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.     

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 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.     

q-Deformed orthogonal and pseudo-orthogonal algebras and their representations

Deformed orthogonal and pseudo-orthogonal Lie algebras are constructed which differ from deformations of Lie algebras in terms of Cartan subalgebra and root vectors and which make it possible to construct representations by operators acting according to Gel'fand-Tsetlin-type formulas. Unitary representations of the q-deformed algebras U _q (so _n,1) are found.     

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).     

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].     

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.     

On a general analytic formula for U q(su(3)) Clebsch-Gordan coefficients

We present the projection operator method in combination with the Wigner-Racah calculus of the subalgebra U _q(su(2)) for calculation of Clebsch-Gordan coefficients (CGCs) of the quantum algebra U _q(su(3)). The key formulas of the method are couplings of the tensor and projection operators and also a tensor form for the projection operator of U _q(su(3)). We obtain a very compact general analytic formula for the U _q(su(3)) CGCs in terms of the U _q(su(2)) Wigner 3nj symbols.     

(p,q)-Analog of Two-Dimensional Conformal Field Theory. The Ward Identities and Correlation Functions

Abstract

A (p, q)-analog of two-dimensional conformally invariant field theory based on the quantum algebra U_pq (su(1, 1)) is proposed. The representation of the algebra U_pq (su(1, 1)) on the space of quasi-primary fields is given. The (p, q)-deformed Ward identities of conformal field theory are defined. The two- and three-point correlation functions of quasi-primary fields are calculated.     

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).     

Irreducible highest weight representations of quantum groupsU q (gl(n,C))

Explicit recurrence formulas of canonical realization (boson representation) for quantum enveloping algebrasU _q (gl(n, C)) are given. Using them, irreducible highest weight representations ofU _q (gl(n, C)) are obtained as restriction of representation of Fock space to invariant subspace generated by vacuum as a cyclic vector.     

The q-boson realizations of the quantum groups Uq(B2) and Uq(C2)

We give explicit realization of formulae of canonical realization for the quantum enveloping algebrasU _q (B ₂)~U _q (so(5)) andU _q (C ₂)~U _q (sp(4)). In these formulae the generators of the algebra are expressed by means of 3 canonicalq-boson pairs and one auxiliary representation ofU _q (gl(2)).     

Polynomial relations in the centre ofU q (sl(N))

When the parameter of deformationq is a root of unity, the centre ofU _q (sl(N)) contains, besides the usualq-deformed Casimirs, a set of new generators, which are basically themth powers of all the Cartan generators ofU _q (sl(N)). All these central elements are, however, not independent. In this Letter, generalizing the well-known case ofU _q (sl(2)), we explicitly write polynomial relations satisfied by the generators of the centre. Application to the parametrization of irreducible representations and to fusion rules are sketched.On leave from SPht, CE Saclay, 91191 Gif-sur-Yvette Cedex, France.     

Realization ofU q (so(N)) within the differential algebra onR q N

We realize the Hopf algebraU _q–1 (so(N)) as an algebra of differential operators on the quantum Euclidean spaceR _q ^N . The generators are suitableq-deformed analogs of the angular momentum components on ordinaryR ^N . The algebra Fun(R _q ^N ) of functions onR _q ^N splits into a direct sum of irreducible vector representations ofU _q–1 (so(N)); the latter are explicitly constructed as highest weight representations.     

Vertex operator representations of the quantum affine algebra U q(B r (1) )

We construct the level one vertex operator representations of the q-deformation U _q(B _r ⁽¹⁾ ) of the affine Kac-Moody algebra B _r ⁽¹⁾ . Beside the q-deformed vertex operators introduced by Frenkel and Jing, this construction involves a q-deformation of free fermionic fields.     

q-deformed conformal and Poincaré algebras on quantum 4-spinors

We investigate quantum deformation of conformal algebras by constructing the quantum space forsl _q (4). The differential calculus on the quantum space and the action of the quantum generators are studied. We derive deformedsu(2,2) algebra from the deformedsl(4) algebra using the quantum 4-spinor and its conjugate spinor. The quantum 6-vector inso _q (4,2) is constructed as a tensor product of two sets of 4-spinors. We obtain theq-deformed conformal algebra with the suitable assignment of the generators which satisfy the reality condition. The deformed Poincaré algebra is derived through a contraction procedure.Work partially supported by the Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture (#030083)     

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).     

Braid group action on theq-Weyl algebra

We provide a braid group action on theq-deformed Weyl algebraW _q (n). The restriction of this action to the representations ofU _q (A _n–1 ) andU _q (C _n ) inW _q (n) is seen to agree with the braid group action introduced by Lusztig on these quantum algebras.Supported in part by the National Sciences and Engineering Research Council (NSERC) of Canada.     

Vertex operators of quantum affine Lie algebrasU q (D n (1) )

We give an explicit formula for the vertex operators related to the level 1 representations of the quantum affine Lie algebrasU _q (D _n ⁽¹⁾ ) in terms of bosons. As an application, we derive an integral formula for the correlation functions of the vertex models withU _q (D _n ⁽¹⁾ )-symmetry.NJ was supported in part by NSA grant MDA904-93-H-3005 and University of Kansas General Research allocation.SJK was supported in part by Basic Science Research Institute Program, Ministry of Education of Korea, BSRI-94-1414 and GARC-KOSEF at Seoul National University, Korea.     

Models of Q algebra representations

This article gives a review of various straightforward models ofQ algebra representations. This is done using one and two variable function space models of theq-analogues of Lie enveloping algebras. The algebras considered are the quantum algebraU _q (su ₂ ) and aq analogue of the oscillator algebra. We present only the general framework and refer the reader to references of the joint work of the author and Willard Miller, Jr.Presented at the Colloquium on the Quantum Groups, Prague, 18–20 June 1992.