Asymptotic solutions to the Knizhnik-Zamolodchikov equation and crystal base

The Knizhnik-Zamolodchikov equation associated withsl ₂ is considered. The transition functions between asymptotic solutions to the Knizhnik-Zamolodchikov equation are described. A connection between asymptotic solutions and the crystal base in the tensor product of modules over the quantum groupU _q sl ₂ is established, in particular, a correspondence between the Bethe vectors of the Gaudin model of an inhomogeneous magnetic chain and the Q-basis of the crystal base.Dedicated to the memory of Ansgar SchnizerThe author was supported by NSF Grant DMS-9203929     

Residues of $q$-Hypergeometric Integrals and Characters of Affine Lie Algebras

We study certain subspaces of solutions to the sl₂ rational qKZ equation at level zero. Each subspace is specified by the vanishing of the residue at a certain divisor which stems from models in two dimensional integrable field theories. We determine the character of the subspace which is parametrized by the number of variables and the sl₂ weight of the solutions. The sum of all characters with a fixed weight gives rise to the branching functions of the irreducible representations of sl₂ in the level one integrable highest weight representations of . It is written in the fermionic form.     

On representations of the elliptic quantum groupE τ,η(sl 2)

We describe representation theory of the elliptic quantum groupE _,(sl ₂). It turns out that the representation theory is parallel to the representation theory of the YangianY(sl ₂) and the quantum loop group .We introduce basic notions of representation theory of the elliptic quantum groupE _,(sl ₂) and construct three families of modules: evaluation modules, cyclic modules, one-dimensional modules. We show that under certain conditions any irreducible highest weight module of finite type is isomorphic to a tensor product of evaluation modules and a one-dimensional module. We describe fusion of finite dimensional evaluation modules. In particular, we show that under certain conditions the tensor product of two evaluation modules becomes reducible and contains an evaluation module, in this case the imbedding of the evaluation module into the tensor product is given in terms of elliptic binomial coefficients. We describe the determinant element of the elliptic quantum group. Representation theory becomes special ifN=m+l, whereN,m,l are integers. We indicate some new features in this case.The authors were supported in part by NSF grants DMS-9400841 and DMS-9501290.     

Quasitriangularity of quantum groups at roots of 1

An important property of a Hopf algebra is its quasitriangularity and it is useful for various applications. This property is investigated for quantum groupssl ₂ at roots of 1. It is shown that different forms of the quantum groupsl ₂ at roots of 1 are either quasitriangular or have similar structure which will be called braiding. In the most interesting cases this property means that braiding automorphism is a combination of some Poisson transformation and an adjoint transformation with a certain element of the tensor square of the algebra.This work was supported by an Alfred P. Sloan fellowship and by National Science Foundation grant DMS-9296120.     

Highest weight representation for Sklyanin algebra <Emphasis Type="Italic">sl</Emphasis>(3)(<Emphasis Type="Italic">u</Emphasis>) with application to the Gaudin model

We study the infinite-dimensional Sklyanin algebra sl(3)(u). Specifically we construct the highest weight representation for this algebra in an explicit form. Its application to the Gaudin model is mentioned.     

Completing Bethe's Equations at Roots of Unity

In a previous paper we demonstrated that Bethe's equations are not sufficient to specify the eigenvectors of the XXZ model at roots of unity for states where the Hamiltonian has degenerate eigenvalues. We here find the equations which will complete the specification of the eigenvectors in these degenerate cases and present evidence that the sl ₂ loop algebra symmetry is sufficiently powerful to determine that the highest weight of each irreducible representation is given by Bethe's ansatz.     

Highest weight representations of quantum current algebras

We study the highest weight and continuous tensor product representations ofq-deformed Lie algebras through the mappings of a manifold into a locally compact group. As an example the highest weight representation of theq-deformed algebra sl_q(2,) is calculated in detail.Alexander von Humboldt-Stiftung fellow. On leave from Institute of Physics, Chinese Academy of Sciences, Beijing, P.R. China.     

Free fermions and the Alexander-Conway polynomial

We show how the Conway Alexander polynomial arises from theq deformation of (Z ₂ graded)sl(n, n) algebras. In the simplestsl(1, 1) case we then establish connection between classical knot theory and its modern versions based on quantum groups. We first shown how the crystal and the fundamental group of the complement of a knot give rise naturally to the Burau representation of the braid group. The Burau matrix is then transformed into theU _q sl(1, 1) R matrix by going to the exterior power algebra. Using a det=str identity, this allows us to recover the state model of [K2, 89] as well. We also show how theU _q> sl(1, 1) algebra describes free fermions propagating on the knot diagram. We rewrite the Conway Alexander polynomial as a Berezin integral, and thus as an apparently new determinant.Work supported in part by NSF grant no. DMS-8822602Work supported in part by the NSF: grant nos. PYI PHY 86-57788 and PHY 90-00386 and by CNRS, France     

CyclicL-operator related with a 3-stateR-matrix

We consider the problem of constructing a cyclicL-operator associated with a 3-stateR-matrix related to theU _q (sl(3)) algebra atq ^N =1. This problem is reduced to the construction of a cyclic (i.e. with no highest weight vector) representation of some twelve generating element algebra, which generalizes theU _q (sl(3)) algebra. We found such representation acting inC ^N C ^N C ^N . The necessary conditions of the existence of the intertwining operator for two representations are also discussed.     

Free boson representation ofq-vertex operators and their correlation functions

A bosonization scheme of theq-vertex operators of U_q(sl₂) for arbitrary level is obtained. They act as intertwiners among the highest weight modules constructed in a bosonic Fock space. An integral formula is proposed forN-point functions and explicit calculation for two-point function is presented.Partly supported by the Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture (No. 04245206)A Fellow of the Japan Society of the Promotion of Science for Japanese Junior Scientists. Partly supported by the Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture (No. 04-2297)     

Bethe's Equation Is Incomplete for the XXZ Model at Roots of Unity

We demonstrate for the six vertex and XXZ model parameterized by = –(q+q^-1)/2±1 that when q^2N=1 for integer N2 the Bethe's ansatz equations determine only the eigenvectors which are the highest weights of the infinite dimensional sl₂ loop algebra symmetry group of the model. Therefore in this case the Bethe's ansatz equations are incomplete and further conditions need to be imposed in order to completely specify the wave function. We discuss how the evaluation parameters of the finite dimensional representations of the sl₂ loop algebra can be used to complete this specification.     

Indecomposable representations of the algebra su(2,1)

Representations of the Lie algebra sl(3) with highest weight are analyzed. Invariant subspaces of indecomposable representations are determined. We study the decomposition of these representations with respect to the subalgebras su(2) and su(1,1) (in their obvious imbedding in su(2,1)).For special cases this decomposition gives indecomposable non multiplicity free representations (indecomposable pairs) with highest weight. These were discussed in [1] and appear also in the decomposition so(3,2) su(1,1) of the Rac representation, [7].     

From CFT to graphs

In this paper, we pursue the discussion of the connections between rational conformal field theories (CFT) and graphs. We generalise our recent work on the relations of operator product algebra (OPA) structure constants of sl(2) theories with the Pasquier algebra attached to the graph. We show that in a variety of CFT's built on sl(n) (typically conformal embeddings and orbifolds), similar considerations enable one to write a linear system satisfied by the matrix elements of the Pasquier algebra in terms of conformal data (quantum dimensions and fusion coefficients). In some cases this provides sufficient information for the determination of all the eigenvectors of an adjacency matrix, and hence of a graph.     

Representations of the cyclically symmetric q-deformed algebra U q(so3)

An algebra homomorphism from the nonstandard q-deformed (cyclically symmetric) algebra U _q(so₃) to the extension Û _q(sl₂) of the Hopf algebra U _q(sl₂) is constructed. Not all irreducible representations (IR) of U _q(sl₂) can be extended to representations of Û _q(sl₂). Composing the homomorphism with irreducible representations of Û _q(sl₂) we obtain representations of U _q(so₃). Not all of these representations of U _q(so₃) are irreducible. Reducible representations of U _q(so₃) are decomposed into irreducible components. In this way we obtain all IR of U _q(so₃) when q is not a root of unity. A part of these representations turn into IR of the Lie algebra so₃ when q 1.     

Universal exchange algebra for Bloch waves and Liouville theory

We derive a universal formula for the exchange algebra in the Bloch wave basis. The main tool we use is a lattice version of the Coulomb gas picture of conformal field theory, making its quantum group structure explicit from the very beginning. Calulations are then reduced to a factorization problem inU _q (sl ₂).     

On algebraic equations satisfied by hypergeometric correlators in WZW models. I

It is proven that integral expressions for conformal correlators insl(2) WZW model found in [SV] satisfy certain natural algebraic equations. This implies that the above integrals really take their values in spaces of conformal blocks.The second author was supported in part by the NSF grant DMS-9202280. The third author was supported in part by the NSF grant DMS-9203939     

Topological representations of the quantum groupU q(sl 2)

We define a topological action of the quantum groupU _q(sl ₂) on a space of homology cycles with twisted coefficients on the configuration space of the punctured disc. This action commutes with the monodromy action of the braid groupoid, which is given by theR-matrix ofU _q(sl ₂).Currently visiting the Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA. Supported in part by the NSF under Grant No. PHY89-04035, supplemented by funds from the NASA     

A Coset-Type Construction for the Deformed Virasoro Algebra

An analog of the minimal unitary series representations for the deformed Virasoro algebra is constructed using vertex operators of the quantum affine algebra U_q(sl₂). A similar construction is proposed for the elliptic algebra A_q,p(sl₂).     

Mappin class group actions on quantum doubles

We study representations of the mapping class group of the punctured torus on the double of a finite dimensional possibly non-semisimple Hopf algebra that arise in the construction of universal, extended topological field theories. We discuss how for doubles the degeneracy problem of TQFT's is circumvented. We find compact formulae for theS ^±1-matrices using the canonical, non-degenerate forms of Hopf algebras and the bicrossed structure of doubles rather than monodromy matrices. A rigorous proof of the modular relations and the computation of the projective phases is supplied using Radford's relations between the canonical forms and the moduli of integrals. We analyze the projectiveSL(2, Z)-action on the center ofU _q(sl₂) forq anl=2m+1^st root of unity. It appears that the 3m+1-dimensional representation decomposes into anm+1-dimensional finite representation and a2m-dimensional, irreducible representation. The latter is the tensor product of the two dimensional, standard representation ofSL(2, Z) and the finite,m-dimensional representation, obtained from the truncated TQFT of the semisimplified representation category ofU _q(sl₂).     

New Symmetries for the Uq(slN) 6-j Symbols from the Eigenvalue Conjecture1

In the present paper, we discuss the eigenvalue conjecture, suggested in 2012, in the particular case of U_q(sl_N) 6-j The eigenvalue conjecture provides a certain symmetry for Racah coefficients and we prove that the eigenvalue conjecture is provided by the Regge symmetry for U_q(sl_N) 6-j, when three representations coincide. This in perspective provides us a kind of generalization of the Regge symmetry to arbitrary U_q(sl_N) 6-j.