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).     

Nonstandard U q(so3) and U q(so4): tensor products of representations, oscillator realizations and roots of unity

Nonstandard q-deformed algebras U _q(so₃) and U _q(so₄), which can be embedded into U _q(sl₃) and U _q(sl₄) and are coideals in them, are considered. It is shown how to multiply finite dimensional representations of U _q(so₃) when q is positive. Homomorphisms from U _q(so₃) and U _q(so₄) to the q-oscillator algebras are given. By making use of these homomorphisms, irreducible representations of U _q(so₃) and U _q(so₄) for q equal to a root of unity are obtained.     

Representations of Lie algebras from representations of quantum groups

In paper [*] (P. Moylan: Czech. J. Phys., Vol. 47 (1997), p. 1251) we gave an explicit embedding of the three dimensional Euclidean algebra (2) into a quantum structure associated with U _q(so(2, 1)). We used this embedding to construct skew symmetric representations of (2) out of skew symmetric representations of U _q(so(2, 1)). Here we consider generalizations of the results in [*] to a more complicated quantum group, which is of importance to physics. We consider U _q(so(3, 2)), and we show that, for a particular representation, namely the Rac representation, many of the results in [*] carry over to this case. In particular, we construct representations of so(3, 2), P(2, 2), the Poincaré algebra in 2+2 dimensions, and the Poincaré algebra out of the Rac representation of U _q(so(3, 2)). These results may be of interest to those working on exploiting representations of U _q(so(3, 2)), like the Rac, as an example of kinematical confinement for particle constituents such as the quarks.     

Spin physics with internal targets and the blast detector

The aim of this paper is to give a set of central elements of the algebras U_q(so_m) and U _q(iso _m ) when q is a root of unity. They are surprisingly arise from a single polynomial Casimir element of the algebra U_q(so₃). It is conjectured that the Casimir elements of these algebras under any values of q (not only for q a root of unity) and the central elements for q a root of unity derived in this paper generate the centers of U_q(so_m) and U _q(iso _m ) when q is a root of unity.     

Nonstandard q-deformation of the Euclidean Lie algebra and its representations

A nonstandard q-deformed Euclidean algebra U _q(iso _n ), based on the definition of the twisted q-deformed algebra U _qso_n) (different from the Drinfeld–Jimbo algebra U _q(so _n )), is defined. Infinite dimensional representations R of U _q(iso _n ) are described. Explicit formulas for operators of these representations in the orthonormal basis are given. The spectra of the operators R(T _n) corresponding to a q-analogue of the infinitesimal operator of shifts along the n-th axis are described. Contrary to the case of the classical Euclidean Lie algebra iso _n , these spectra are discrete and spectral points have one point of accumulation.     

Small representations of quantum affine algebras

We characterize the finite-dimensional representations of the quantum affine algebra U _q ( _n+1) (whereq ^× is not a root of unity) which are irreducible as representations of U _q (sl _n+1). We call such representations small. In 1986, Jimbo defined a family of homomorphismsev _a from U _q (sl _n+1) to (an enlargement of) U _q (sl_,n+1), depending on a parametera ^·. A second family,ev ^a can be obtained by a small modification of Jimbo's formulas. We show that every small representation of U _q ( _n+1) is obtained by pulling back an irreducible representation of U _q (sl _n+1) byev _a orev ^a for somea ^·.     

Quantum deformations of singletons and of free zero-mass fields

We consider quantum deformations of the real symplectic (or anti-De Sitter) algebra sp(4), spin(3, 2) and of its singleton and (4-dimensional) zero-mass representations. For q a root of –1, these representations admit finite-dimensional unitary subrepresentations. It is pointed out that U_q (sp(4, )), unlike U_q (su(2, 2)), contains U_q (sl ₂ ) as a quantum subalgebra.To Asim Barut, with all our friendship.     

Discrete series of unitary irreducible representations of the Uq(u(3, 1)) and Uq(u(n, 1)) noncompact quantum algebras

The structure of all discrete series of unitary irreducible representations of the U _q (u(3, 1)) and U _q (u(n, 1)) noncompact quantum algebras are investigated with the aid of extremal projection operators and the q-analog of the Mickelsson-Zhelobenko algebra Z(g, g′) _q . The orthonormal basis constructed in the infinite-dimensional space of irreducible representations of the U _q (u(n, 1)) ⊇ U _q (u(n)) algebra is the q-analog of the Gelfand-Graev basis in the space of the corresponding irreducible representations of the u(n, 1) ⊇ u(n) classical algebra.     

Spectra,eigenvectors and overlap functions for representation operators ofq-deformed algebras

Operators of representations corresponding to symmetric elements of theq-deformed algebrasU _q (su_1,1),U _q (so_2,1),U _q (so_3,1),U _q (so _n ) and representable by Jacobi matrices are studied. Closures of unbounded symmetric operators of representations of the algebrasU _q (su_1,1) andU _q (so_2,1) are not selfadjoint operators. For representations of the discrete series their deficiency indices are (1,1). Bounded symmetric operators of these representations are trace class operators or have continuous simple spectra. Eigenvectors of some operators of representations are evaluated explicitly. Coefficients of transition to eigenvectors (overlap coefficients) are given in terms ofq-orthogonal polynomials. It is shown how results on eigenvectors and overlap coefficients can be used for obtaining new results in representation theory ofq-deformed algebras.     

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].     

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.     

Positive-energy irreps of the quantum anti de Sitter algebra

We obtain positive-energy irreducible representations of theq-deformed anti de Sitter algebraU _q (so(3, 2)) by deformation of the classical ones. When the deformation parameterq isN-th root of unity, all these irreducible representations become unitary and finite-dimensional. Generically, their dimensions are smaller than those of the corresponding finite-dimensional non-unitary representations ofso(3, 2). We discuss in detail the singleton representations, i.e. the Di and Rac. WhenN is odd, the Di has dimension 1/2(N ²–1) and the Rac has dimension 1/2(N ²+1), while ifN is even, both the Di and Rac have dimension 1/2N ². These dimensions are classical only forN=3 when the Di and Rac are deformations of the two fundamental non-unitary representations ofso(3, 2).Presented at the 4th Colloquium Quantum groups and integrable systems, Prague, 22–24 June 1995.On leave from Bulgarian Acad. Sci., Institute of Nuclear Research and Nuclear Energy, 72 Tsarigradsko Chaussee, 1784 Sofia, Bulgaria.On leave from Pennsylvania State University (Fulbright scholar).     

∗-Representations of the Quantum Algebra U q(sl(3))

Abstract

Studied in this paper are real forms of the quantum algebra U _q(sl(3)). Integrable operator representations of ?-algebras are defined. Irreducible representations are classified up to a unitary equivalence.     

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).     

Representations of the nonstandard (twisted) deformationU′ q(so n ) forq a root of unity

Irreducible representations of the algebrasU′ _q(so _n ) forq a root of unityq ^p=1 are given. The main class of these representations act onp ^N-dimensional linear space (whereN is a number of positive roots of the Lie algebra so _n ) and are given byr = dim so _n complex parameters. Some classes of degenerate irreducible representations are also described. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000. The research described in this publication was made possible in part by Grant UP1-2115 of the U.S. Civilian Research and Development Foundation for the Independent States of the Former Soviet Union (CRDF).     

Centre and representations of small quantum superalgebras at roots of unity

When the deformation parameter is a root of unity, the centre of a quantum group can be described by a set of generators and non trivial relations. In the case ofU _q (sl(N)), these relations simply derive from the expressions of the deformed Casimir operators. In the case ofU _q (osp(1|2)), the relation is simple if we use an operator which anticommutes with the fermionic generators and whose square is the quadratic Casimir. This operator also simplifies the classification of finite dimensional irreducible representations. In the case ofU _q (sl(1|2)), the relations derive from the (infinite set of) standard Casimir operators.Presented at the 5th International Colloquium on Quantum Groups: Quantum Groups and Integrable Systems, Prague, 20–22 June 1996.     

The nonstandardq-deformation of enveloping algebraU(so n ): results and problems

We describe properties of the nonstandardq-deformationU _q ^/′ (so _n ) of the universal enveloping algebraU(so _n ) of the Lie algebra so _n which does not coincide with the Drinfeld-Jimbo quantum algebraU _q(so _n ) and is important for quantum gravity. Many unsolved problems are formulated. Some of these problems are solved in special cases. The research of this paper was made possible in part by Award UP1-2115 of U.S. Civilian Research and Development Foundation. Presented at DI-CRM Workshop held in Prague, 18–21 June 2000.     

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).