem due to r steinberg

the drinfeld realization of the elliptic quantum group bq λ a2 2 we construct a realization of the l operator satisfying the rll relation of the face type elliptic quantum group bq a2 2 the construction is based on the elliptic analog of the drinfeld currents of uq a2 2 which forms the elliptic algebra uq p a2 2 we give a realization of the elliptic currents e z f z and k z as a tensor product of the drinfeld currents of uq a2 2 and a heisenberg algebra in the level one representation we also give a free field realization of the elliptic currents applying these results we derive a free field realization of the uq p a2 2 analog of the bq a2 2 intertwining operators the resultant operators coincide with those of the vertex operators in the dilute al model which is known to be a rsos restriction of the a2 2 face model

dynamical centers for the elliptic quantum algebra we identify generating functionals that satisfy dynamical exchange relations with the lax matrices defining the face type elliptic quantum algebra when the central charge takes the two possible values these generating functionals are constructed as quadratic trace like objects in terms of the lax matrices the obtained structures are characterized as dynamical centers i e the centrality property is deformed by dynamical shifts for these values the functionals define genuine abelian subalgebras of

wigner weyl correspondence in quantum mechanics for continuous and discrete systems a dirac inspired view drawing inspiration from dirac s work on functions of non commuting observables we develop an approach to phase space descriptions of operators and the wigner weyl correspondence in quantum mechanics complementary to standard formulations this involves a two step process introducing phase space descriptions based on placing position dependences to the left of momentum dependences or the other way around then carrying out a natural transformation to eliminate a kernel which appears in the expression for the trace of the product of two operators the method works uniformly for both continuous cartesian degrees of freedom and for systems with finite dimensional state spaces it is interesting that the kernel encountered is naturally expressible in terms of geometric phases and its removal involves extracting its square root in a suitable manner

a q deformation of wakimoto modules primary fields and screening operators theq vertex operators of frenkel and reshetikhin are studied by means of aq deformation of the wakimoto module for the quantum affine algebrau q widehat mathfrak s mathfrak l 2 at an arbitrary levelk 0 2 a fock module version of theq deformed primary field of spinj is introduced as well as the screening operators which anti commute with the action ofu q widehat mathfrak s mathfrak l 2 up to a total difference of a field a proof of the intertwining property is given for theq vertex operators corresponding to the primary fields of spinj 1 2z 0 a sample calculation of the correlation function is also given

a commutative algebra on degenerate cp1 and macdonald polynomials we introduce a unital associative algebra a associated with degenerate cp1 we show that a is a commutative algebra and whose poincare series is given by the number of partitions thereby we can regard a as a smooth degeneration limit of the elliptic algebra introduced by feigin and odesskii int math res notices 11 531 1997 then we study the commutative family of the macdonald difference operators acting on the space of symmetric functions a canonical basis is proposed for this family by using a and the heisenberg representation of the commutative family studied by shiraishi commun math phys 263 439 2006 it is found that the ding iohara algebra lett math phys 41 183 1997 provides us with an algebraic framework for the free field construction an elliptic deformation of our construction is discussed showing connections with the drinfeld quasi hopf twisting leningrad math j 1 1419 1990 in the sence of babelon bernard billey phys lett b 375 89 1996 the ruijsenaar

property testing and its connection to learning and approximation in this paper we consider the question of determining whether a function f has property p or is e far from any function with property p a property testing algorithm is given a sample of the value of f on instances drawn according to some distribution in some cases it is also allowed to query f on instances of its choice we study this question for different properties and establish some connections to problems in learning theory and approximation in particular we focus our attention on testing graph properties given access to a graph g in the form of being able to query whether an edge exists or not between a pair of vertices we devise algorithms to test whether the underlying graph has properties such as being bipartite k colorable or having a p clique clique of density p with respect to the vertex set our graph property testing algorithms are probabilistic and make assertions that are correct with high probability while making a number of queries that is independent of the size of the graph moreover the property testing algorithms can be used to efficiently i e in time linear in the number of vertices construct partitions of the graph that correspond to the property being tested if it holds for the input graph

algebra of screening operators for the deformed wn algebra we construct a family of intertwining operators screening operators between various fock space modules over the deformed wn algebra they are given as integrals involving a product of screening currents and elliptic theta functions we derive a set of quadratic relations among the screening operators and use them to construct a felder type complex in the case of the deformed w3 algebra

the elliptic algebra and the drinfeld realization of the elliptic quantum group by using the elliptic analogue of the drinfeld currents in the elliptic algebra we construct a l operator which satisfies the rll relations characterizing the face type elliptic quantum group for this purpose we introduce a set of new currents k j v 1 leq j leq n in as in the n 2 case we find a structure of as a certain tensor product of and a heisenberg algebra in the level one representation we give a free field realization of the currents in using the coalgebra structure of and the above tensor structure we derive a free field realization of the analogue of intertwining operators the resultant operators coincide with those of the vertex operators in the a n 1 1 type face model

commutation relations of vertex operators for uq sl m n we consider commutation relations and invertibility relations of vertex operators for the quantum affine superalgebra uq sl m n by using bosonization we show that vertex operators give a representation of the graded zamolodchikov faddeev algebra by direct computation invertibility relations of type ii vertex operators for n m are very similar to those of type i for m n we consider commutation relations and invertibility relations of vertex operators for the quantum affine superalgebra uq sl m n by using bosonization we show that vertex operators give a representation of the graded zamolodchikov faddeev algebra by direct computation invertibility relations of type ii vertex operators for n m are very similar to those of type i for m n

local monotonicity reconstruction we investigate the problem of monotonicity reconstruction as defined by ailon et al 2004 in a localized setting we have oracle access to a nonnegative real valued function f defined on the domain n d 1 dots n d where d is viewed as a constant we would like to closely approximate f by a monotone function g this should be done by a procedure a filter that given as input a point x in n d outputs the value of g x and runs in time that is polylogarithmic in n the procedure can indeed must be randomized but we require that all of the randomness be specified in advance by a single short random seed we construct such an implementation where the time and space per query is log n o 1 and the size of the seed is polynomial in log n and d furthermore with high probability the ratio of the hamming distance between g and f to the minimum possible hamming distance between a monotone function and f is bounded above by a function of d independent of n this allows for a local implementation one can initialize many copies of the filter with the same short random seed and they can autonomously handle queries while producing outputs that are consistent with the same approximating function g

bosonization of vertex operators for the face model we present a free boson realization of the vertex operators and their duals for the solvable sos lattice model of type we discuss a possible connection with the calculation of correlation functions

the elliptic quantum algebra u_ q hbox scriptsize p widehat sl_n and its vertex operators we construct a realization of the elliptic quantum algebra for any given level k in terms of free boson fields and their twisted partners it can be considered as the elliptic deformation of the wakimoto realization of the quantum affine algebra we also construct a family of screening currents which commute with the currents of up to total q differences and we give explicit twisted expressions for the type i and type ii vertex operators of by twisting the known results of the type i vertex operators of the quantum affine algebra and the new results of the type ii vertex operators of we obtained in this paper

characterization of informational completeness for covariant phase space observables in the nonrelativistic setting with finitely many canonical degrees of freedom a shift covariant phase space observable is uniquely characterized by a positive operator of trace one and in turn by the fourier weyl transform of this operator we study three properties of such observables and characterize them in terms of the zero set of this transform the first is informational completeness for which it is necessary and sufficient that the zero set has dense complement the second is a version of informational completeness for the hilbert schmidt class equivalent to the zero set being of measure zero and the third known as regularity is equivalent to the zero set being empty we give examples demonstrating that all three conditions are distinct the three conditions are the special cases for p 1 2 of a more general notion of p regularity defined as the norm density of the span of translates of the operator in the schatten p class we show that the relation between zero sets and p regularity can be mapped completely to the corresponding relation for functions in classical harmonic analysis

quantum scr w algebras and elliptic algebras we define a quantum open image in new window algebra associated to mathfrak s mathfrak l n as an associative algebra depending on two parameters for special values of the parameters this algebra becomes the ordinary open image in new window algebra of mathfrak s mathfrak l n or theq deformed classical open image in new window algebra algebra of mathfrak s mathfrak l n we construct free field realizations of the quantum open image in new window algebra and the screening currents we also point out some interesting elliptic structures arising in these algebras in particular we show that the screening currents satisfy elliptic analogues of the drinfeld relations in open image in new window

towards a cladistics of double yangians and elliptic algebras a self contained description of algebraic structures obtained by combinations of various limit procedures applied to vertex and face sl 2 elliptic quantum affine algebras is given new double yangian structures of dynamical type are defined connections between these structures are established a number of them take the form of twist like actions these are conjectured to be evaluations of universal twists

on the strength of comparisons in property testing an e test for a property p of functions from d 1 d to the positive integers is a randomized algorithm which makes queries on the value of an input function at specified locations and distinguishes with high probability between the case of the function satisfying p and the case that it has to be modified in more than ed places to make it satisfy p we prove that an e test for a property of integer sequences such as the property of the sequence being a monotone non decreasing sequence that depends in a strict sense only on the order relations between the sequence members cannot perform less queries in the worst case than the best e test which uses only comparisons between the queried values in addition we show that an adaptive algorithm for testing that a sequence is monotone non decreasing performs no better than the best non adaptive one with respect to query complexity from this follows a tight lower bound on tests for this property

elliptic algebra drinfeld currents and vertex operators we investigate the structure of the elliptic algebra introduced earlier by one of the authors our construction is based on a new set of generating series in the quantum affine algebra which are elliptic analogs of the drinfeld currents they enable us to identify with the tensor product of and a heisenberg algebra generated by p q with q p 1 in terms of these currents we construct an l operator satisfying the dynamical rll relation in the presence of the central element c the vertex operators of lukyanov and pugai arise as intertwiners of for the level one representation in the sense to be elaborated on in the text we also present vertex operators with higher level spin in the free field representation

vertex operators for quantum groups and application to integrable systems starting with any r matrix with spectral parameters and obeying the yang baxter equation and a unitarity condition we construct the corresponding infinite dimensional quantum group r in terms of a deformed oscillator algebra r the realization we present is an infinite series very similar to a vertex operator then considering the integrable hierarchy naturally associated with r we show that r provides its integrals of motion the construction can be applied to any infinite dimensional quantum group e g yangians or elliptic quantum groups taking as an example the r matrix of y n the yangian based on gl n using this construction we recover the nonlinear schrodinger equation and its y n symmetry

free field approach to the dilute al models we construct a free field realization of vertex operators of the dilute al models along with the felder complex for l 3 we also study an e8 structure in terms of the deformed virasoro currents

twisting invariance of link polynomials derived from ribbon quasi hopf algebras the construction of link polynomials associated with finite dimensional representations of ribbon quasi hopf algebras is discussed in terms of the formulation of an appropriate markov trace we then show that this markov trace is invariant under twisting of the quasi hopf structure which in turn implies twisting invariance of the associated link polynomials

sequential measurement of conjugate variables as an alternative quantum state tomography it is shown how it is possible to reconstruct the initial state of a one dimensional system by sequentially measuring two conjugate variables the procedure relies on the quasicharacteristic function the fourier transform of the wigner quasiprobability the proper characteristic function obtained by fourier transforming the experimentally accessible joint probability of observing position then momentum or vice versa can be expressed as a product of the quasicharacteristic function of the two detectors and that unknown of the quantum system this allows state reconstruction through the sequence 1 data collection 2 fourier transform 3 algebraic operation and 4 inverse fourier transform the strength of the measurement should be intermediate for the procedure to work

the deformed virasoro algebra at roots of unity we discuss some aspects of the representation theory of the deformed virasoro algebra virpq in particular we give a proof of the formula for the kac determinant and then determine the center of virp q for q a primitive nth root of unity we derive explicit expressions for the generators of the center in the limit t qp 1 and elucidate the connection to the hall littlewood symmetric functions furthermore we argue that for the algebra describes gentile statistics of order n 1 i e a situation in which at most n 1 particles can occupy the same state

diagonalization of infinite transfer matrix of boundary uq p an 1 1 face model we study infinitely many commuting operators tb z which we call infinite transfer matrix of boundary uq p an 1 1 face model we diagonalize the infinite transfer matrix tb z by using free field realizations of the vertex operators of the elliptic quantum group uq p an 1 1

on the vertex operators of the elliptic quantum algebra uq p sl2 k a realization of the elliptic quantum algebra uq p sl2 for any given level k is constructed in terms of three free boson fields and their accompanying twisted partners it can be viewed as the elliptic deformation of the wakimoto realization two screening currents are constructed they commute or anticommute with uq p sl2 modulo total q differences the free field realization for two types of vertex operators nominated as the type i and the type ii vertex operators is presented the twisted version of the two types of vertex operators are also obtained they all play crucial roles in calculating correlation functions

the elliptic algebra and the deformation of wn algebra after reviewing the recent results on the drinfeld realization of the face type elliptic quantum group b q sl n by the elliptic algebra u q p sl n we investigate a fusion of the vertex operators of u q p sl n the basic generating functions j z 1 j n 1 of the deformed w n algebra are derived explicitly