On the representations of quantum oscillator algebra

It is shown that for q<1, the quantum oscillator algebra has a supplementary family of representations inequivalent to the usual q-Fock representation, with no counterpart at the limit q=1. They are used to build representations of SU _q (1,1) and E(2) in Schwinger's way.     

Operator representations of U q (sl2(ℝ))

Integrable (well-behaved) operator representations of the *-algebra U _q (sl₂()), |q|= 1, q ² 1, |q|=1, q² 1, in Hilbert space are defined and classified up to unitary equivalence.     

q-transformation theory

We obtain the complete set of states of theq-oscillator in both configuration space and momentum space as well as the transformation between these spaces. The states as well as the matrix elements lie in the SU _q (2) algebra. To obtain transition probabilities, one must take the Woronowicz square.     

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.     

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.     

Projective quantum spaces

Associated to the standard SU _q (n) R-matrices, we introduce quantum spheresS _q ^2n-1 , projective quantum spaces _q ^n-1 , and quantum Grassmann manifoldsG _k( _q ⁿ ). These algebras are shown to be homogeneous spaces of standard quantum groups and are also quantum principle bundles in the sense of T. Brzeziski and S. Majid.     

Coordinates of the quantum plane as q-tensor operators in (\mathfrak{s}\mathfrak{u}(2)*\mathfrak{s}\mathfrak{u}(2))

The relation between the set of transformations of the quantum plane and the quantum universal enveloping algebra U _q (u(2)) is investigated by constructing representations of the factor algebra U _q (u(2))* . The noncommuting coordinates of , on which U _q (2) * U _q (2) acts, are realized as q-spinors with respect to each U _q (u(2)) algebra. The representation matrices of U _q (2) are constructed as polynomials in these spinor components. This construction allows a derivation of the commutation relations of the noncommuting coordinates of directly from properties of U _q (u(2)). The generalization of these results to U _q (u(n)) and is also discussed.     

The q-boson operator algebra and q-Hermite polynomials

The q-boson algebra is defined as an associative algebra with generators and relations. Some examples are given, and then the q-boson algebra is extended such that the roots of the diagonal generators are also defined. It is shown that a family of transformations exist mapping one set of standard generators of the q-boson algebra to another set of standard generators. Using such a transformation, one obtains expressions for q-bosons for which the kth q-boson state is expressed in terms of a q-Hermite polynomial p _k (x; q) which reduces to the ordinary Hermite polynomial of degree k when q=1.     

Continued fractions and fermionic representations for characters of M(p,p′) minimal models

We present fermionic sum representations of the characters _{, s} ^{(p, p)} of the minimal M(p,p) models for all relatively prime integers p>p for some allowed values of r and s. Our starting point is biomial (q-binomial) identities derived from a truncation of the state counting equations of the XXZ spin 1/2 chain of anisotropy –=–cos((p/p)). We use the Takahashi-Suzuki method to express the allowed values of r (and s) in terms of the continued fraction decomposition of {p/p} (and p/p), where {x} stands for the fractional part of x. These values are, in fact, the dimensions of the Hermitian irreducible representations of SU _q- (2) (and SU _q+ (2)) with q–=exp(i{p/p}) (and q+=exp(i(p/p))). We also establish the duality relation M(p,p) M(p–p,p) and discuss the action of the Andrews-Bailey transformation in the space of minimal models. Many new identities of the Rogers-Ramanujan type are presented.Dedicated to Prof. Vladimir Rittenberg on his 60th birthday     

gl q (n) and quantum monodromy

A generalized Toda lattice based on gl(n) is considered. The Poisson brackets are expressed in terms of a Lax connection, L=–() and a classical r-matrix, {₁,₂}=[r,₁+₂}. The essential point is that the local lattice transfer matrix is taken to be the ordinary exponential, T=e; this assures the intepretation of the local and the global transfer matrices in terms of monodromy, which is not true of the T-matrix used for the sl(n) Toda lattice. To relate this exponential transfer matrix to the more manageable and traditional factorized form, it is necessary to make specific assumptions about the equal time operator product expansions. The simplest possible assumptions lead to an equivalent, factorized expression for T, in terms of operators in (an extension of) the enveloping algebra of gl(n). Restricted to sl(n), and to multiplicity-free representations, these operators satisfy the commutation relations of sl _q (n), which provides a very simple injection of sl _q (n) into the enveloping algebra of sl(n). A deformed coproduct, similar in form to the familiar coproduct on sl _q (n), turns gl(n) into a deformed Hopf algebra gl _q (n). It contains sl _q (n) as a subalgebra, but not as a sub-Hopf algebra.     

Diagonalization of the braid generator on unitary irreps of quantum supergroups

It is shown that the braid generator associated with the universalR-matrix is diagonalizable on all unitary representations of quantum supergroups. An example is considered using U _q (gl(2|1)) and a family of eight-dimensional typical representations.     

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.     

q-Weyl coefficients forU q (3) andq-Racah coefficients forSU q (2)

We show that theq-Weyl coefficients of the quantum algebraSU _q (3) are equal to theq-Racah coefficients of the quantum algebraSU _q (2) (up to a simple phase factor). Using aq-analog of the resummation procedure we obtain also theq-analogues of all known general analytical expressions for the 6j-symbols (or the Racah coefficients) of the quantum algebraSU _q (2) starting from one such formula.Presented at the 4th Colloquium Quantum groups and integrable systems, Prague, 22–24 June 1995.The research described in this publication was supported in part by Grants No. MB1000 and No. NRC000 from International Science Foundation.     

On the hopf structure of U p,q (gl(1∣1)) and the universal T-matrix of fun p,q (GL(1∣1))

The dually conjugate Hopf superalgebras Fun _p,q (GL(11)) and U _p,q (gl(11)) are studied using the Frønsdal-Galindo approach and the full Hopf structure of U _p,q (gl(11)) is extracted. A finite expression for the universal T-matrix, identified with the dual form and expressing the generalization of the exponential map of the classical groups, is obtained for Fun _p,q (GL(11)). In a representation with a colour index, the T-matrix assumes a form that satisfies a coloured graded Yang-Baxter equation.     

Unitary representations of the quantum group SU q (1, 1): II — Matrix elements of unitary representations and the basic hypergoemetric functions

Some series of unitary representations of the quantum group SU _q (1, 1) are introduced. Their matrix elements are expressed in terms of the basic hypergeometric functions. Operator realization of the coordinate elements of SU _q (1, 1) and aq-analogue of some classical identities are discussed.     

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.     

Coherent states of theq-Weyl algebra

New coherent states of theq-Weyl algebraAA –qA A = 1,0 <q < 1, are constructed. They are defined as eigenstates of the operatorA which is the lowering operator for nonhighest weight representations describing positive energy states. Depending on whether the positive spectrum is discrete or continuous, these coherent states are related either to the bilateral basic hypergeometric series or to some integrals over them. The free particle realization of theq-Weyl algebra whenA A d²/dx² is used for illustrations.On leave of absence from the Institute for Nuclear Research, Russian Academy of Sciences, Moscow, Russia.     

Nonclassical type representations of the q-deformed algebra U′q(son)

The nonstandard q-deformation U_q(so_n) of the universal enveloping algebra U(so _n ) has irreducible finite dimensional representations which are a q-deformation of the well-known irreducible finite dimensional representations of U(so _n ). But U_q(so_n) also has irreducible finite dimensional representations which have no classical analogue. The aim of this paper is to give these representations which are called nonclassical type representations. They are given by explicit formulas for operators of the representations corresponding to the generators of U_q(so_n).     

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.     

Bicovariant differential calculus on quantum groups and wave mechanics

The bicovariant differential calculus on quantum groups being defined by Woronowicz and later worked out explicitly by Carow-Watamura et at. and Juro for the real quantum groupsSU _q (N) andSO _q (N) through a systematic construction of the bicovariant bimodules of these quantum groups is reviewed forSU _q (2) andSO _q (N). The resulting vector fields build representations of the quantized universal enveloping algebras acting as covariant differential operators on the quantum groups and their associated quantum spaces. As an application a free particle stationary wave equation on quantum space is formulated and solved in terms of a complete set of energy eigenfunctions.Presented at the Colloquium on the Quantum Groups, Prague, 18–20 June 1992.