deformed algebras from elliptic sl n algebras we extend to the case the results that we previously obtained on the construction of algebras from the elliptic algebra the elliptic algebra at the critical level c n has an extended center containing trace like operators t z families of poisson structures indexed by n n 1 2 integers defining q deformations of the algebra are constructed the operators t z also close an exchange algebra when for it becomes abelian when in addition p qnh where h is a non zero integer the poisson structures obtained in these classical limits contain different q deformed algebras depending on the parity of h characterizing the exchange structures at p qnh as new algebras

a vertex operator approach for form factors of belavin s mathbb z n mathbb z symmetric model belavin s symmetric model is considered on the basis of bosonization of vertex operators in the a 1 n 1 model and vertex face transformation free field representations of nonlocal tail operators are constructed for off diagonal matrix elements with respect to the ground state sectors as a result integral formulae for form factors of any local operators in the symmetric model can in principle be obtained

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

an elliptic algebra and the fusion rsos model we introduce an elliptic algebra with and present its free boson representation at generic level k we show that this algebra governs a structure of the space of states in the k fusion rsos model specified by a pair of positive integers r k or equivalently a q deformation of the coset conformal field theory extending the work by lukyanov and pugai corresponding to the case k 1 we give a full set of screening operators for k 1 the algebra has two interesting degeneration limits p 0 and p 1 the former limit yields the quantum affine algebra whereas the latter yields the algebra the scaling limit of the elliptic algebra using this correspondence we also obtain the highest component of two types of vertex operators which can be regarded as q deformations of the primary fields in the coset conformal field theory

exact evaluation of the simple cubic lattice green function for a general lattice point the simple cubic lattice green function is investigated where l m n denotes a set of integers and w u iv is a complex variable in the u v plane in particular it is shown that the modified green function can be expressed in the parametric form where k k and e k are complete elliptic integrals of the first and second kind respectively with a modulus the connection between the parameter and the variable w is given by and rj l m n j 0 1 2 3 is a set of rational functions of it is found that the complete elliptic integral formulae for the green functions n n n w n 1 2 3 4 and 2n n n w n 1 2 3 4 can be factorized as a product of two linear forms in k k and e k whose coefficients are rational functions of the parameter on the basis of these explicit results it is conjectured that this factorization propery is valid for all integer values of n

universal vertex irf transformation for quantum affine algebras we construct a universal solution of the generalized coboundary equation in the case of quantum affine algebras which is an extension of our previous work to uq ar 1 this universal solution has a simple gauss decomposition which is constructed using sevostyanov s characters of twisted quantum borel algebras we show that in the evaluation representations it gives a vertex face transformation between a vertex type solution and a face type solution of the quantum dynamical yang baxter equation in particular in the evaluation representation of uq a1 1 it gives baxter s well known transformation between the 8 vertex model and the interaction round faces irf height model

integration of quantum many body problems by affine knizhnik zamolodchikov equations abstract an equivalence of the affine analogues of knizhnik zamolodchikov and dunkl equations is established for arbitrary root systems as a corollary an isomorphism is obtained from the solutions of the above kz equations in the induced representations of hecke algebras to the eigenfunctions for the corresponding matrix quantum calogero sutherland operators which generalizes the matsuo map

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

the integrals of motion for the elliptic deformation of the virasoro and wn algebra abstract we review the free field realization of the deformed virasoro algebra v i r q t and the deformed w albegra w q t g l n we explicitly construct two classes of infinitely many commutative operators i m g m m n in terms of these algebras they can be regarded as the elliptic deformation of the local and nonlocal integrals of motion for the conformal field theory this review is based on the works b feigin t kojima j shiraishi h watanbe the integrals of motion for the deformed virasoro algebra submitted for publication 2007 preprint 54 pages arxiv 0705 0427 b feigin t kojima j shiraishi h watanbe the integrals of motion for the deformed w algebra w q t g l n in proceedings of representation theory 2006 atami japan 2006 pp 102 114 t kojima and j shiraishi the integrals of motion for the deformed w algebra w q t g l n ii proof of the commutation relations comm math phys 283 2008 795 851

the integrals of motion for the deformed w algebra w_ q t widehat gl_n ii proof of the commutation relations we explicitly construct two classes of infinitely many commutative operators in terms of the deformed w algebra w q t widehat gl n and give proofs of the commutation relations of these operators we call one of them local integrals of motion and the other nonlocal since they can be regarded as elliptic deformations of local and nonlocal integrals of motion for the virasoro algebra and the w 3 algebra 1 2

current redistribution in resistor networks fat tail statistics in regular and small world networks the redistribution of electrical currents in resistor networks after single bond failures is analyzed in terms of current redistribution factors that are shown to depend only on the topology of the network and on the values of the bond resistances we investigate the properties of these current redistribution factors for regular network topologies e g d dimensional hypercubic lattices as well as for small world networks in particular we find that the statistics of the current redistribution factors exhibits a fat tail behavior which reflects the long range nature of the current redistribution as determined by kirchhoff s circuit laws

some combinatorial properties of jack symmetric functions if 1 l n then write 2 n or 1 n if p is another partition then write p c j if pi 6 for all i i e if the diagram of 1 contains the diagram of p if ipl ii 1 then write p 2 1 reverse lexicographic order if either j i or the first nonvanishing difference e pi is positive for instance writing 2 1 i for ai i it 5 5 41 32 5 311 221 2111 11111 finally write p i if ipl lal and p1 pl2 pli a1 a2 li for all i macdonald m p 63 call the partial ordering the natural ordering but we will call it the dominance ordering let x x x2 be an infinite set of indeterminates as in mi we

on the cubic lattice green functions wheatstone physics laboratory king s college university of london strand london wc2r 2ls u k it is proved that k k 4 eta frac 1 2 1 eta frac 1 2 k k where eta is a complex variable which lies in a certain region scr r 2 of the eta plane and k k pm are complete elliptic integrals of the first kind with moduli k pm which are given by k pm 2 equiv k pm 2 eta frac 1 2 pm frac 1 4 eta 4 eta frac 1 2 frac 1 4 2 eta 1 eta frac 1 2 this basic result is then used to express the face centred cubic and simple cubic lattice green functions at the origin in terms of the square of a complete elliptic integral of the first kind several new identities involving the heun function f a b alpha beta gamma delta eta are also derived next it is shown that the three cubic lattice green functions all have parametric representations which involve the green function for the two dimensional honeycomb lattice finally the results are applied to a variety of problems in lattice statistics in particular a new simplified formula for the generating function of staircase polygons on a four dimensional hypercubic lattice is derived

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

on the cubic modular transformation and the cubic lattice green functions the elliptic modular transformation of order 3 is used to prove that the lattice green functions at the origin for the face centred cubic and simple cubic lattices can both be expressed in terms of the square of a complete elliptic integral of the first kind

dynamically twisted algebra as current algebra generalizing screening currents of q deformed virasoro algebra in this paper we propose an elliptic algebra which is based on the relations where r and are the dynamical r matrices of type face model with the elliptic moduli shifted by the centre of the algebra from the ding frenkel correspondence we find that its corresponding drinfeld current algebra at level one is the algebra of screening currents for q deformed virasoro algebra we realize the elliptic algebra at level one by miki s construction from the bosonization for the type i and type ii vertex operators we also show that the algebra is related to the algebra by dynamically twisting

equivalence after extension and matricial coupling coincide with schur coupling on separable hilbert spaces it is known that two banach space operators that are schur coupled are also equivalent after extension or equivalently matricially coupled the converse implication that operators which are equivalent after extension or matricially coupled are also schur coupled was only known for fredholm hilbert space operators and fredholm banach space operators with index 0 we prove that this implication also holds for hilbert space operators with closed range generalizing the result for fredholm operators and banach space operators that can be approximated in operator norm by invertible operators the combination of these two results enables us to prove that the implication holds for all operators on separable hilbert spaces

recursive evaluation of space time lattice green s functions up to a multiplicative constant the lattice green s function lgf as defined in condensed matter physics and lattice statistical mechanics is equivalent to the z domain counterpart of the finite difference time domain green s function gf on a lattice expansion of a well known integral representation for the lgf on a dimensional hyper cubic lattice in powers of z 1 and application of the chu vandermonde identity results in 1 nested finite sum representations for discrete space time gfs due to severe numerical cancellations these nested finite sums are of little practical use for 2 the finite sum may be evaluated in closed form in terms of a generalized hypergeometric function for special lattice points that representation simplifies considerably while on the other hand the finite difference stencil may be used to derive single lattice point second order recurrence schemes for generating 2d discrete space time gf time sequences on the fly for arbitrary symbolic lattice points zeilberger s algorithm produces a third order recurrence operator with polynomial coefficients of the sixth degree the corresponding recurrence scheme constitutes the most efficient numerical method for the majority of lattice points in spite of the fact that for explicit numeric lattice points the associated third order recurrence operator is not the minimum recurrence operator as regards the asymptotic bounds for the possible solutions to the recurrence scheme perron s theorem precludes factorial or exponential growth along horizontal lattices directions rapid initial growth does occur but poses no problems in augmented dynamic range fixed precision arithmetic by analysing long distance wave propagation along a horizontal lattice direction we have concluded that the chirp up oscillations of the discrete space time gf are the root cause of grid dispersion anisotropy with each factor of ten increase in the lattice distance one would have to roughly double the pulse width of the source signature to keep pulse distortion at bay the gf time sequences can also be used for an efficient computation of discrete space frequency lgfs especially if one employs aitken s 2 process for the acceleration of the convergence of the consecutive partial sums

schur coupling and related equivalence relations for operators on a hilbert space abstract for operators on hilbert spaces of any dimension we show that equivalence after extension coincides with equivalence after one sided extension thus obtaining a proof of their coincidence with schur coupling we also provide a concrete description of this equivalence relation in several cases in particular for compact operators

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

transfer functions and operator theory abstract the notion of a transfer function from systems theory serves as an important tool in dealing with certain problems in operator and matrix theory this is illustrated with a variety of material taken from existing literature a striking feature of the results is their high degree of explicitness

quantum 401 1401 1401 1algebras and macdonald polynomials we derive a quantum deformation of thew n algebra and its quantum miura transformation whose singular vectors realize the macdonald polynomials

a j spectral factorization approach to h control it is shown that necessary and sufficient conditions for the existence of sub optimal solutions to the model matching problem associated with h control are that two coupled j spectral factorizations exist the second j spectral factor provides a parameterization of all solutions to the model matching problem the existence of the j spectral factors is then shown to be equivalent to the existence of stabilizing non negative definite solutions to two algebraic riccati equations allowing a state space formula for a linear fractional representation of all controllers to be given

exact values for the cubic lattice green functions it is shown that the value of the cubic lattice green functions g z l m n at the upper critical point z 3 for the simple cubic and face centred cubic lattices or z 1 for the body centred cubic lattice is expressible rationally in terms of the known value of g 3 0 0 0 or g 1 0 0 0 and

the elliptic quantum algebra aq p sln and its bosonization at level one we extend the work of foda et al and propose an elliptic quantum algebra aq p sln similar to the case of aq p sl2 our presentation of the algebra is based on the relation rll llr where r and r are zn symmetric r matrices with the elliptic moduli chosen differently and a scalar factor is also involved with the help of the results obtained by asai et al we realize type i and type ii vertex operators in terms of bosonic free fields for the zn symmetric belavin model we also give a bosonization for the elliptic quantum algebra aq p sln at level one

the q agt w relations via shuffle algebras we construct the action of the q deformed w algebra on its level r representation geometrically using the moduli space of u r instantons on the plane and the double shuffle algebra we give an explicit ldu decomposition for the action of w algebra currents in the fixed point basis of the level r representation and prove a relation between the carlsson okounkov ext operator and intertwiners for the deformed w algebra we interpret this result as a q deformed version of the agt w relations

elliptic quantum groups e_ tau eta mathfrak s mathfrak l _2 and quasi hopf algebras we construct an algebra morphism from the elliptic quantum group e tau eta mathfrak s mathfrak l 2 to a certain elliptic version of the quantum loop groups in higher genus studied by v rubtsov and the first author this provides an embedding of e tau eta mathfrak s mathfrak l 2 in an algebra with central extension in particular we construct l operators obeying a dynamical version of the reshetikhin semenov tian shansky relations to do that we construct the factorization of a certain twist of the quantum loop algebra that automatically satisfies the twisted cocycle equation of o babelon d bernard and e billey and therefore provides a solution of the dynamical yang baxter equation

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

a j spectral factorization approach to control necessary and sufficient conditions for the existence of suboptimal solutions to the standard model matching problem associated with mathcal h infty control j control are derived using j spectral factorization theory the existence of solutions to the model matching problem is shown to be equivalent to the existence of solutions to two coupled j spectral factorization problems with the second factor providing a parametrization of all solutions to the model matching problem the existence of the j spectral factors is then shown to be equivalent to the existence of nonnegative definite stabilizing solutions to two indefinite algebraic riccati equations allowing a state space formula for a linear fractional representation of all controllers to be given a virtue of the approach is that a very general class of problems may be tackled within a conceptually simple framework and no additional auxiliary riccati equations are required

integral representations of the macdonald symmetric polynomials multiple integral representations of the skew macdonald symmetric polynomials are obtained some bosonization schemes for the integral representations are also constructed

complete integrability of relativistic calogero moser systems and elliptic function identities poincare invariant generalizations of the galilei invariant calogero moser a particle systems are studied a quantization of the classical integrals s1 sn is presented such that the operators si9 sn mutually commute as a corollary it follows that si9 9sn poisson commute these results hinge on functional equations satisfied by the weierstrass and 0 functions a generalized cauchy identity involving the function leads to an n x n matrix l whose symmetric functions are proportional to sl5 9sn

elliptic modular double and 4d partition functions we consider 4d supersymmetric special unitary gamma quiver gauge theories on compact manifolds which are t 2 fibrations over s 2 we show that their partition functions are correlators of vertex

macdonald operators and homological invariants of the colored hopf link using a power sum boson realization for the macdonald operators we investigate the gukov iqbal kozcaz and vafa gikv proposal for the homological invariants of the colored hopf link which include khovanov rozansky homology as a special case we prove the polynomiality of the invariants obtained by gikv s proposal for arbitrary representations we derive a closed formula of the invariants of the colored hopf link for antisymmetric representations we argue that a little amendment of gikv s proposal is required to make all the coefficients of the polynomial non negative integers

exact product forms for the simple cubic lattice green function i the analytical properties of the lattice green function are investigated where n is an integer and w is a complex variable in particular it is demonstrated that g n n n w is a solution of a fourth order linear differential equation of the fuchsian type from this differential equation it is found that g n n n w can be evaluated in terms of a product of two heun functions hj n v j 1 2 where a detailed discussion of the properties of hj n v j 1 2 is then given the heun function results are used to prove that the product form for g n n n w can be expressed in terms of complete elliptic integrals of the first and second kinds it is also shown that g n n n w can be written in the hypergeometric form where this formula is valid for varying values of w in the neighbourhood of w provided that the argument function w does not take real values in the interval 1 finally this 2f1 product form is used to determine the asymptotic behaviour of g n n n w as n

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

a probabilistic interpretation of the macdonald polynomials the two parameter macdonald polynomials are a central object of algebraic combinatorics and representation theory we give a markov chain on partitions of k with eigenfunctions the coefficients of the macdonald polynomials when expanded in the power sum polynomials the markov chain has stationary distribution a new two parameter family of measures on partitions the inverse of the macdonald weight rescaled the uniform distribution on cycles of permutations and the ewens sampling formula are special cases the markov chain is a version of the auxiliary variables algorithm of statistical physics properties of the macdonald polynomials allow a sharp analysis of the running time in natural cases a bounded number of steps suffice for arbitrarily large k

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

deformed virasoro algebras from elliptic quantum algebras we revisit the construction of deformed virasoro algebras from elliptic quantum algebras of vertex type generalizing the bilinear trace procedure proposed in the 1990s it allows us to make contact with the vertex operator techniques that were introduced separately at the same period as a by product the method pinpoints two critical values of the central charge for which the center of the algebra is extended as well as in the gl 2 case a liouville formula

exact product form for the anisotropic simple cubic lattice green function the analytical properties of the lattice green function are investigated where n is an integer w is a complex variable and is a real parameter in the interval 0 in particular it is shown that g 2n n n w is a solution of a fourth order linear differential equation of the fuchsian type from this differential equation it is proved that g 2n n n w can be expressed in the hypergeometric form where and n denotes the pochhammer symbol this formula is valid for varying values of w in the neighbourhood of w provided that the argument function w does not take real values in the interval 1 the 2f1 product form is used to determine the asymptotic behaviour of g 2n n n w as n finally a five term linear recurrence relation is given for g 2n n n w

brief necessary and sufficient conditions for existence of j spectral factorization for para hermitian rational matrix functions for a para hermitian rational matrix function g l j c li p a 1b where j j is invertible and which has no poles and zeros on the imaginary line we give necessary and sufficient conditions in terms of a b c and j for the existence of a j spectral factorization as well as an algorithm to obtain the j spectral factor in case it exists

stabilisation of matrix polynomials a state feedback is proposed to analyse the stability of a matrix polynomial in closed loop first it is shown that a matrix polynomial is stable if and only if a state space realisation of a ladder form of certain transfer matrix is stable following the ideas of the routh hurwitz stability procedure for scalar polynomials certain continued fraction expansions of polynomial matrices are carrying out by unimodular matrices to achieve the euclid s division algorithm which leads to an extension of the well known routh hurwitz stability criteria but this time in terms of matrix coefficients after that stability of the closed loop matrix polynomial is guaranteed based on a corollary of a lyapunov theorem the sufficient stability conditions are i the matrices of one column of the presented array must be symmetric and positive definite and ii the matrices of the cascade realisation must satisfy a commutative condition these stability conditions are also necessary for matrix polynomial of second order

elliptic algebra in the scaling limit the scaling limit of the elliptic algebra is investigated the limiting algebra is defined in terms of a continuous family of generators being fourier harmonics of gauss coordinates of the l operator the ding frenkel isomorphism between l operator s and current descriptions of the algebra is established and is identified with the riemann problem on a strip the representations coalgebraic structure and intertwining operators of the algebra are studied

integrable connections related to zonal spherical functions we define a system of differential equations of first order for a function valued in the group algebra of the weyl group associated with an arbitrary root system this is equivalent to the system of differential equations given by heckman and opdam which is a deformation of the system satisfied by the zonal spherical function of the riemannian symmetric spaceg k of non compact type when the root system is a n type our equation is related to the knizhnik zamolodchikov equation in conformal field theory

free field construction for the abf models in regime ii the wakimoto construction for the quantum affine algebra u q widehat mathfrak s mathfrak l 2 admits a reduction to the q deformed parafermion algebras we interpret the latter theory as a free field realization of the andrews baxter forrester models in regime ii we give multi particle form factors of some local operators on the lattice and compute their scaling limit where the models are described by a massive field theory with mathbb z k symmetric minimal scattering matrices

classical lattice w algebras and integrable systems several aspects of the lattice algebra are studied motivated by the fact that the lotka volterra model can be written in terms of a current of the lattice virasoro algebra the faddeev takhtajan volkov algebra integrable dynamical models on the lattice have been formulated as a model associated with the lattice algebra

wiener hopf factorization inverse fourier transforms and exponentially dichotomous operators abstract for a certain class of matrix functions that are analytic on the real line but not at infinity an explicit method of factorization and of constructing inverse fourier transforms is developed this method is applied to invert wiener hopf integral equations on the half line and the full line the results obtained extend analogous results of the authors for rational matrix functions and for functions that are analytic on the real line and at infinity the analysis is based on an infinite dimensional realization theorem which involves operators that are a direct sum of two infinitesimal generators of c 0 semigroups of negative exponential type one of which has support on the negative half line and the other on the positive half line the latter operators are called exponentially dichotomous and the study of their properties forms an essential part of the paper

schur complements and state space realizations motivated by state space realizations of transfer functions from system theory a number of operations on schur complements are introduced and studied these operations are equivalence extension multiplication inversion and factorization together they form an algebraic framework which is of independent interest and also useful in solving problems in analysis

equivalence after extension for compact operators on banach spaces in recent years the coincidence of the operator relations equivalence after extension and schur coupling was settled for the hilbert space case by showing that equivalence after extension implies equivalence after one sided extension in the present paper we investigate consequences of equivalence after extension for compact banach space operators we show that generating the same operator ideal is necessary but not sufficient for two compact operators to be equivalent after extension in analogy with the necessary and sufficient conditions for compact hilbert space operators to be equivalent after extension in terms of their singular values we prove under certain additional conditions the necessity of a similar relationship between the s numbers of two compact banach space operators that are equivalent after extension for arbitrary s functions we investigate equivalence after extension for operators on lp spaces we show that two operators that act on different lp spaces cannot be equivalent after one sided extension such operators can still be equivalent after extension for instance all invertible operators are equivalent after extension however if one of the two operators is compact then they cannot be equivalent after extension this contrasts the hilbert space case where equivalence after one sided extension and equivalence after extension are in fact identical relations finally for general banach spaces x and y we investigate consequences of an operator on x being equivalent after extension to a compact operator on y we show that in this case a closed finite codimensional subspace of y must embed into x and that certain general banach space properties must transfer from x to y we also show that no operator on x can be equivalent after extension to an operator on y if x and y are essentially incomparable banach spaces

an a r threesome matrix models 2d conformal field theories and 4d n 2 gauge theories we explore the connections between three classes of theories a quiver matrix models d 2 conformal a r toda field theories and d 4 n 2 supersymmetric conformal a r quiver gauge theories in particular we analyze the quiver matrix models recently introduced by dijkgraaf and vafa unpublished and make detailed comparisons with the corresponding quantities in the toda field theories and the n 2 quiver gauge theories we also make a speculative proposal for how the matrix models should be modified in order for them to reproduce the instanton partition functions in quiver gauge theories in five dimensions c 2010 american institute of physics doi 10 1063 1 3449328

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

complementary schur complements abstract for square matrices the relationship is discussed between the notion of schur complement and the equivalent concepts of matricial coupling and equivalence after extension special attention is paid to the hermitian and skew hermitian cases

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