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

on rational addition theorems abstract we investigate the functions which admit an addition theorem of the form f t s i 1 n y i t u i s j 1 m z j t v j s our approach is based on the reduction to a system of differential equations all functions are supposed to be continuously differentiable on some interval the concepts of joint linear dependence and joint quadratic dependence of two families of functions are introduced it s proved that there are only two possibilities in the case of jointly linearly independent ui and vj a the function f is a ratio of quasi polynomials b the families yi and zj are jointly quadratically dependent the second possibility is studied for m n 2 we apply our results to the solving of the functional equation f t s y 1 t y 2 s y 2 t y 1 s z 1 t z 2 s z 2 t z 1 s this equation and its special cases have important applications in mathematical physics and were studied by several authors with the assumption of analyticity of the solutions we show that the class of solutions remains the same if one looks for the differentiable solutions on an interval

on a generalised connes hochschild kostant rosenberg theorem the central result of this paper is an explicit computation of the hochschild and cyclic homologies of a natural smooth subalgebra of stable continuous trace algebras having smooth manifolds x as their spectrum more precisely the hochschild homology is identified with the space of differential forms on x and the periodic cyclic homology with the twisted de rham cohomology of x thereby generalising some fundamental results of connes and hochschild kostant rosenberg the connes chern character is also identified here with the twisted chern character

boundary quantum knizhnik zamolodchikov equations and bethe vectors solutions to boundary quantum knizhnik zamolodchikov equations are constructed as bilateral sums involving off shell bethe vectors in case the reflection matrix is diagonal and only the 2 dimensional representation of mathcal u q widehat mathfrak sl 2 is involved we also consider their rational and classical degenerations

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

weyl groups and elliptic solutions of the wdvv equations abstract a functional ansatz is developed which gives certain elliptic solutions of the witten dijkgraaf verlinde verlinde or wdvv equations this ansatz is based on the elliptic trilogarithm function introduced by beilinson and levin for this to be a solution results in a number of purely algebraic conditions on the set of vectors that appear in the ansatz this providing an elliptic version of the idea introduced by veselov of a system rational and trigonometric limits are studied together with examples of elliptic systems based on various weyl groups jacobi group orbit spaces are studied these carry the structure of a frobenius manifold the corresponding almost dual structure is shown in the a n and b n cases and conjecturally for an arbitrary weyl group to correspond to the elliptic solutions of the wdvv equations transformation properties under the jacobi group of the elliptic trilogarithm are derived together with various functional identities which generalize the classical frobenius stickelberger relations

integral representation for the eigenstates of the spin system with inverse square interactions we consider the calogero sutherland moser system with spin 1 2 we give the integral representation for the ground state wavefunction excited states of the calogero spin model confined in the harmonic potential are also given

a unification of knizhnik zamolodchikov and dunkl operators via affine hecke algebras some generalizations of the lusztig lascoux schutzenberger operators for affine hecke algebras are considered as corollaries we obtain lusztig s isomorphisms from affine hecke algebras to their degenerate versions a natural interpretation of the dunkl operators and a new class of differential difference operators generalizing dunkl s ones and the knizhnik zamolodchikov operators from the two dimensional conformal field theory

elliptic ruijsenaars operators and functional equations we study mutually commutative difference operators introduced by ruijsenaars which are regarded as elliptic analogs of macdonald operators and consist of the jacobi theta functions and shift operators these operators are constructed in terms of r operators due to shibukawa and ueno or root algebras introduced by cherednik which give rise to a set of functional equations we investigate solutions of these functional equations for generalizations of elliptic ruijsenaars operators

the classical fluid of associating hard rods in an external field two models of one dimensional fluids of associating hard rods in an arbitrary external field are investigated in the first model particles can only form dimers while in the second model which has been solved previously by percus aggregates of any size coexist in both cases the grand canonical potential and the external potential are found exactly as functionals of the density it is shown that wertheim s thermodynamic perturbation theory of polymerization provides a straightforward route to the exact solution by expanding the functional space to include more density parameters this suggests that wertheim s theory should be used also for studying the structure and not only the thermo dynamics of real associating fluids

quantum calogero moser models integrability for all root systems the issues related to the integrability of quantum calogero moser models based on any root systems are addressed for the models with degenerate potentials i e the rational with without the harmonic confining force the hyperbolic and the trigonometric we demonstrate the following for all the root systems i construction of a complete set of quantum conserved quantities in terms of a total sum of the lax matrix l i e sigma mu nu is an element ofr l n mu nu in which r is a set of r r vectors invariant under the action of the coxeter group they form a single coxeter orbit ii proof of liouville integrability iii triangularity of the quantum hamiltonian and the entire discrete spectrum generalized jack polynomials are defined for all root systems as unique eigenfunctions of the hamiltonian iv equivalence of the lax operator and the dunkl operator v algebraic construction of all excited states in terms of creation operators these are mainly generalizations of the results known for the models based on the a series i e su n type root systems

boundary k matrix elliptic dunkl operator and quantum many body systems the infinite dimensional representation of the boundary k operator the solution of the reflection equation is considered the trigonometric k matrix is studied as a degenerate case of the elliptic operator a method to construct the elliptic dunkl operator is proposed and the relationship with the quantum many body problem is also discussed

quantum integrability of the generalized elliptic ruijsenaars models the quantum integrability of the generalized elliptic ruijsenaars models is shown these models are mathematically related to the macdonald operator and the macdonald koornwinder operator which appeared in the q orthogonal polynomial theories we construct these integrable families by using the yang baxter equation and the reflection equation

knizhnik zamolodchikov equations and the calogero sutherland moser integrable models with exchange terms it is shown that from some solutions of generalized knizhnik zamolodchikov equations one can construct eigenfunctions of the calogero sutherland moser hamiltonians with exchange terms which are characterized by any given permutational symmetry under particle exchange this generalizes some results previously derived by matsuo 1992 and cherednik 1991 for the ordinary calogero sutherland moser hamiltonians

differential difference operators associated to reflection groups there is a theory of spherical harmonics for measures invariant under a finite reflection group the measures are products of powers of linear functions whose zero sets are the mirrors of the reflections in the group times the rotation invariant measure on the unit sphere in rn a commutative set of differential difference operators each homogeneous of degree 1 is the analogue of the set of first order partial derivatives in the ordinary theory of spherical harmonics in the case of r2 and dihedral groups there are analogues of the cauchy riemann equations which apply to gegenbauer and jacobi polynomial expansions the analysis of orthogonality structures for polynomials in several variables is a problem of vast dimensions this paper is part of an ongoing program to establish a workable theory for one particular class the underlying structure is based on finite coxeter groups these are finite groups acting on euclidean space generated by reflections in the zero sets of a collection of linear functions the roots the weight functions for the orthogonality are products of powers of these linear functions restricted to the surface of the unit sphere in addition the weight function is required to be invariant under the action of the group the resulting theory has strong similarities to the theory of spherical harmonics this was established in previous papers of the author 3 4 5 most notably a homogeneous polynomial is orthogonal to all polynomials of lower degree if and only if it is annihilated by a certain second order differential difference operator ordinary partial differentiation acts as an endomorphism on ordinary harmonic functions the use of such operators and their adjoints leads to recurrence formulas and orthogonal decompositions for harmonic polynomials in this paper we construct a commutative set of first order differentialdifference operators associated to the second order operator previously mentioned received by the editors june 1 1987 1980 mathematics subject classification 1985 revision primary 33a45 33a65 20h15 secondary 20c30 42c10 51f15

conserved operators of the generalized elliptic ruijsenaars models we construct the integrable families of the generalized elliptic ruijsenaars models by use of the yang baxter equation and the reflection equation the higher order conserved operators of these models are explicitly presented