International Journal of Modern Physics A

We present the Wakimoto construction of the super OSP(1,2) and SL(2,1) Kac-Moody algebras and the free field representation of the corresponding WZW models. After imposing suitable constraints, we can lead the Feigin-Fuchs representation of Virasoro algebras and coadjoint actions of the N=1 and N=2 conformal symmetries. This formulation corresponds to a supercovariant extension of the Drinfeld and Sokolov Hamiltonian reduction.

The problem of decomposing the symmetric and exterior algebras of the adjoint representation of the Lie algebra into -irreducible components are considered. The exact formula for the principal specialization of the internal product of the Schur functions (similar to the formula for Kostka-Foulkes polynomials) is obtained by the purely combinatorial approach, based on the theory of rigged configurations. The stable behaviour of some polynomials is studied. Different examples are presented.

The properties of the Seifert circles in an oriented tangle diagram are exploited to prove a theorem that asserts that every (n,n)-tangle diagram is isotopic to a partially closed braid, and a second one that facilitates the assignment of wrong-way edges, one on each Seifert circle, in a tangle diagram. These result are used to identify the structure of an abstract algebra on which a functor for the isotopy of general tangles may be constructed. Any finite dimensional irreducible representation of a quasitriangular Hopf algebra is a realization of this algebra.

The Clebsch-Gordan and Racah-Wigner coefficients for the positive (or negative) discrete series of irreducible representations for the noncompact form U_q(SU(1,1)) of the algebra U_q(sl(2)) are computed.

We prove existence of BGG resolution of an irreducible highest weight module over a quantum group, classify morphisms of Verma modules over a quantum group and find formulas for singular vectors in Verma modules. As an application we find cohomology of the quantum group of the type with coefficients in a finite-dimensional module.

We prove that an N-state vertex model representation of the A_n Hecke algebra quotient NH_n(q) is faithful for all q. We use the result to examine the indecomposable content of these representations, and hence the structure of the centraliser algebra, which is generically a quotient of U_qsl(N), at q a root of unity. We achieve a complete analysis in the case N=2, finding a number of Morita self-dual algebraic structures.

For M a finite dimensional complex vector space and A a certain type of (unital) subalgebra of End(M) (including some specific types of physical significance in the field of quantum spin chains) we give an algorithm for constructing the centraliser or commutant B of A on M. We give examples, and discuss the conditions for centralising to be an involution, i.e. A, B a dual pair, and for B and A to be Morita equivalent. A special case of one example shows that H_n(q), U_q(sl₂) act as a dual pair on the tensored vector representation for all q.

We consider a series of N-state L(≥N) site quantum spin chains, characterised by the ordered partition of N into 2 parts, N=P+M. These (P/M) chains are invariant under an action of U_qSU(P/M), and are built from a representation of the Hecke algebra H_L-1(q). We establish that the intersection of the spectra of a (P/M) and (P'/M') chain of fixed length L is the spectrum of the (min(P,P')/min(M,M')) chain of that length. We establish that the spectrum of the (P/M) chain breaks into blocks corresponding to irreducible representations of H_L-1(q) (or equivalently irreducible representations of U_qSU(P/M)) characterised by Young diagrams with no rectangular subdiagrams of dimension (P+1)×(M+1) (height × width resp.). We give the corresponding quotient relations for the Hecke algebra. We discuss several implications of these results.

We study the nonunitary diagonal cosets constructed from admissible representations of Kač-Moody algebras at fractional level, with an emphasis on the question of field identification. Generic classes of field identifications are obtained from the analysis of the modular S matrix. These include the usual class related to outer automorphisms, as well as some intrinsically nonunitary field identifications. They allow for a simple choice of coset field representatives where all field components of the coset are associated with integrable weights.

The free–fermion model in presense of fields discussed by Felderhof is reformulated as a solution of the Yang–Baxter equation associated with a family of 2–dimensional vector spaces. The trigonometric limit of this solution is equal to the Yang–Baxterization of the state model related to the multi–variable Alexander polynomial of links. This trigonometric limit is also equal to the R–matrix with respect to a family of representations of at of affine type.

4-d self-dual gravity and Yang-Mills theories are identified with 2-d sigma models taking values in infinite dimensional groups. This allows us to view 4-d self-dual theories as "large N limits" of 2-d conformal field theories. We also find a possible "integrable deformation" of 4-d self-dual gravity leading to the Einstein-Maxwell system and discuss its implication in the context of higher dimensional integrability.

Determinantal functional equations satisfied by the row transfer matrix eigenvalues of critical A–D–E lattice spin models are presented. These are obtained for models associated with the Lie algebras , , A_L, D_L and E_6,7,8 by exploiting connections with functional equations satisfied by the row transfer matrix eigenvalues of the six-vertex model at rational values of the crossing parameter λ=sπ/h where h is the Coxeter number. In addition, fusion is used to derive special functional equations, called inversion identity hierarchies, which provide the key to the direct calculation of finite-size corrections, central charges and conformal weights for the critical A–D–E lattice models.

The particles interchanging operator and hamiltonian for two particles on a quantum sphere are constructed. Though there is no interaction hamiltonian, "interaction energy" occurs. It suggests a possibility that in a future quantum theory, all particles might be free.

Some recent results in the representation theory of the algebras of functions on compact quantum groups are presented. Applications to different subjects are given.

An analogue of the KP hierarchy, the SDiff(2) KP hierarchy, related to the group of area-preserving diffeomorphisms on a cylinder is proposed. An improved Lax formalism of the KP hierarchy is shown to give a prototype of this new hierarchy. Two important potentials, S and τ, are introduced. The latter is a counterpart of the tau function of the ordinary KP hierarchy. A Riemann-Hilbert problem relative to the group of area-diffeomorphisms gives a twistor theoretical description (nonlinear gravition construction) of general solutions. A special family of solutions related to topological minimal models are identified in the framework of the Riemann-Hilbert problem. Further, infinitesimal symmetries of the hierarchy are constructed. At the level of the tau function, these symmetries obey anomalous commutation relations, hence leads to a central extension of the algebra of infinitesimal area-preserving diffeomorphisms (or of the associated Poisson algebra).

The general Zakharov-Shabat type equations without the L operators are shown to be equivalent to the KP and the Toda lattice hierarchies. That is, the Zakharov-Shabat type equations automatically guarantee the existence of the L operators.

We describe the Killing form of the quantum algebra using the duality pairing between the plus and the minus parts, and give a structure theorem for the center. A detailed proof of the existence of the universal R-matrix (Drinfeld's theorem) is also given.

Irreducible cyclic representations of the algebra of monodromy matrices corresponding to the R-matrix of the six-vertex model are described. As a consequence, the direct computation of spectra for transfer-matrices of the chiral Potts model with special fixed-spin boundary conditions is done. The generalization of simple Baxter's Hamiltonian is proposed.

A q-analogue of the Frobenius formula is proved by means of the quantum groups U_q(gl_n+1), A_q(GL_n+1) and Iwahori's Hecke algebra of type A_N-1, and then, the character table of this Hecke algebra is investigated.

The original motivation for studying unitarizable representations of the braid group was to construct interesting examples of subfactors of II₁ von Neumann factors. This is possible for representations factoring through Hecke algebras or q-Brauer algebras only if the deformation parameter is a root of unity. Information gained from this work can be applied to studying tensor products of representations of quantum groups at roots of unity and invariants of 3-manifolds.

It is shown that an (N_ρ,N_σ) chiral Potts model, which is a generalization of the Ashkin-Teller model and consists of two chiral Potts models which are coupled together by four-spin interactions, can always be mapped to a single chiral Potts model of N_ρN_σ states if N_ρ and N_σ are relative prime. Moreover, if on every lattice site there are d spins with N_ρ,…,N_σ states, respectively, similar mappings exist: If there are chiral two-spin interactions between nearest neighbor spins of the same kind and if the d sublattices are coupled together by chiral 2j-spin interactions for j≤d between the j pairs of spins, this defines a composite (N_ρ,…,N_σ) state chiral Potts model. If (N_i,N_j)=1, for i≠j, i,j=1,…,d, then the composite model with (N_ρ,…,N_σ) states can be mapped into a -state chiral Potts model. Finally, it is shown that if one or more of the spins of a unit cell sits on the dual lattice whereas the other spins sit on the original lattice, so that this is a generalization of the eight-vertex model in the spin language, such a mapping also exists. This mean that results obtained for the chiral Potts models can be used for many such composite models.

In this paper we present the solutions of the star-triangle equation for Boltzmann weights of the (N_α,N_β) model. This model consists of two scalar Potts models (of N_α and N_β states, respectively) coupled together by four-spin interaction terms and was introduced by Domany and Riedel to study the phase transitions in adsorbed thin films. If N_α,N_β≠2, the star-triangle equations have nontrivial self-dual and non-self-dual solutions only for N_α=N_β; whereas if N_α≠N_β=2, we have other self-dual and non-self-dual solutions. After applying partial duality transformations several of our solutions can be identified with known solutions for Interaction-Round-a-Face (IRF) models. Our results also mean that our high-genus solutions found earlier are not the only solutions to the star-triangle equations in chiral Potts models with six or more states per site.

There exists the wonderful integral transform on complex semisimple Lie groups, which assigns to a function on the group the set of its integrals over "generalized horospheres" — some specific submanifolds of the Lie group. The local inversion formula for this integral transform, discovered in 50's for by Gel'fand and Graev, made it possible to decompose the regular representation on into irreducible ones.

In case of real semisimple Lie group the situation becomes more complicated, and usually there is no reasonable analogous integral transform at all. Nevertheless, in the present paper we succeed to define the integral transforms on the Lorentz group and some other real semisimple Lie groups, which are in a sense analogous to the integration over "horospheres". We obtain the inversion formulas for these integral transforms.