Finite dimensional representations of the quantum Lorentz group

All finite dimensional irreducible representations of the quantum Lorentz group SL _q (2,) are described explicitly and it is proved all finite dimensional representations of SL _q (2,) are completely reducible. The conjecture of Podle and Woronowicz will be answered affirmatively.     

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     

Duality for a Lorentz quantum group

We give the algebra _q ^/* dual to the matrix Lorentz quantum group _q of Podles-Woronowicz, and Watamuraet al. As a commutation algebra, it has the classical form _q ^/* U _q (sl(2, )) U _q (sl(2, )). However, this splitting is not preserved by the coalgebra structure which we also give. For the derivation, we use a generalization of the approach of Sudbery, viz. tangent vectors at the identity.     

Associated quantum vector bundles and symplectic structure on a quantum space

We define a quantum generalization of the algebra of functions over an associated vector bundle of a principal bundle. Here the role of a quantum principal bundle is played by a Hopf-Galois extension. Smash products of an algebra times a Hopf algebra H are particular instances of these extensions, and in these cases we are able to define a differential calculus over their associated vector bundles without requiring the use of a (bicovariant) differential structure over H. Moreover, if H is coquasitriangular, it coacts naturally on the associated bundle, and the differential structure is covariant.We apply this construction to the case of the finite quotient of the SL _q(2) function Hopf algebra at a root of unity (q ³ = 1) as the structure group, and a reduced 2-dimensional quantum plane as both the base manifold and fibre, getting an algebra which generalizes the notion of classical phase space for this quantum space. We also build explicitly a differential complex for this phase space algebra, and find that levels 0 and 2 support a (co)representation of the quantum symplectic group. On this phase space we define vector fields, and with the help of the Sp _q structure we introduce a symplectic form relating 1-forms to vector fields. This leads naturally to the introduction of Poisson brackets, a necessary step to do classical mechanics on a quantum space, the quantum plane.     

Radiactiv corrections to deeply virtual compton scattering

The non-commuting matrix elements of matrices from the quantum group GL _q(2;C) with q = being the n-th root of unity are given a representation as operators in Hilbert space with help of C ₄ ⁽ⁿ⁾ generalized Clifford algebra generators.The case of q C, |q| = 1 is treated parallelly.     

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.     

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.     

On a possible origin of quantum groups

A Poisson bracket structure having the commutation relations of the quantum group SL _q (2) is quantized by means of the Moyal star-product on C (²), showing that quantum groups are not exactly quantizations, but require a quantization (with another parameter) in the background. The resulting associative algebra is a strongly invariant nonlinear star-product realization of the q-algebra U _q (sl(2)). The principle of strong invariance (the requirement that the star-commutator is star-expressed, up to a phase, by the same function as its classical limit) implies essentially the uniqueness of the commutation relations of U _q (sl(2)).     

Cyclic homology of differential operators,the Virasoro algebra and aq-analogue

We show how methods from cyclic homology give easily an explicit 2-cocycle on the Lie algebra of differential operators of the circle such that restricts to the cocycle defining the Virasoro algebra. The same methods yield also aq-analogue of as well as an infinite family of linearly independent cocycles arising when the complex parameterq is a root of unity. We use an algebra ofq-difference operators andq-analogues of Koszul and the Rham complexes to construct these quantum cocycles.     

Noncommutative Ricci Curvature and Dirac Operator on C q [SL2] at Roots of Unity

We find a unique torsion free Riemannian spin connection for the natural Killing metric on the quantum group C _q [ SL₂], using a recent frame bundle formulation. We find that its covariant Ricci curvature is essentially proportional to the metric (i.e. an Einstein space). We compute the Dirac operator and find for q an odd rth root of unity that its eigenvalues are given by q-integers [m] _q for m=0,1...,r–1 offset by the constant background curvature. We fully solve the Dirac equation for r=3.     

Differential algebras for quantum groups of type B, C, and D

We study higher order bicovariant differential calculi on the quantum groups O_q(N) and Sp _q (N). We show that the second antisymmetrizer exterior algebra _u is the quotient of the universal exterior algebra _u by the principal ideal generated by . Here denotes the unique up to scalars biinvariant 1-form. Moreover is central in _u and _u is an inner differential calculus. We show that the quadratic dual to the left-invariant algebra _{s L} is isomorphic to the reflection equation algebra. Let be an arbitrary left-covariant first order differential calculus. We show that the dimension of the space of left-invariant 2-forms in the universal exterior algebra equals the number of linearly independent quadratic-linear relations in the quantum tangent space.     

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.     

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)     

Weight Function Approach to q-Difference Equations for the q-Hypergeometric Polynomials

In this paper we define a new algebra generated by the difference operators D _q and D _q-1 with two analytic functions (x) and (x). Also, we define an operator M = J ₁ J ₂–J ₃ J ₄ s.t. all q-hypergeometric orthogonal polynomials Y _n(x), x cos(), are eigenfunctions of the operator M with eigenvalues _q [n] _q . The choice of (x) and (x) depend on the weight function of Y _n (x).     

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.     

Unitary representations of suq(2) on the plane for q ∈ R+ or generic q ∈ S1

Some time ago, Rideau and Winternitz introduced a realization of the quantum algebra su _q(2) on a real two-dimensional sphere, or a real plane, and constructed a basis for its representations in terms of q-special functions, which can be expressed in terms of q-Vilenkin functions. In their study, the values of q were implicitly restricted to q R⁺. In the present paper, we extend their work to the case of generic values of q S ¹ (i.e., q values different from a root of unity). In addition, we unitarize the representations for both types of q values, q R⁺ and generic q S ¹, by determining some appropriate scalar products.     

Non-commutative Low-dimension Spaces and Superspaces Associated with Contracted Quantum Groups and Supergroups

Quantum planes, which correspond to all one-parameter solutions of Quantum Yang-Baxter Equation (QYBE) for the two-dimensional case of GL-groups, are summarized and their geometrical interpretations are given. It is shown that the quantum dual plane is associated with an exotic solution of QYBE and the well-known quantum h-plane may be regarded as the quantum analog of the flag (or fiber) plane. Contractions of the quantum supergroup G L _q (12) and corresponding quantum superspace C _q (12) are considered in Cartesian basis. The contracted quantum superspace C _h (12);) is interpreted as the non-commutative analog of the superspace with the fiber odd part.     

Generalized Transvectants-Rankin–Cohen Brackets

We introduce o(p+1q+1)-invariant bilinear differential operators on the space of tensor densities on Rⁿ generalizing the well-known bilinear sl₂-invariant differential operators in the one-dimensional case, called Transvectants or Rankin–Cohen brackets. We also consider already known linear o(p+1q+1)-invariant differential operators given by powers of the Laplacian.     

Disentangling q-Exponentials: A General Approach

We revisit the q-deformed counterpart of the Zassenhaus formula, expressing the Jackson q-exponential of the sum of two non-q-commuting operators as an (in general) infinite product of q-exponential operators involving repeated q-commutators of increasing order, E_q(A+B) = E_q0(A)E_q1 (B) _i=2 E_qi. By systematically transforming the q-exponentials into exponentials of series and using the conventional Baker–Campbell–Hausdorff formula, we prove that one can make any choice for the bases ^qi, i=0, 1, 2, ..., of the q-exponentials in the infinite product. An explicit calculation of the operators C _i in the successive factors, carried out up to sixth order, also shows that the simplest q-Zassenhaus formula is obtained for ₀ = ₁ =1, and ₂ = 2, and ₃ = 3. This confirms and reinforces a result of Sridhar and Jagannathan, on the basis of fourth-order calculations.     

Braided matrix structure of the Sklyanin algebra and of the quantum Lorentz group

Braided groups and braided matrices are novel algebraic structures living in braided or quasitensor categories. As such they are a generalization of super-groups and super-matrices to the case of braid statistics. Here we construct braided group versions of the standard quantum groupsU _q (g). They have the same FRT generatorsl ^± but a matrix braided-coproductL=LL, whereL=l ⁺ Sl ^–, and are self-dual. As an application, the degenerate Sklyanin algebra is shown to be isomorphic to the braided matricesBM _q(2); it is a braided-commutative bialgebra in a braided category. As a second application, we show that the quantum doubleD(U _q (sl ₂)) (also known as the quantum Lorentz group) is the semidirect product as an algebra of two copies ofU _q (sl ₂), and also a semidirect product as a coalgebra if we use braid statistics. We find various results of this type for the doubles of general quantum groups and their semi-classical limits as doubles of the Lie algebras of Poisson Lie groups.