Latest in q-alg

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On the classification of hyperbolic root systems of the rank three. Part IIDec 30 1997Mar 23 1998Here we prove classification results announced in Part I (alg-geom/9711032). We classify maximal hyperbolic root systems of the rank 3 having restricted arithmetic type and a generalized lattice Weyl vector $\rho$ with $\rho^2\ge 0$ (i.e. of elliptic ... More
Yangian Algebras and Classical Riemann ProblemsDec 30 1997Jan 12 1998We investigate different Hopf algebras associated to Yang's solution of quantum Yang-Baxter equation. It is shown that for the precise definition of the algebra one needs the commutation relations for the deformed algebra of formal currents and the specialization ... More
Solutions of the qKZB Equation in Tensor Products of finite dimensional modules over the elliptic quantum group $E_{τ,η}sl_2$Dec 26 1997We consider the quantized Knizhnik-Zamolodchikov-Bernard difference equation (qKZB) with step $p$ and values in a tensor product of finite dimensional evaluation modules over the elliptic quantum group $E_{\tau,\eta}(sl_2)$, the equation defined in terms ... More
The $q$-Fourier transform of $q$-distributionsDec 25 1997Jan 12 1998We consider functions on the lattice generated by the integer powers of $q^2$ for $0<q<1$ and construct the $q$-analog of Fourier transform based on the Jackson integral in the space of distributions on the lattice.
Eigenfunctions of Macdonald's $q$-difference operator for the root system of type $C_n$Dec 25 1997We construct an integral representation of eigenfunctions for Macdonald's $q$-difference operator associated with the root system of type $C_n .$ It is given in terms of a restriction of a $q$-Jordan-Pochhammer integral. Choosing a suitable cycle of the ... More
Skew divided difference operators and Schubert polynomialsDec 25 1997We study an action of the skew divided difference operators on the Schubert polynomials and give an explicit formula for structural constants for the Schubert polynomials in terms of certain weighted paths in the Bruhat order on the symmetric group. We ... More
A characterization of interpolation Macdonald polynomialsDec 23 1997In this elementary paper we prove that the extra vanishing property characterizes the BC interpolation Macdonald polynomials inside a very general class of multivariate interpolation polynomials. It follows that they are the only polynomials in this class ... More
On Cherednik-Macdonald-Mehta identitiesDec 23 1997In this note we give a short proof of Cherednik's generalization of Macdonald-Mehta identities for the root system $A_{n-1}$ using the representation theory of quantum groups. These identities, suggested and proved by Cherednik, give an explicit formula ... More
The perturbations $φ_{2,1}$ and $φ_{1,5}$ of the minimal models $M(p,p')$ and the trinomial analogue of Bailey's lemmaDec 23 1997Jan 02 1998We derive the fermionic polynomial generalizations of the characters of the integrable perturbations $\phi_{2,1}$ and $\phi_{1,5}$ of the general minimal $M(p,p')$ conformal field theory by use of the recently discovered trinomial analogue of Bailey's ... More
Noncommutative Geometry and Spacetime Gauge Symmetries of String TheoryDec 22 1997We illustrate the various ways in which the algebraic framework of noncommutative geometry naturally captures the short-distance spacetime properties of string theory. We describe the noncommutative spacetime constructed from a vertex operator algebra ... More
Centralizer construction for twisted YangiansDec 22 1997For each of the classical Lie algebras $g(n)=o(2n+1), sp(2n), o(2n)$ of type B, C, D we consider the centralizer of the subalgebra $g(n-m)$ in the universal enveloping algebra $U(g(n))$. We show that the $n$th centralizer algebra can be naturally projected ... More
On hypergeometric functions connected with quantum cohomology of flag spacesDec 20 1997Givental's recursion relations for the flag varieties $G/B$ are established.
Detecting knot invertibilityDec 19 1997We discuss the consequences of the possibility that Vassiliev invariants do not detect knot invertibility as well as the fact that quantum Lie group invariants are known not to do so. On the other hand, finite group invariants, such as the set of homomorphisms ... More
Non-involutory Hopf algebras and 3-manifold invariantsDec 19 1997We present a definition of an invariant #(M,H), defined for every finite-dimensional Hopf algebra (or Hopf superalgebra or Hopf object) H and for every closed, framed 3-manifold M. When H is a quantized universal enveloping algebra, #(M,H) is closely ... More
Web bases for sl(3) are not dual canonicalDec 19 1997Sep 28 1998We compare two natural bases for the invariant space of a tensor product of irreducible representations of A_2, or sl(3). One basis is the web basis, defined from a skein theory called the combinatorial A_2 spider. The other basis is the dual canonical ... More
Finite type invariants, the mapping class group and blinksDec 17 1997Dec 18 1997The goal of the present paper is to find higher genus surgery formulae for the set of finite-type invariants of homology spheres, and to develop a companion theory of finite-type invariants to be applied, in a subsequent publication, to the study of subgroups ... More
Theory of linear G-difference equationsDec 17 1997We introduce the notion of difference equation defined on a structured set. The symmetry group of the structure determines the set of difference operators. All main notions in the theory of difference equations are introduced as invariants of the symmetry ... More
Deformed Harmonic Oscillator Algebras defined by their Bargmann representationsDec 17 1997Deformed Harmonic Oscillator Algebras are generated by four operators, two mutually adjoint $a$ and $a^\dagger$, and two self-adjoint $N$ and the unity $1$ such as: $[a,N] = a, [a^\dagger, N]= -a^\dagger, a^\dagger a = \psi(N)$ and $aa^\dagger =\psi(N+1)$. ... More
Tensor product representations of the quantum double of a compact groupDec 17 1997Apr 20 1998We consider the quantum double D(G) of a compact group G, following an earlier paper. We use the explicit comultiplication on D(G) in order to build tensor products of irreducible *-representations. Then we study their behaviour under the action of the ... More
Affine Sergeev Algebra and $q$-Analogues of the Young Symmetrizers for Projective Representations of the Symmetric GroupDec 16 1997Mar 20 2000We study a $q$-deformation for the semi-direct 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 found by the second author ... More
Double Lie algebras and Manin triplesDec 15 1997The construction of Lie bialgebra from double Lie algebra is presented. It is used to relate some types of cobracket on inhomogenous so(p,q) algebras with double Lie algebra structures on so(p+1,q) or so(p,q+1). Also it is shown that the cobracket corresponding ... More
Gauge-Invariant Operators for Singular Knots in Chern-Simons Gauge TheoryDec 15 1997We construct gauge invariant operators for singular knots in the context of Chern-Simons gauge theory. These new operators provide polynomial invariants and Vassiliev invariants for singular knots. As an application we present the form of the Kontsevich ... More
Genus-zero modular functors and intertwining operator algebrasDec 13 1997In [H5] (q-alg/9512024) and [H7] (q-alg/9704008), the author introduced the notion of intertwining operator algebra, a nonmeromorphic generalization of the notion of vertex operator algebra involving monodromies. The problem of constructing intertwining ... More
Braid group approach to the derivation of universal Ř matricesDec 13 1997A new method for deriving universal \v{R} matrices from braid group representation is discussed. In this case, universal \v{R} operators can be defined and expressed in terms of products of braid group generators. The advantage of this method is that ... More
Universal Solutions of Quantum Dynamical Yang-Baxter EquationsDec 12 1997We construct a universal trigonometric solution of the Gervais-Neveu-Felder equation in the case of finite dimensional simple Lie algebras and finite dimensional contragredient simple Lie superalgebras.
The Cremmer-Gervais solution of the Yang-Baxter equationDec 12 1997A direct proof is given of the fact that the Cremmer-Gervais R-matrices satisfy the Yang-Baxter equation.
A solution of the quantum Knizhnik Zamolodchikov equation of type $C_n$Dec 12 1997We construct a solution of Cherednik's quantum Knizhnik Zamolodchikov equation associated with the root system of type $C_n$. This solution is given in terms of a restriction of a $q$-Jordan-Pochhammer integral. As its applicaton, we give an explicit ... More
The XXC ModelsDec 11 1997A class of recently introduced multi-states XX models is generalized to include a deformation parameter. This corresponds to an additional nearest-neighbor CC interaction in the defining quadratic hamiltonian. Complete integrability of the one-dimensional ... More
Nonstandard solutions of the Yang-Baxter equationDec 11 1997Explicit solutions of the quantum Yang-Baxter equation are given corresponding to the non-unitary solutions of the classical Yang-Baxter equation for sl(5).
Some Properties of Finite-Dimensional Semisimple Hopf AlgebrasDec 11 1997Kaplansky conjectured that if H is a finite-dimensional semisimple Hopf algebra over an algebraically closed field k of characteristic 0, then H is of Frobenius type (i.e. if V is an irreducible representation of H then dimV divides dimH). It was proved ... More
The Potts Model with a Reflecting BoundaryDec 11 1997A Potts model with a reflecting boundary is introduced and it is shown that its partition function can be expressed as a Markov trace on the Temperley-Lieb Algebra of Coxeter type B.
Actions of Tensor Categories and Cylinder BraidsDec 11 1997Categorial actions of braided tensor categories are defined and shown to be the right framework for a discussion of the categorial structure related to the group of braids in the cylinder. A Kauffman polynomial of links in the solid torus is constructed. ... More
An Ariki-Koike like extension of the Birman-Murakami-Wenzl AlgebraDec 11 1997We introduce an Ariki-Koike like extension of the Birman-Murakami-Wenzl Algebra and show it to be semi-simple. This algebra supports a faithful Markov trace that gives rise to link invariants of closures of Coxeter type B braids.
Quasi-Hopf twistors for elliptic quantum groupsDec 11 1997Oct 14 1998The Yang-Baxter equation admits two classes of elliptic solutions, the vertex type and the face type. On the basis of these solutions, two types of elliptic quantum groups have been introduced (Foda et al., Felder). Fronsdal made a penetrating observation ... More
Two-parameter deformation of loop algebras and superalgebrasDec 11 1997We discuss two-parameter deformations of an universal enveloping algebra $U(g[u])$ of a polynomial loop algebra $g[u]$, where $g$ is a finite-dimensional complex simple Lie algebra (or superalgebra). These deformations are Hopf algebras. One deformation ... More
Coherence Constraints for Operads, Categories and AlgebrasDec 10 1997Jul 14 2000Coherence phenomena appear in two different situations. In the context of category theory the term `coherence constraints' refers to a set of diagrams whose commutativity implies the commutativity of a larger class of diagrams. In the context of algebra ... More
Hecke algebraic properties of dynamical R-matrices. Application to related quantum matrix algebrasDec 10 1997The quantum dynamical Yang-Baxter (or Gervais-Neveu-Felder) equation defines an R-matrix R(p), where $p$ stands for a set of mutually commuting variables. A family of SL(n)-type solutions of this equation provides a new realization of the Hecke algebra. ... More
Dynamical Correlation Functions for an Impenetrable Bose gas with open boundary conditionsDec 10 1997We study the time and temperature dependent correlation functions for an impenetrable bose gas with open boundary conditions. We derive the Fredholm determinant formulae for the correlation functions, by means of the Bethe Ansatz. In the case of time ... More
Symmetry Algebras of Large-N Matrix Models for Open StringsDec 09 1997Jun 08 1999We have discovered that the gauge invariant observables of matrix models invariant under U($N$) form a Lie algebra, in the planar large-N limit. These models include Quantum Chromodynamics and the M(atrix)-Theory of strings. We study here the gauge invariant ... More
Embedding Diagrams of N=2 Verma Modules and Relaxed ^sl(2) Verma ModulesDec 09 1997We classify and explicitly construct the embedding diagrams of Verma modules over the N=2 supersymmetric extension of the Virasoro algebra. The essential ingredient of the solution consists in drawing the distinction between two different types of submodules ... More
The quantum Euler class and the quantum cohomology of the GrassmanniansDec 09 1997The Poincare duality of classical cohomology and the extension of this duality to quantum cohomology endows these rings with the structure of a Frobenius algebra. Any such algebra possesses a canonical ``characteristic element;'' in the classical case ... More
The Character of the Infinite Wedge RepresentationDec 09 1997Dec 16 1997We study the character of the infinite wedge projective representation of the algebra of differential operators on the circle. We prove quasi-modularity of this character and also compute certain generating functions for traces of differential operators ... More
Probabilistic measures and algorithms arising from the Macdonald symmetric functionsDec 09 1997The Macdonald symmetric functions are used to define measures on the set of all partitions of all integers. Probabilistic algorithms are given for growing partitions according to these measures. The case of Hall-Littlewood polynomials is related to the ... More
The Rogers-Ramanujan Identities, the Finite General Linear Groups, and the Hall-Littlewood PolynomialsDec 09 1997The Rogers-Ramanujan identities have been studied from the viewpoints of combinatorics, number theory, affine Lie algebras, statistical mechanics, and quantum field theory. This note connects the Rogers-Ramanujan identities with the finite general linear ... More
Past the Highest-Weight, and What You Can Find ThereDec 09 1997The properties of highest-weight representations of the N=2 superconformal algebra in two dimensions can be considerably simplified when re-expressed in terms of relaxed ^sl(2) representations. This applies to the appearance of submodules and hence, of ... More
On modules associated to coalgebra Galois extensionsDec 09 1997Oct 09 1998For a given entwining structure $(A,C)_\psi$ involving an algebra $A$, a coalgebra $C$, and an entwining map $\psi: C\otimes A\to A\otimes C$, a category $\M_A^C(\psi)$ of right $(A,C)_\psi$-modules is defined and its structure analysed. In particular, ... More
Transmutations between Singular and Subsingular Vectors of the N=2 Superconformal AlgebrasDec 08 1997Dec 13 1999We present subsingular vectors of the N=2 superconformal algebras other than the ones which become singular in chiral Verma modules, reported recently by Gato-Rivera and Rosado. We show that two large classes of singular vectors of the Topological algebra ... More
Screening Current Representation of Quantum SuperalgebrasDec 07 1997In this letter a screening current or contour representation is given of certain quantum superalgebras. The Gomez-Sierra construction of quantum groups in conformal field theory is generalized to cover superalgebras and illustrated using recent results ... More
Capelli Identities for Classical Lie AlgebrasDec 07 1997Feb 25 1999We extend the Capelli identities (1890) from the Lie algebra $gl_N$ to the other two classical Lie algebras $so_N$ and $sp_N$. We employ the theory of reductive dual pairs due to Howe. Our technique comes from the representation theory of Yangians.
Classification of eight-vertex solutions of the colored Yang-Baxter equationDec 05 1997In this paper all eight-vertex type solutions of the colored Yang-Baxter equation dependent on spectral as well as color parameter are given. It is proved that they are composed of three groups of basic solutions, three groups of their degenerate forms ... More
The New Identity for the Scattering Matrx of Exactly Solvable ModelsDec 04 1997We discovered a simple quadratic equation, which relates scattering phases of particles on Fermi surface. We consider one dimensional Bose gas and XXZ Heisenberg spin chain.
Cellular algebras arising from Hecke algebras of type H_nDec 04 1997We study a finite-dimensional quotient of the Hecke algebra of type $H_n$ for general $n$, using a calculus of diagrams. This provides a basis of monomials in a certain set of generators. Using this, we prove a conjecture of C.K. Fan about the semisimplicity ... More
Generalized Temperley-Lieb algebras and decorated tanglesDec 04 1997We give presentations, by means of diagrammatic generators and relations, of the analogues of the Temperley-Lieb algebras associated as Hecke algebra quotients to Coxeter graphs of type B and $D$. This generalizes Kauffman's diagram calculus for the Temperley-Lieb ... More
Unification of parastatistics defined as triple operator algebrasDec 04 1997Dec 18 1997We unify parastatistics, defined as triple operator algebras represented on Fock space, in a simple way using the transition number operators. We express them as a normal ordered expansion of creation and annihilation operators. We discuss several examples ... More
Dynamical systems related to the Cremmer-Gervais R-matrixDec 04 1997The generalized Cremmer-Gervais R-matrix being a twist of the standard R-matrix of $SL_q(3)$, depends on two extra parameters. Properties of this R-matrix are discussed and two dynamical systems, the quantum group covariant $q$-oscillator and an integrable ... More
Homological infiniteness of Torelli groupsDec 03 1997We prove that rational homology of the Torelli group of genus g is infinite dimensional, provided g>6. This means that rational homology of the Torelli space of genus g>6 is infinite dimensional. The Torelli groups with marked points are also considered. ... More
Classification of Low Dimensional Lie Super-BialgebrasDec 03 1997A thorough analysis of Lie super-bialgebra structures on Lie super-algebras osp(1|2) and super-e(2) is presented. Combined technique of computer algebraic computations and a subsequent identification of equivalent structures is applied. In all the cases ... More
Commutation relations of vertex operators related with the spin representation of $U_q(D_n^{(1)})$Dec 03 1997We calculate commutation relations of vertex operators for the spin representation of $U_q(D_n^{(1)})$ by using recursive formulae of R-matrices. In quantum symmetry approach, we obtain the energy and momentum spectrum of the quantum spin chain model ... More
Dual Affine Quantum GroupsDec 03 1997Sep 13 1999Let $\hat{\frak g}$ be an untwisted affine Kac-Moody algebra, with its Sklyanin-Drinfel'd structure of Lie bialgebra, and let $\hat{\frak h}$ be the dual Lie bialgebra. By dualizing the quantum double construction - via formal Hopf algebras - we construct ... More
Dual Affine Quantum GroupsDec 03 1997May 15 2017Let $\hat{\mathfrak{g}}$ be an untwisted affine Kac-Moody algebra, with its Sklyanin-Drinfel'd structure of Lie bialgebra, and let $\hat{\mathfrak{h}}$ be the dual Lie bialgebra. By dualizing the quantum double construction - via formal Hopf algebras ... More
Perfect Crystals of $U_q(G_2^{(1)})$Dec 03 1997The notion of perfect crystals was introduced in "Perfect Crystal and Vertex Models", (Internat. J. Modern Phys. A7(1992)449-484) by S-J. Kang et al. In this paper, we give a series of perfect crystals of $U_q(G_2^{(1)})$.
On Super RS algebra and Drinfeld Realization of Quantum Affine SuperalgebrasDec 03 1997We describe the realization of the super Reshetikhin-Semenov-Tian-Shansky (RS) algebra in quantum affine superalgebras, thus generalizing the approach of Frenkel-Reshetikhin to the supersymmetric (and twisted) case. The algebraic homomorphism between ... More
A proof of Feigin's conjectureDec 02 1997The paper is devoted to the proof of the following conjecture due to B. Feigin. Let $\frak u_\ell$ be the small quantum group a the primitive $\ell$-th root of unity. Then it is known that the usual $Ext$ algebra of the trivial $\frak u_\ell$-module is ... More
q_R-conformal symmetries in 2D nonlocal quantum field theory, categorical representation theory, and Virasoro algebraDec 02 1997Apr 18 1998The paper is devoted to the symmetry aspects of 2D nonlocal field theory, which is the simplest deformation of the conformally invariant quantum field theory with one free bosonic field. The inverse problem of representation theory is solved for $q_R$-conformal ... More
Noncommutative analogues of q-special polynomials and q-integral on a quantum sphereDec 02 1997The q-Legendre polynomials can be treated as some special "functions in the quantum double cosets $U(1)\setminus SU_q(2)/U(1)$". They form a family (depending on a parameter $q$) of polynomials in one variable. We get their further generalization by introducing ... More
Cohomology of the Lie algebras of differential operators: lifting formulasDec 01 1997We construct the explicit formula for the (2n+1)-cocycle of the Lie algebra of (pseudo)differential operators on a n-dimensional space. We prove that this formula in fact defines a cocycle for n=1 and n=2.
Twist Deformation of the rank one Lie SuperalgebraDec 01 1997The Drinfeld twist is applyed to deforme the rank one orthosymplectic Lie superalgebra $osp(1|2)$. The twist element is the same as for the $sl(2)$ Lie algebra due to the embedding of the $sl(2)$ into the superalgebra $osp(1|2)$. The R-matrix has the ... More
Factorizable sheaves and quantum groupsDec 01 1997Apr 15 1998The category $\cal{C}$ (studied by Andersen-Jantzen-Soergel) of representations of a Lusztig's small quantum group at a root of unity, together with its modular structure, is defined geometrically, using configuration spaces.
Five Lectures on Soliton EquationsNov 30 1997This is a self-contained review of a new approach to soliton equations of KdV type developed by the author together with B. Feigin and B. Enriquez.
High-speed Contraction of Transverse Rotations to Gauge TransformationsNov 30 1997The Inonu-Wigner contraction is applied to special relativity and the little groups of the Lorentz group. If the O(3) symmetry group for massive particle is boosted to an infinite-momentum frame, it becomes contracted to a combination of the cylindrical ... More
On Irreducibility of Tensor Products of Yangian ModulesNov 30 1997We study the tensor product $V$ of any number of "elementary" irreducible modules over the Yangian of the general linear Lie algebra. An elementary module is determined by a skew Young diagram and by a complex parameter, and contains a vector called singular. ... More
Spiders for rank 2 Lie algebrasNov 29 1997A spider is an axiomatization of the representation theory of a group, quantum group, Lie algebra, or other group or group-like object. We define certain combinatorial spiders by generators and relations that are isomorphic to the representation theories ... More
On solutions of the KZ and qKZ equations at level zeroNov 29 1997We discuss relations between different formulae for solutions of the Knizhnik-Zamolodchikov differential and the quantum Knizhnik-Zamolodchikov difference equations at level 0 and associated with rational solutions of the Yang-Baxter equation.
Yang-Baxter systems, solutions and applicationsNov 28 1997Two types of Yang-Baxter systems play roles in the theoretical physics -- constant and colour dependent. The constant systems are used mainly for construction of special Hopf algebra while the colour or spectral dependent for construction of quantum integrable ... More
Lie Algebra of Noncommutative Inhomogeneous Hopf AlgebraNov 28 1997We construct the vector space dual to the space of right-invariant differential forms construct from a first order differential calculus on inhomogeneous quantum group. We show that this vector space is equipped with a structure of a Hopf algebra which ... More
Poisson harmonic forms, Kostant harmonic forms, and the $S^1$-equivariant cohomology of $K/T$Nov 26 1997We characterize the harmonic forms on a flag manifold $K/T$ defined by Kostant in 1963 in terms of a Poisson structure. Namely, they are ``Poisson harmonic" with respect to the so-called Bruhat Poisson structure on $K/T$. This enables us to give Poisson ... More
On some Hochschild cohomology classes of fusion algebrasNov 26 1997Dec 16 1997The obstructions for an arbitrary fusion algebra to be a fusion algebra of some semisimple monoidal category are constructed. Those obstructions lie in groups which are closely related to the Hochschild cohomology of fusion algebras with coefficients ... More
FRT Quantization Theory for the Nonsemisimple Cayley-Klein GroupsNov 26 1997The quantization theory of the simple Lie groups and algebras was developed by Faddeev-Reshetikhin-Takhtadjan (FRT). In group theory there is a remarkable set of groups, namely the motion groups of n-dimensional spaces of constant curvature or the orthogonal ... More
Exotic Differential Operators on Complex Minimal Nilpotent OrbitsNov 26 1997Jun 30 1998Let O be the minimal nilpotent adjoint orbit in a classical complex semisimple Lie algebra g. O is a smooth quasi-affine variety stable under the Euler dilation action $C^*$ on g. The algebra of differential operators on O is D(O)=D(Cl(O)) where the closure ... More
Finite-dimensional irreducible representations of twisted YangiansNov 25 1997We study quantized enveloping algebras called twisted Yangians. They are analogues 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 ... More
On the classification of hyperbolic root systems of the rank three. Part INov 25 1997It was recently understood that from the point of view of automorphic Lorentzian Kac-Moody algebras and some aspects of Mirror Symmetry, interesting hyperbolic root systems should have restricted arithmetic type and a generalized lattice Weyl vector. ... More
The consistent reduction of the differential calculus on the quantum group $GL_{q}(2,C)$ to the differential calculi on its subgroups and $σ$-models on the quantum group manifolds $SL_{q}(2,R)$, $SL_{q}(2,R)/U_{h}(1)$, $C{q}(2|0)$ and infinitesimal transformations ... MoreNov 24 1997Explicit construction of the second order left differential calculi on the quantum group and its subgroups are obtained with the property of the natural reduction: the differential calculus on the quantum group $GL_q(2,C)$ has to contain the 3-dimensional ... More
Poincaré Series of Quantum Spaces Associated to Hecke OperatorsNov 23 1997We study the Poincar\'e series of the quantum spaces associated to a Hecke operator, i.e., a Yang-Baxter operator satisfying the equation $(x+1)(x-q)=0$. The Poincar\'e series of the corresponding matrix bialgebra is also considered. Using an old result ... More
A skein theoretic proof of the hook formula for quantum dimensionNov 20 1997We give a skein theoretic proof the Reshetikhin hook length formula for quantum dimension for the quantum group U_q(sl(N)).
A one-dimensional many-body integrable model from $Z_n$ Belavin model with open boundary conditionsNov 20 1997We use factorized $L$ operator to construct an integrable model with open boundary conditions. By taking trigonometic limit($\tau \to \sqrt{-1}\infty$) and scaling limit($\omega \to 0$), we get a Hamiltonian of a classical integrable system. It shows ... More
Screening Currents in Affine Current AlgebraNov 19 1997In this paper screening currents of the second kind are considered. They are constructed in any affine current algebra for directions corresponding to simple roots with multiplicity one in a decomposition of the highest root on a set of simple roots. ... More
Homogeneous Fedosov Star Products on Cotangent Bundles II: GNS Representations, the WKB Expansion, and ApplicationsNov 19 1997This paper is part II of a series of papers on the deformation quantization on the cotangent bundle of an arbitrary manifold $Q$. For certain homogeneous star products of Weyl ordered type (which we have obtained from a Fedosov type procedure in part ... More
Adams operators and knot decorationsNov 19 1997We use an explicit isomorphism from the representation ring of the quantum group U_q(sl(N)) to the Homfly skein of the annulus, to determine an element of the skein which is the image of the mth Adams operator, \psi_m, on the fundamental representation, ... More
Equations of the moduli of pointed curves in the infinite GrassmannianNov 19 1997Feb 24 1999The main result of this paper is the explicit computation of the equations defining the moduli space of triples $(C,p,z)$ (where $C$ is an integral and complete algebraic curve, $p$ a smooth rational point and $z$ a formal trivialization around $p$) in ... More
A spin network generalization of the Jones Polynomial and Vassiliev invariantsNov 18 1997We apply the ideas of Alvarez and Labastida to the invariant of spin networks defined by Witten and Martin based on Chern-Simons theory. We show that it is possible to define ambient invariants of spin networks that (for the case of SU(2)) can be considered ... More
Fourier KnotsNov 14 1997Dec 03 1997This paper introduces the concept of a Fourier knot. A Fourier knot is a knot that is represented by a parametrized curve in three dimensional space such that the coordinate functions are finite Fourier series in the parameter. The previously studied ... More
Hubbard Models as Fusion Products of Free FermionsNov 14 1997Dec 23 1997A class of recently introduced su(n) `free-fermion' models has recently been used to construct generalized Hubbard models. I derive an algebra defining the `free-fermion' models and give new classes of solutions. I then introduce a conjugation matrix ... More
Duals of coloured quantum universal enveloping algebras and coloured universal $\cal T$-matricesNov 14 1997We extend the notion of dually conjugate Hopf (super)algebras to the coloured Hopf (super)algebras ${\cal H}^c$ that we recently introduced. We show that if the standard Hopf (super)algebras ${\cal H}_q$ that are the building blocks of ${\cal H}^c$ have ... More
Coloured Hopf algebras and their dualsNov 14 1997Coloured Hopf algebras, related to the coloured Yang-Baxter equation, are reviewed, as well as their duals. The special case of coloured quantum universal enveloping algebras provides a coloured extension of Drinfeld and Jimbo formalism. The universal ... More
The nondynamical r-matrix structure for the elliptic $A_{n-1}$ Calogero-Moser modelNov 14 1997In this paper, we construct a new Lax operator for the elliptic $A_{n-1}$ Calogero-Moser model with general $n(2\leq n$) from the classical dynamical twisting,in which the corresponding r-matrix is purely numeric (nondynamical one). The nondynamical r-matrix ... More
Skew Young diagram method in spectral decomposition of integrable lattice models II: Higher levelsNov 13 1997The spectral decomposition of the path space of the vertex model associated to the level $l$ representation of the quantized affine algebra $U_q(\hat{sl}_n)$ is studied. The spectrum and its degeneracy are parametrized by skew Young diagrams and what ... More
Support varieties for quantum groupsNov 12 1997Feb 09 1998For any module $M$ over small quantum group one defines the support variety using construction from the theory of restricted Lie algebras. It is a closed conical subset of nilpotent cone of the corresponding Lie algebra. If module $M$ is a module over ... More
Gaudin Model, KZ Equation, and Isomonodromic Problem on TorusNov 10 1997Dec 05 1997This paper presents a construction of isospectral problems on the torus. The construction starts from an SU(n) version of the XYZ Gaudin model recently studied by Kuroki and Takebe in the context of a twisted WZW model. In the classical limit, the quantum ... More
Effective actions, relative cohomology and Chern Simons formsNov 10 1997Feb 19 1998The explicit expression of all the WZW effective actions for a simple group G broken down to a subgroup H is established in a simple and direct way, and the formal similarity of these actions to the Chern-Simons forms is explained. Applications are also ... More
A Factorization of the Conway PolynomialNov 08 1997A string link S can be closed in a canonical way to produce an ordinary closed link L. We also consider a twisted closing which produces a knot K. We give a formula for the Conway polynomial of L as a product of the Conway polynomial of K times a power ... More