gaudin models with irregular singularities abstract we introduce a class of quantum integrable systems generalizing the gaudin model the corresponding algebras of quantum hamiltonians are obtained as quotients of the center of the enveloping algebra of an affine kac moody algebra at the critical level extending the construction of higher gaudin hamiltonians from b feigin et al 1994 17 to the case of non highest weight representations of affine algebras we show that these algebras are isomorphic to algebras of functions on the spaces of opers on p 1 with regular as well as irregular singularities at finitely many points we construct eigenvectors of these hamiltonians using wakimoto modules of critical level and show that their spectra on finite dimensional representations are given by opers with trivial monodromy we also comment on the connection between the generalized gaudin models and the geometric langlands correspondence with ramification

on kostant s theorem for the lie superalgebra q n abstract in this paper we study finite w algebras for basic superalgebras and q n associated with the regular even nilpotent coadjoint orbits we prove that this algebra satisfies the amitsur levitzki identity and therefore all its irreducible representations are finite dimensional in the case of q n we give an explicit description of the w algebra in terms of generators and relations and realize it as a quotient of the super yangian of q 1

homological and monodromy representations of framed braid groups in this paper we introduce two new classes of representations of the framed braid groups one is the homological representation constructed as the action of a mapping class group on a certain homology group the other is the monodromy representation of the confluent kz equation which is a generalization of the kz equation to have irregular singularities we also give a conjectural equivalence between these two classes of representations

homological stability for automorphism groups abstract given a family of groups admitting a braided monoidal structure satisfying mild assumptions we construct a family of spaces on which the groups act and whose connectivity yields via a classical argument of quillen homological stability for the family of groups we show that stability also holds with both polynomial and abelian twisted coefficients with no further assumptions this new construction of a family of spaces from a family of groups recovers known spaces in the classical examples of stable families of groups such as the symmetric groups general linear groups and mapping class groups by making systematic the proofs of classical stability results we show that they all hold with the same type of coefficient systems obtaining in particular without any further work new stability theorems with twisted coefficients for the symmetric groups braid groups automorphisms of free groups unitary groups mapping class groups of non orientable surfaces and mapping class groups of 3 manifolds our construction can also be applied to families of groups not considered before in the context of homological stability as a byproduct of our work we construct the braided analogue of the category fi of finite sets and injections relevant to the present context and define polynomiality for functors in the context of pre braided monoidal categories

on the finite w algebra for the lie superalgebra q n in the non regular case in this paper we study the finite w algebra for the queer lie superalgebra q n associated with the non regular even nilpotent coadjoint orbits in the case when n nl and the corresponding nilpotent element has jordan blocks each of size l we prove that this finite w algebra is isomorphic to a quotient of the super yangian of q n

some new results in the theory of negative imaginary systems with symmetric transfer matrix function abstract this note represents a first attempt to provide a definition and characterisation of negative imaginary systems for not necessarily rational transfer functions via a sign condition expressed in the entire domain of analyticity along the same lines of the classic definition of positive real systems under the standing assumption of symmetric transfer functions we then derive a necessary and sufficient condition that characterises negative imaginary transfer functions in terms of a matrix sign condition restricted to the imaginary axis once again following the same line of argument of the standard positive real case using this definition even transfer functions with a pole at the origin with double multiplicity as well as with a possibly negative relative degree can be negative imaginary

stable bundles and integrable systems on considere la geometrie symplectique des fibres cotangents aux espaces de modules de fibres vectoriels stables sur une surface de riemann on montre que ce sont des systemes dynamiques hamiltoniens algebriquement completement integrables

quantization of the shift of argument subalgebras in type a abstract given a simple lie algebra g and an element g the corresponding shift of argument subalgebra of s g is poisson commutative in the case where is regular this subalgebra is known to admit a quantization that is it can be lifted to a commutative subalgebra of u g we show that if g is of type a then this property extends to arbitrary thus proving a conjecture of feigin frenkel and toledano laredo the proof relies on an explicit construction of generators of the center of the affine vertex algebra at the critical level

on pseudo spectral factorization over the complex numbers and quaternions abstract this paper is a continuation of the research of our previous work 5 and considers quaternionic generalized caratheodory functions and the related family of generalized positive functions it is addressed to a wide audience which includes researchers in complex and hypercomplex analysis in the theory of linear systems but also electric engineers for this reason it includes some results on generalized caratheodory functions and their factorization in the classic complex case which might be of independent interest an important new result is a pseudo spectral factorization and we also discuss some interpolation problems in the class of quaternionic generalized positive functions

gaudin models for m n we establish the basics of the bethe ansatz for the gaudin model associated to the lie superalgebra x1d524 x1d529 m n in particular we prove the completeness of the bethe ansatz in the case of tensor products of fundamental representations

confluent kz equations for mathfrak sl _n with poincare rank 2 at infinity we construct confluent kz equations with poincar rank 2 at infinity for the case of and the integral representation for the solutions hamiltonians of these confluent kz equations are derived from suitable quantization of dlog constructed in the theory of monodromy preserving deformation mpd in jimbo et al 1981 physica d 2 306 52 our confluent kz equations can be viewed as a quantization of mpd with poincar rank 2 at infinity

on the structure of quantum vertex algebras a definition of a quantum vertex algebra which is a deformation of a vertex algebra was proposed by etingof and kazhdan in 1998 sel math 6 1 105 130 2000 in a nutshell a quantum vertex algebra is a braided state field correspondence that satisfies associativity and braided locality axioms we develop a structure theory of quantum vertex algebras parallel to that of vertex algebras in particular we introduce braided n products for a braided state field correspondence and prove for quantum vertex algebras a version of the borcherds identity a definition of a quantum vertex algebra which is a deformation of a vertex algebra was proposed by etingof and kazhdan in 1998 sel math 6 1 105 130 2000 in a nutshell a quantum vertex algebra is a braided state field correspondence that satisfies associativity and braided locality axioms we develop a structure theory of quantum vertex algebras parallel to that of vertex algebras in particular we introduce braided n products for a braided state field correspondence and prove for quantum vertex algebras a version of the borcherds identity

multiple orthogonal polynomials and a counterexample to the gaudin bethe ansatz conjecture jacobi polynomials are polynomials whose zeros form the unique solution of the bethe ansatz equation associated with two irreducible modules we study sequences of polynomials whose zeros form the unique solution of the bethe ansatz equation associated with two highest weight irreducible modules with the restriction that the highest weight of one of the modules is a multiple of the first fundamental weight we describe the recursion which can be used to compute these polynomials moreover we show that the first polynomial in the sequence coincides with the jacobi pineiro multiple orthogonal polynomial and others are given by wronskian type determinants of jacobi pineiro polynomials as a byproduct we describe a counterexample to the bethe ansatz conjecture for the gaudin model

bosonic representations of yangian double with on the basis of the rtt ttr formalism we introduce the quantum double of the yangian for with a central extension the gauss decomposition of the t matrices gives us the so called drinfel d generators using these generators we present some examples of both finite and infinite dimensional representations that are quite natural deformations of their corresponding affine counterpart

a dual formulation of mixed spl mu and on the losslessness of d g scaling this paper studies the mixed structured singular value spl mu and the well known d g scaling upper bound spl nu a dual characterization of spl mu and spl nu is derived which intimately links the two values using the duals it is shown that spl nu is guaranteed to be lossless i e equal to spl mu if and only if 2 m sub r m sub e m sub c spl les 3 where m sub r m sub c and m sub c are the numbers of repeated real scalar blocks repeated complex scalar blocks and full complex blocks respectively the losslessness result further leads to a variation of the well known kalman yakubovich popov lemma and lyapunov inequalities

affine kac moody algebras and semi infinite flag manifolds we study representations of affine kac moody algebras from a geometric point of view it is shown that wakimoto modules introduced in 18 which are important in conformal field theory correspond to certain sheaves on a semi infinite flag manifold with support on its schhubert cells this manifold is equipped with a remarkable semi infinite structure which is discussed in particular the semi infinite homology of this manifold is computed the cousin grothendieck resolution of an invertible sheaf on a semi infinite flag manifold gives a two sided resolution of an irreducible representation of an affine algebras consisting of wakimoto modules this is just the brst complex as a byproduct we compute the homology of an algebra of currents on the real line with values in a nilpotent lie algebra

on the kalman yakubovich popov lemma the purpose of this note is to present a new elementary proof for the multivariable k y p lemma a minimum of linear algebra and finite dimensional convexity theory is used

affine sergeev algebra and q analogues of the young symmetrizers for projective representations of the symmetric group we study a q deformation for the crossed product of the symmetric group s n with the clifford algebra on n anticommuting generators we obtain a q version of the projective analogue for the classical young symmetrizer introduced by the second author our main tool is an analogue of the hecke algebra of complex valued functions on the group mbox gl n over a p adic field relative to the iwahori subgroup

staggered polarization of vertex models with u_q widehat sl n symmetry in this paper we give an explicit formula for level 1 vertex operators related to u q widehat sl n as operators on the fock spaces we derive also their commutation relations as an application we calculate with the vector representation of u q widehat sl n thereby extending the recent work on the staggered polarization of the xxz model

semidefinite programming duality and linear time invariant systems several important problems in control theory can be reformulated as semidefinite programming problems i e minimization of a linear objective subject to linear matrix inequality lmi constraints from convex optimization duality theory conditions for infeasibility of the lmis as well as dual optimization problems can be formulated these can in turn be reinterpreted in control or system theoretic terms often yielding new results or new proofs for existing results from control theory we explore such connections for a few problems associated with linear time invariant systems

free boson representation of dyℏ gln k and dyℏ sln k we construct a realization of the yangian double dy h over bar gl n k and dy h over bar sl n k for an arbitrary level k in terms of free boson fields with a continuous parameter in the h over bar 0 limit this realization becomes the wakimoto realization of affine lie algebras gl n and sl n respectively the vertex operators and the screening currents are also constructed with the same spirits the screening currents commute with dy h over bar sl n module total difference c 1998 american institute of physics s0022 2488 98 00104 2

fock representations of the affine lie algebra a 1 1 the aim of this note is to show that the affine lie algebraa1 1 has a natural family v of fock representations on the spacec xi yj i andj parametrized by v c2 by corresponding the highest weight of to each the parameter spacec2 forms a double cover of the weight spacec 0 c 1 with singularities at linear forms of level 2 this number is 1 times the dual coxeter number our results contain explicit realizations of irreducible non integrable highest wieghta1 1 modules for generic v

free field representation for massive integrable models a new approach to massive integrable models is considered it allows one to find symmetry algebras which define the spaces of local operators and to get general integral representations for form factors in thesu 2 thirring and sine gordon models

twisted affine lie superalgebra of type q and quantization of its enveloping superalgebra we introduce a new quantum group which is a quantization of the enveloping superalgebra of a twisted affine lie superalgebra of type q we study generators and relations for superalgebras in the finite and twisted affine cases and also universal central extensions afterwards we apply the frt formalism to a certain solution of the quantum yang baxter equation to define that new quantum group and we study some of its properties we construct a functor of schur weyl type which connects it to affine hecke clifford algebras and prove that it provides an equivalence between two categories of modules

stability of the homology of the mapping class groups of orientable surfaces the mapping class group of f fgs r is f rgs wo a where a is the topological group of orientation preserving diffeomorphisms of f which are the identity on df and fix the s punctures when r 0 and 2g s 3 an important feature of f is that it acts properly discontinuously on the teichmiller space us the quotient is the moduli space f of isometry classes of complete hyperbolic metrics of finite area on f since us is homeomorphic to euclidean space and f is virtually torsion free we have

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

opers with irregular singularity and spectra of the shift of argument subalgebra joint work with boris feigin and edward frenkel math qa 0712 1183 let g be a semisimple complex lie algebra and u g its universal enveloping algebra the algebra u g bears a natural filtration by the degree with respect to the generators the associated graded algebra is the symmetric algebra s g c g by the poincare birkhoff witt theorem the commutator on u g defines the poisson bracket on s g let zs g s g be the center of s g with respect to the poisson bracket and let g g be a regular semisimple element due to the result of mischenko and fomenko 1978 the algebra a s g generated by the elements where zs g is commutative with respect to the poisson bracket and has maximal possible transcendence degree equal to 12 dim g rk g if is a regular element contained in the cartan subalgebra h g then h a and if f is the principal nilpotent element then the subalgebra af contains the principal nilpotent subalgebra zg f it was recently shown that the shift of argument subalgebras can be quantized

confluent primary fields in the conformal field theory for any complex simple lie algebra we generalize the primary fields in the wess zumino novikov witten conformal field theory for a case with irregular singularities we refer to these generalized primary fields as confluent primary fields we present the screening currents ward identity a recursion rule for computing the expectation values of the products of confluent primary fields in the case of the expectation values of the products of confluent primary fields are integral formulas of solutions to confluent knizhnik zamolodchikov kz equations given in jimbo et al 2008 j phys a math theor 41 175205 by computing the operator product expansion of the energy momentum tensor t z and the confluent primary fields we obtain new differential operators moreover in the case of these differential operators are the same as those of the confluent kz equations jimbo et al 2008

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 theory of passive linear systems with no assumptions we present two linked theorems on passivity the passive behavior theorem parts 1 and 2 part 1 provides necessary and sufficient conditions for a general linear system described by a set of high order differential equations to be passive part 2 extends the positive real lemma to include uncontrollable and unobservable state space systems

extended trigonometric cherednik algebras and nonstationary schrodinger equations with delta potentials we realize an extended version of the trigonometric cherednik algebra as affine dunkl operators involving heaviside functions we use the quadratic casimir element of the extended trigonometric cherednik algebra to define an explicit nonstationary schrodinger equation with delta potential we use coordinate bethe ansatz methods to construct solutions of the nonstationary schrodinger equation in terms of generalized bethe wave functions it is shown that the generalized bethe wave functions satisfy affine difference knizhnik zamolodchikov equations as functions of the momenta the relation to the vector valued root system analogs of the quantum bose gas on the circle with delta function interactions is indicated

finite dimensional irreducible representations of twisted yangians we study quantized enveloping algebras called twisted yangians they are analogs of the yangian y gl n for the classical lie algebras of b c and d series the twisted yangians are subalgebras in y gl n and coideals with respect to the coproduct in y gl n we give a complete description of their finite dimensional irreducible representations every such representation is highest weight and we give necessary and sufficient conditions for an irreducible highest weight representation to be finite dimensional the result is analogous to drinfeld s theorem for the ordinary yangians its detailed proof for the a series is also reproduced for the simplest twisted yangians we construct an explicit realization for each finite dimensional irreducible representation in tensor products of representations of the corresponding lie algebras

isomorphism between two realizations of the yangian y mathfrak so _3 the isomorphism between drinfeld s new realization and the frt realization is proved for the yangian algebra using gauss decomposition

yangian of the queer lie superalgebra consider the complex matrix lie superalgebra with the standard generators where define an involutory automorphism of by the twisted polynomial current lie superalgebra has a natural lie co superalgebra structure we quantise the universal enveloping algebra as a co poisson hopf superalgebra for the quantised algebra we give a description of the centre and construct the double in the sense of drinfeld we also construct a wide class of irreducible representations of the quantised algebra

iterated monoidal categories abstract we develop a notion of an n fold monoidal category and show that it corresponds in a precise way to the notion of an n fold loop space specifically the group completion of the nerve of such a category is an n fold loop space and free n fold monoidal categories give rise to a finite simplicial operad of the same homotopy type as the classical little cubes operad used to parametrize the higher h space structure of an n fold loop space we also show directly that this operad has the same homotopy type as the n th smith filtration of the barratt eccles operad and the n th filtration of berger s complete graph operad moreover this operad contains an equivalent preoperad which gives rise to milgram s small model for 2 2 x when n 2 and is very closely related to milgram s model of n n x for n 2

on the edge of the stable range we prove a general homological stability theorem for certain families of groups equipped with product maps followed by two theorems of a new kind that give information about the last two homology groups outside the stable range these last two unstable groups are the edge in our title applying our results to automorphism groups of free groups yields a new proof of homological stability with an improved stable range a description of the last unstable group up to a single ambiguity and a lower bound on the rank of the penultimate unstable group we give similar applications to the general linear groups of the integers and of the field of order 2 this time recovering the known stability range the results can also be applied to general linear groups of arbitrary principal ideal domains symmetric groups and braid groups our methods require us to use field coefficients throughout

parabolic presentations of the yangian we introduce some new presentations for the yangian associated to the lie algebra open image in new window these presentations are parametrized by tuples of positive integers summing to n at one extreme for the tuple n the presentation is the usual rtt presentation of y n at the other extreme for the tuple 1 n the presentation is closely related to drinfeld s presentation in general the presentations are useful for understanding the structure of the standard parabolic subalgebras of y n

super yangian double dy gl 1 1 and its gauss decomposition we extend the yangian double to the super or graded case and give its drinfel d generators a realization by gauss decomposition

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

positive real lemma necessary and sufficient conditions for the existence of solutions under virtually no assumptions in this note the celebrated positive real lemma equations are considered with the purpose of obtaining necessary and sufficient conditions for the existence of solutions under the mildest system theoretic assumptions more precisely necessary and sufficient conditions are established under the very weak assumption of sign controllability it is then shown that the order of the equations may be suitably reduced by restricting attention to a subspace related to a certain observable subsystem this reduction beside being interesting per se permits us to weaken the assumption of sign controllability finally the assumptions are further weakened as to include the case when the state matrix has uncontrollable eigenvalues on the imaginary axis

a rigid irregular connection on the projective line in this paper we construct a connection on the trivial g bundle on p 1 for any simple complex algebraic group g which is regular outside of the points 0 and has a regular singularity at the point 0 with principal unipotent monodromy and has an irregular singularity at the point with slope 1 h the reciprocal of the coxeter number of g the connection which admits the structure of an oper in the sense of beilinson and drinfeld appears to be the characteristic 0 counterpart of a hypothetical family of l adic representations which should parametrize a specific automorphic representation under the global langlands correspondence these l adic representations and their characteristic 0 counterparts have been constructed in some cases by deligne and katz our connection is constructed uniformly for any simple algebraic group and characterized using the formalism of opers it provides an example of the geometric langlands correspondence with wild ramification we compute the de rham cohomology of our connection with values in a representation v of g and describe the differential galois group of as a subgroup of g

compatibility of the feigin frenkel isomorphism and the harish chandra isomorphism for jet algebras let g be a simple finite dimensional complex lie algebra with a cartan subalgebra h and weyl group w let g denote the lie algebra of n jets on g a theorem of rais and tauvel and geoffriau identifies the centre of the category of g modules with the algebra of functions on the variety of n jets on the affine space h w on the other hand a theorem of feigin and frenkel identifies the centre of the category of critical level smooth modules of the corresponding affine kac moody algebra with the algebra of functions on the ind scheme of opers for the langlands dual group we prove that these two isomorphisms are compatible by defining the higher residue of opers with irregular singularities we also define generalized verma and wakimoto modules and relate them by a nontrivial morphism

homological stability for moduli spaces of high dimensional manifolds i s galatius was partially supported by nsf grants dms 1105058 and dms 1405001 the european research council erc under the european union s horizon 2020 research and innovation pro gramme grant agreement no 682922 as well as the danish national research foundation through the centre for symmetry and deformation dnrf92 and erc 682992 o randal williams was partially supported by the herchel smith fund and epsrc grant ep m027783 1

twisted yangians twisted quantum loop algebras and affine hecke algebras of type bc we study twisted yangians of type aiii which have appeared in the literature under the name of reflection algebras they admit q versions which are new twisted quantum loop algebras we explain how these can be defined equivalently either via the reflection equation or as coideal subalgebras of yangians of gln resp of quantum loop algebras of gln the connection with affine hecke algebras of type bc comes from a functor of schur weyl type between their module categories

stabilization for mapping class groups of 3 manifolds we prove that the homology of the mapping class group of any 3manifold stabilizes under connected sum and boundary connected sum with an arbitrary 3 manifold when both manifolds are compact and orientable the stabilization also holds for the quotient group by twists along spheres and disks and includes as particular cases homological stability for symmetric automorphisms of free groups automorphisms of certain free products and handlebody mapping class groups our methods also apply to manifolds of other dimensions in the case of stabilization by punctures the main result of this paper is a homological stability theorem for mapping class groups of 3 manifolds where the stabilization is by connected sum with an arbitrary 3 manifold more precisely we show that given any two compact connected oriented 3 manifolds n and p with n 6 the homology group hi 0diff n p p rel n z is independent of the number n of copies of p in the connected sum as long as n 2i 2 i e each homology group stabilizes with p we also prove an analogous result for boundary connected sum and a version for the quotient group of the mapping class group by twists along spheres and disks a group closely related to the automorphism group of the fundamental group of the manifold homological stability theorems were first found in the sixties for symmetric groups by nakaoka 36 and linear groups by quillen and now form the foundation of modern algebraic k theory see for example 28 part iv and 42 stability theorems for mapping class groups of surfaces were obtained in the eighties by harer and ivanov 14 25 and recently turned out to be a key ingredient to a solution of the mumford conjecture about the homology of the riemann moduli space 30 the other main examples of families of groups for which stability has been known are braid groups 1 and automorphism groups of free groups 17 18 the present paper extends significantly the class of groups for which homological stability is known to hold it suggests that it is a widespread phenomenon among families of groups containing enough symmetries in addition to the already mentioned stability theorems for mapping class groups of 3 manifolds corollaries of our main result include stability for handlebody subgroups of surface mapping class groups symmetric automorphism groups of free groups and automorphism groups of free products ng for many groups g using similar techniques we obtain stability results also for mapping class groups 0diff m n points rel m for m any m dimensional manifold with boundary m 2 even the case m 2 is new here as well as for the fundamental group 1conf m n of the configuration space of n unordered points in m our paper thus also unifies previous known results as we recover stability for braid groups as 1conf d n symmetric groups as 1conf d n and automorphism groups of free groups as aut nz

manin matrices and talalaev s formula in this paper we study properties of lax and transfer matrices associated with quantum integrable systems our point of view stems from the fact that their elements satisfy special commutation properties considered by yu i manin some 20 years ago at the beginning of quantum group theory these are the commutation properties of matrix elements of linear homomorphisms between polynomial rings more explicitly these read 1 elements of the same column commute 2 commutators of the cross terms are equal mij mkl mkj mil e g m11 m22 m21 m12 the main aim of this paper is twofold on the one hand we observe and prove that such matrices which we call manin matrices in short behave almost as well as matrices with commutative elements namely the theorems of linear algebra e g a natural definition of the determinant the cayley hamilton theorem the newton identities and so on and so forth have a straightforward counterpart in the case of manin matrices on the other hand we remark that such matrices are somewhat ubiquitous in the theory of quantum integrability for instance manin matrices and their q analogs include matrices satisfying the yang baxter relation rtt ttr and the so called cartier foata matrices also they enter talalaev s remarkable formulae for the quantum spectral curve and appear in the separation of variables problem and capelli identities we show that theorems of linear algebra after being established for such matrices have various applications to quantum integrable systems and lie algebras e g in the construction of new generators in and in general in the construction of quantum conservation laws in the knizhnik zamolodchikov equation and in the problem of wick ordering we propose in the appendix a construction of quantum separated variables for the xxx heisenberg system

on the solvability of the positive real lemma equations abstract in this paper we consider the classical equations of the positive real lemma under the sole assumption that the state matrix a has unmixed spectrum a a without any other system theoretic assumption observability reachability stability etc we derive a necessary and sufficient condition for the solvability of the positive real lemma equations

invariants of the vacuum module associated with the lie superalgebra mathfrak gl 1 1 we describe the algebra of invariants of the vacuum module associated with an affinization of the lie superalgebra we give a formula for its hilbert poincare series in a fermionic cancellation free form which turns out to coincide with the generating function of the plane partitions over the hook our arguments are based on a super version of the beilinson drinfeld rais tauvel theorem which we prove by producing an explicit basis of invariants of the symmetric algebra of polynomial currents associated with we identify the invariants with affine supersymmetric polynomials via a version of the chevalley theorem

isomorphism of two realizations of quantum affine algebra u_q widehat mathfrak g mathfrak l text n text we establish an explicit isomorphism between two realizations of the quantum affine algebra u q widehat mathfrak g mathfrak l text n text given previously by drinfeld and reshetikhin semenov tian shansky our result can be considered as an affine version of the isomorphism between the drinfield jimbo and the faddeev reshetikhin takhtajan constructions of the quantum algebra u q mathfrak g mathfrak l n

wakimoto modules opers and the center at the critical level abstract wakimoto modules are representations of affine kac moody algebras in fock modules over infinite dimensional heisenberg algebras in this paper we present the construction of the wakimoto modules from the point of view of the vertex algebra theory we then use wakimoto modules to identify the center of the completed universal enveloping algebra of an affine kac moody algebra at the critical level with the algebra of functions on the space of opers for the langlands dual group on the punctured disc giving another proof of the theorem of b feigin and the author

yang baxter equation in long range interacting systems we consider the su p spin chains with long range interactions and the spin generalization of the calogero sutherland models we show that their properties derive from a transfer matrix obeying the yang baxter equation we obtain the expression of the conserved quantities of the dynamical models and we diagonalise them in the spin chain case we establish the connection between the degeneracies of the spectrum and the representation theory of the yangians we use a correspondence with the dynamical models to diagonalise the hamiltonian finally we extend the previous results to the case of a trigonometric r matrix

quantum current algebras associated with rational r matrix abstract we study quantum current algebra a r associated with the rational r matrix of gl n and we give explicit formulae for the elements of its center at the critical level due to etingof kazhdan s construction the level c vacuum module v c r for the algebra a r possesses a quantum vertex algebra structure for any complex number c we prove that any module for the quantum vertex algebra v c r is naturally equipped with a structure of restricted a r module of level c and vice versa

reflection equation and twisted yangians with any involutive antialgebra and coalgebra automorphism of a quasitriangular bialgebra we associate a reflection equation algebra a hopf algebraic treatment of the reflection equation of this type and its universal solution is given applications to the twisted yangians are considered

feigin frenkel center in types b c and d for each simple lie algebra mathfrak g consider the corresponding affine vertex algebra v mathrm crit mathfrak g at the critical level the center of this vertex algebra is a commutative associative algebra whose structure was described by a remarkable theorem of feigin and frenkel about two decades ago however only recently simple formulas for the generators of the center were found for the lie algebras of type a following talalaev s discovery of explicit higher gaudin hamiltonians we give explicit formulas for generators of the centers of the affine vertex algebras v mathrm crit mathfrak g associated with the simple lie algebras mathfrak g of types b c and d the construction relies on the schur weyl duality involving the brauer algebra and the generators are expressed as weighted traces over tensor spaces and equivalently as traces over the spaces of singular vectors for the action of the lie algebra mathfrak sl 2 in the context of howe duality this leads to explicit constructions of commutative subalgebras of the universal enveloping algebras mathrm u mathfrak g t and mathrm u mathfrak g and to higher order hamiltonians in the gaudin model associated with each lie algebra mathfrak g we also introduce analogues of the bethe subalgebras of the yangians mathrm y mathfrak g and show that their graded images coincide with the respective commutative subalgebras of mathrm u mathfrak g t

yangian doubles of classical types and their vertex representations the yangian double dy gn is introduced for the classical types of gn o2n 1 sp2n and o2n via the gauss decomposition of the generator matrix the yangian double is given the drinfeld presentation in addition bosonization of level 1 realizations for the yangian double dy gn of non simply laced types are explicitly constructed

hbar adic quantum vertex algebras and their modules this is a paper in a series to study vertex algebra like structures arising from various algebras including quantum affine algebras and yangians in this paper we study notions of hbar adic nonlocal vertex algebra and hbar adic weak quantum vertex algebra slightly generalizing etingof kazhdan s notion of quantum vertex operator algebra for any topologically free mathbb c lbrack lbrack hbar rbrack rbrack module w we study hbar adically compatible subsets and hbar adically mathcal s local subsets of end w x x 1 we prove that any hbar adically compatible subset generates an hbar adic nonlocal vertex algebra with w as a module and that any hbar adically mathcal s local subset generates an hbar adic weak quantum vertex algebra with w as a module a general construction theorem of hbar adic nonlocal vertex algebras and hbar adic quantum vertex algebras is obtained as an application we associate the centrally extended double yangian of mathfrak s mathfrak l 2 to hbar adic quantum vertex algebras

principal realization of the yangian y gl n motivated to simplify the structure of tensor representations we give a new set of generators for the yangian y sl n using the principal realization in simple lie algebras the isomorphism between our new basis and the standard cartan weyl basis is also given we show by example that the principal basis simplifies the yangian action significantly in the tensor product of the fundamental representation and its dual

remarks on the confluent kz equation for mathfrak sl _2 and quantum painleve equations for the lie algebra we give an explicit form of an irregular singular version of the kz equation in the simplest non trivial cases these kz equations reproduce the quantum painlev equations possessing affine weyl group symmetry found previously by one of the authors

bethe subalgebras in twisted yangians we study analogues of the yangian of the lie algebra mathfrak g mathfrak l n for the other classical lie algebras mathfrak s mathfrak o n and mathfrak s mathfrak p n we call them twisted yangians they are coideal subalgebras in the yangian of mathfrak g mathfrak l n and admit homomorphisms onto the universal enveloping algebras u mathfrak s mathfrak o n and u mathfrak s mathfrak p n respectively in every twisted yangian we construct a family of maximal commutative subalgebras parametrized by the regular semisimple elements of the corresponding classical lie algebra the images in u mathfrak s mathfrak o n and u mathfrak s mathfrak p n of these subalgebras are also maximal commutative

on kalman yakubovich popov lemma for stabilizable systems the kalman yakubovich popov kyp lemma has been a cornerstone in system theory and network analysis and synthesis it relates an analytic property of a square transfer matrix in the frequency domain to a set of algebraic equations involving parameters of a minimal realization in time domain this note proves that the kyp lemma is also valid for realizations which are stabilizable and observable

an observation on the positive real lemma abstract we give a necessary and a sufficient condition that the transfer function of an exponentially stable linear finite dimensional system be a real positive matrix the condition does not assume controllability observability properties

center of the quantum affine vertex algebra in type a abstract we consider the quantum vertex algebra associated with the double yangian in type a as defined by etingof and kazhdan we show that its center is a commutative associative algebra and construct algebraically independent families of topological generators of the center at the critical level

on a class of representations of the yangian and moduli space of monopoles a new class of infinite dimensional representations of the yangians yopen image in new window and yopen image in new window corresponding to a complex semisimple algebra open image in new window and its borel subalgebra open image in new window is constructed it is based on the generalization of the drinfeld realization of open image in new window in terms of quantum minors to the case of an arbitrary semisimple lie algebra open image in new window the poisson geometry associated with the constructed representations is described in particular it is shown that the underlying symplectic leaves are isomorphic to the moduli spaces of g monopoles defined as the components of the space of based maps of 1 into the generalized flag manifold open image in new window thus the constructed representations of the yangian may be considered as a quantization of the moduli space of the monopoles

vertex representations of quantum affine algebras we construct vertex representations of quantum affine algebras of ade type which were first introduced in greater generality by drinfeld and jimbo the limiting special case of our construction is the untwisted vertex representation of affine lie algebras of frenkel kac and segal our representation is given by means of a new type of vertex operator corresponding to the simple roots and satisfying the defining relations in the case of the quantum affine algebra of type a we introduce vertex operators corresponding to all the roots and determine their commutation relations this provides an analogue of a chevalley basis of the affine lie algebra unk n in the basic representation

poset fiber theorems suppose that f p q is a poset map whose fibers f 1 qless than or equal to q are sufficiently well connected our main result is a formula expressing the homotopy type of p in terms of q and the fibers several fiber theorems from the literature due to babson baclawski and quillen are obtained as consequences or special cases homology cohen macaulay and equivariant versions are given and some applications are discussed

hopf structure of the yangian y sln in the drinfel d realization the yangian of the lie algebra sln is known to have different presentations in particular the rtt realization and the drinfel d realization using the isomorphism between them the explicit expressions of the comultiplication the antipode and the counit in the drinfel d realization of the yangian y sln are given as examples the cases of y sl2 and y sl3 are worked out

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

segal sugawara vectors for the lie algebra of type g2 explicit formulas for segal sugawara vectors associated with the simple lie algebra mathfrak g of type g 2 are found by using computer assisted calculations this leads to a direct proof of the feigin frenkel theorem describing the center of the corresponding affine vertex algebra at the critical level as an application we give an explicit solution of vinberg s quantization problem by providing formulas for generators of maximal commutative subalgebras of u mathfrak g we also calculate the eigenvalues of the hamiltonians on the bethe vectors in the gaudin model associated with mathfrak g