Quantum group (CO)actions onG-spaces and quantum modules

A generalized transformation theory is introduced by using quantum (non-commutative) spaces transformed by quantum Lie groups (Hopf algebras). In our method dual pairs of -quantum groups/algebras (co)act on quantum spaces equipped with the structure of a -comodule algebra. We use the quantized groupSU _q (2) as a show case, and we determine its action on modules such as theq-oscillator and the quantum sphere. We also apply our method for the quantized Euclidean groupF _q (E(2)) acting on a quantum homogeneous space. For the sphere case the construction leads to an analytic pseudodifferential vector field realization of the deformed algebra su _q (2) on the quantum projective plane for north and south pole.Presented by A.A. at the 5th International Colloquium on Quantum Groups: Quantum Groups and Integrable Systems, Prague, 20–20 June 1996 and by D.E. at the 4th International Congress of Geometry, Thessaloniki.     

Quantum groupSU(1, 1) ⋊ Z2 and “super-tensor” products

A quantum analogue of the groupSU(1,1)Z ₂—the normalizer ofSU(1, 1) inSL ₂(C)—is introduced and studied. Although there isno correctly defined tensor product in the category of *-representations of the quantum algebraC[SU(1, 1)] _q of regular functions, some categories of *-representations ofC[SU(1, 1)Z ₂] _q turn out to be endowed with a certainZ ₂-graded structure which can be considered as a super-generalization of the monoidal category structure. This quantum effect may be considered as a step to understanding the concept of quantum topological locally compact group.In fact, there seems to be afamily of quantum groupsSU(1, 1)Z ₂ parameterized by unitary characters T ¹ of the fundamental group of the two-dimensional symplectic leaf ofSU(1, 1)/T, whereT is the subgroup of diagonal matrices.It is shown that thequasi-classical analogues of the results of the paper are connected with the decomposition of Schubert cells of the flag manifoldSL ₂(C)_R/B (whereB is the Borel subgroup of upper-triangular matrices) into symplectic leaves.Supported by the Rosenbaum Fellowship.     

Periodic and flat irreducible representations ofSU(3) q

We construct all the periodic irreducible representations ofU(SU(3)) _q forq am-root of unity. Their dimensions arek(2m) ² fork=1,...,m (onlyk=1,...,m/2 for evenm). Their interest is that they could be a tool to generalize the chiral Potts model. By truncation of these representations, we construct flat representations ofU(SU(3)) _q , in which all the multiplicities of the weights are set to 1.     

The exponential map for representations ofU p,q (gl(2))

For the quantum groupGL _p,q (2) and the corresponding quantum algebraU _p,q (gl(2)) Fronsdal and Galindo [Lett. Math. Phys.27 (1993) 59] explicitly constructed the so-called universalT-matrix. In a previous paper [J. Phys. A28 (1995) 2819] we showed how this universalT-matrix can be used to exponentiate representations from the quantum algebra to get representations (left comodules) for the quantum group. Here, further properties of the universalT-matrix are illustrated. In particular, it is shown how to obtain comodules of the quantum algebra by exponentiating modules of the quantum group. Also the relation with the universalR-matrix is discussed.Presented at the 4th International Colloquium Quantum Groups and Integrable Systems, Prague, 22–24 June 1995.     

On a nonstandard two-parametric quantum algebra and its connections withU p,q (gl(2)) andU p,q(gl(1/1))

A quantum algebraU _{p, q} (,H,X _±) associated with a nonstandardR-matrix with two deformation parameters (p, q) is studied and, in particular, its universal -matrix is derived using Reshetikhin's method. Explicit construction of the (p, q)-dependent nonstandardR-matrix is obtained through a coloured generalized boson realization of the universal -matrix of the standardU _{p, q}(gl(2)) corresponding to a nongeneric case. General finite dimensional coloured representation of the universal -matrix ofU _{p, q}(gl(2)) is also derived. This representation, in nongeneric cases, becomes a source for various (p, q)-dependent nonstandardR-matrices. Superization ofU _{p, q}(,H,X _±) leads to the super-Hopf algebraU _{p, q}(gl(1/1)). A contraction procedure then yields a (p, q)-deformed super-Heisenberg algebraU _{p, q}(sh(1)) and its universal -matrix.     

SU q (2) covariant\hat R-matrices for reducible representations

We consider SU _q (2) covariant -matrices for the reducible3 1 representation. There are three solutions to the Yang-Baxter equation. They coincide with the previously known -matrices for SO _q (3) and SO _q (3, 1). Also, they are the three -matrices which can be constructed by using four different SU _q (2) doublets. Only two of the three -matrices allow a differential structure on the reducible four-dimensional quantum space.     

Three-dimensional representation of the double quantum algebrasu q(η(J)) and theq-deformation of the double complex ernst equation

A three-dimensional representation of the double quantum algebrasu _q((J)) is given. By the use of this representation and a Lax pair, we obtain a nonlinear Ernst equation system. By the harmonic function method, a solution of theq-deformed double complex Ernst equation is given.     

The Hamiltonian systems on the Poisson structure of the quasi-classical limit of GL q (∞)

We consider the Hamiltonian systems on the Poisson structure of GL() which is introduced from the quantum group GL _q () by the so-called quasi-classical limit of GL _q (). Furthermore, we show that the Toda lattice hierarchy is a Hamiltonian system of this structure.     

Deformation of maximal subalgebras of SU(6)

The Cartan-Chevalley generators of a L, L being a maximal subalgebra of SU(6), are written in terms of the generators of SU(6) using a boson realization and then are deformed introducing q-bosons. A procedure to obtain a deformed SU(6) starting from L _q is presented. The deformed SU(6) is not equivalent as Hopf structure to Drinfeld-Jimbo SU _q(6). This scheme provides a way to deform the embedding chain SU(6) L.     

Bicovariant differential calculus on quantum groupsSU q (N) andSO q (N)

Following Woronowicz's proposal the bicovariant differential calculus on the quantum groupsSU _q (N) andSO _q (N) is constructed. A systematic construction of bicovariant bimodules by using the matrix is presented. The relation between the Hopf algebras generated by the linear functionals relating the left and right multiplication of these bicovariant bimodules, and theq-deformed universal enveloping algebras is given. Imposing the conditions of bicovariance and consistency with the quantum group structure the differential algebras and exterior derivatives are defined. As an application the Maurer-Cartan equations and theq-analogue of the structure constants are formulated.Address after 1 Dec. 1990, Institute of Theoretical Physics, University of München.     

Heisenberg realization for U q (sl n ) on the flag manifold

We give the Heisenberg realization for the quantum algebra U _q (sl _n ), which is written by theq-difference operator on the flag manifold. We construct it from the action of U _q (sl _n ) on theq-symmetric algebraA _q (Mat _n ) by the Borel-Weil-like approach. Our realization is applicable to the construction of the free field realization for U _q [2].     

Mass operator in theSU(3)-symmetric strong coupling theory with pseudoscalar mesons

In this paper theSU(3)-symmetric model of a static baryon octet source interacting with pseudoscalar meson octet fields by the coupling of Yukawa type is considered in the strong coupling limit. Using the result derived earlier that the isobaric states form the basis of the unitary irreducible representation of the dynamical group G=T ₂₄ [SU(3) SU(2)], the mass operator is specified by the kinetic part of the Hamiltonian as a particular element of the universal enveloping algebra of the symmetry groupSU(3) SU(2) acting in the space of isobaric states.     

Aq-deformation of Wakimoto modules,primary fields and screening operators

Theq-vertex operators of Frenkel and Reshetikhin are studied by means of aq-deformation of the Wakimoto module for the quantum affine algebraU _q at an arbitrary levelk0, –2. A Fock-module version of theq-deformed primary field of spinj is introduced, as well as the screening operators which (anti-)commute with the action ofU _q up to a total difference of a field. A proof of the intertwining property is given for theq-vertex operators corresponding to the primary fields of spinj1/2Z ₀. A sample calculation of the correlation function is also given.This is a revised version of the preprint distributed in December, 1992, with the title Free Field Realization ofq-deformed Primary Fields forU _q (sl ₂)     

Complex quantum group,dual algebra and bicovariant differential calculus

The method used to construct the bicovariant bimodule in ref. [CSWW] is applied to examine the structure of the dual algebra and the bicovariant differential calculus of the complex quantum group. The complex quantum group Fun _q (SL(N, C)) is defined by requiring that it contains Fun _q (SU(N)) as a subalgebra analogously to the quantum Lorentz group. Analyzing the properties of the fundamental bimodule, we show that the dual algebra has the structure of the twisted product Fun _q (SU(N))Fun _q (SU(N)) _reg ^* . Then the bicovariant differential calculi on the complex quantum group are constructed.     

SU(2) q in a Hilbert space of analytic functions

The algebraSU(2) _q is realized in a Hilbert spaceH _q ² of analytic functions; the starting point is the differential realization of operators that satisfyq-algebra in a Hilbert spaceH _q. The Weyl realization ofSU(2) _q is constructed exhibiting the reproducing kernel and the principal vectors; the noncommutativity of the matrix elements of a 2×2 linear representation ofSU(2) _q is obtained as consistency conditions for couplingj1=j2=1/2 toj=0, 1; the derivation of Clebsch-Gordan coefficients is sketched and theq-generalization of the rotation matrices is included. The unitary correspondence ofH _q with a Hilbert space of complex functions of a real variable is also studied. The study presented in this paper follows Bargmann's formalism for the rotation group as closely as possible.     

Calculation of fractional parentage coefficients for the (1d-2s)-shell nuclei with the projection-operator method

The projection operators for the groupsSU _n are used for constructing the noncanonical basis vectors of irreducible representations of these groups as linear combinations of the Gel'fand-Tseitlin canonical basis vectors. The structure of the basis vectors of the irreducible representation of the groupsSU ₄,SU ₃,SU ₆ in the case of the reductionSU ₄ SU ₂×SU ₂,SU ₃R ₃ andSU ₆ SU ₃, respectively, is discussed. A number of formulae have been obtained for the fractional parentage coefficients for the (1d-2s)-shell nuclei.     

Operator spectra inq-oscillator algebras

Spectra of the operatorX _p =q ^pN (a ⁺ +a)q ^pN ,pR, for theq-deformed oscillator algebra are studied. This operator tends to the position operator of the usual quantum harmonic oscillator ifq 1. These spectra depend on the values ofq andp. The operatorX _p can be bounded or unbounded. In the unbounded case, it is not selfadjoint and has infinitely many selfadjoint extensions.Presented at the 4th Colloquium Quantum Groups and Integrable Systems, Prague, 22–24 June 1995.This research was supported in part by the International Science Foundation under Grant U4J200 and by the Ukrainian State Foundation for Fundamental Research (1.1.3/66).     

Macdonald polynomials from Sklyanin algebras: A conceptual basis for thep-adics-quantum group connection

We establish a previously conjectured connection betweenp-adics and quantum groups. We find in Sklyanin's two parameter elliptic quantum algebra and its generalizations, the conceptual basis for the Macdonald polynomials, which interpolate between the zonal spherical functions of related real andp-adic symmetric spaces. The elliptic quantum algebras underlie theZ _n -Baxter models. We show that in then limit, the Jost function for the scattering offirst level excitations in the 1+1 dimensional field theory model associated to theZ _n -Baxter model coincides with the Harish-Chandra-likec-function constructed from the Macdonald polynomials associated to the root systemA ₁. The partition function of theZ ₂-Baxter model itself is also expressed in terms of this Macdonald-Harish-Chandrac-function, albeit in a less simple way. We relate the two parametersq andt of the Macdonald polynomials to the anisotropy and modular parameters of the Baxter model. In particular thep-adic regimes in the Macdonald polynomials correspond to a discrete sequence of XXZ models. We also discuss the possibility of q-deforming Euler products.Work supported in part by the NSF: PHY-9000386     

HeiddenSL(n) symmetry in conformal field theories

We prove that an irreducible representation of the Virasoro algebra can be extracted from an irreducible representation space of theSL(2, ) current algebra by putting a constraint on the latter using the Becchi-Rouet-Stora-Tyutin formalism. Thus there is aSL(2, ) symmetry in the Virasoro algebra, but it is gauged and hidden. This construction of the Virasoro algebra is the quantum analogue of the Hamiltonian reduction. We then are naturally lead to consider a constrainedSL(2, ) Wess-Zumino-Witten model. This system is also related to quantum field theory of coadjoint orbit of the Virasoro group. Based on this result, we present a canonical derivation of theSL(2, ) current algebra in Polyakov's theory of two-dimensional gravity; it is a manifestation of theSL(2, ) symmetry in conformal field theory hidden by the quantum Hamiltonian reduction. We also discuss the quantum Hamiltonian reduction of theSL(2, ) current algebra and its relation to theW _n -algebra of Zamolodchikov. This makes it possible to define a natural generalization of the geometric action for theW _n -algebra despite its non-Lie-algebraic nature.This paper is dedicated to the memory of Vadik G. Knizhnik     

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.