Real complexified quantum groups,their vector fields and quantum enveloping algebras

We construct complex quantum groups associated with the Lie algebras of typeA _n–1 ,B _n ,C _n andD _n which are considered as real algebras. Following the ideas of Faddeev, Reshetikhin and Takhtayan, we obtain the Hopf algebras of regular functionalsU _R , on these real complexified quantum groups. Theq-analogues of the left invariant vector fields of the quantum enveloping algebras are defined. These quantum vector fields are functionals over the corresponding real formA of the complex quantum groupA. The equivalence of the Hopf algebra of regular functionals and the algebra of complex quantum vector fields is shown by factorizing the vector fields uniquely into a triangular and a unitary part and identifying them with the corresponding elements of the algebra of regular functionals. In the special exampleA ₁ , we derive theq-deformed real complexified enveloping algebraU _q sl(2, ) with six generators.Presented at the Colloquium on the Quantum Groups, Prague, 18–20 June, 1992.Based on the papers: [i]Drabant B., Schlieker M., Weich W., and Zumino B.: PreprintLMU-TPW 1991-5 (to appear in Commun. Math. Phys.) [ii]Chryssomalakos C., Drabant B., Schlieker M., Weich W., and Zumino B.: Preprint UCB 92/03 (to appear in Commun. Math. Phys.) [iii]Drabant B., Juro B., Schlieker M., Weich W., and Zumino B.: Preprint MPI-Ph/92-39 (submitted to Lett. Math. Phys.)     

The hopf algebra of vector fields on complex quantum groups

We derive the equivalence of the complex quantum enveloping algebra and the algebra of complex quantum vector fields for the Lie algebra types A _n , B _n , C _n , and D _n by factorizing the vector fields uniquely into a triangular and a unitary part and identifying them with the corresponding elements of the algebra of regular functionals.Humboldt Fellow.     

Differential calculus onISO q(N), quantum Poincaré algebra and q-gravity

We present a general method to deform the inhomogeneous algebras of theB _n,C_n,D_n type, and find the corresponding bicovariant differential calculus. The method is based on a projection fromB _n+1,C_n+1,D_n+1. For example we obtain the (bicovariant) inhomogeneousq-algebraISO _q(N) as a consistent projection of the (bicovariant)q-algebraSO _q(N=2). This projection works for particular multiparametric deformations ofSO(N+2), the so-called minimal deformations. The case ofISO _q(4) is studied in detail: a real form corresponding to a Lorentz signature exists only for one of the minimal deformations, depending on one parameterq. The quantum Poincaré Lie algebra is given explicitly: it has 10 generators (no dilatations) and contains theclassical Lorentz algebra. Only the commutation relations involving the momenta depend onq. Finally, we discuss aq-deformation of gravity based on the gauging of thisq-Poincaré algebra: the lagrangian generalizes the usual Einstein-Cartan lagrangian.     

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.     

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.     

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].     

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.     

Modules-at-Infinity for Quantum Vertex Algebras

This is a sequel to [Li4] and [Li5] in a series to study vertex algebra-like structures arising from various algebras such as quantum affine algebras and Yangians. In this paper, we study two versions of the double Yangian , denoted by DY _q (sl ₂) and with q a nonzero complex number. For each nonzero complex number q, we construct a quantum vertex algebra V _q and prove that every DY _q (sl ₂)-module is naturally a V _q -module. We also show that -modules are what we call V _q -modules-at-infinity. To achieve this goal, we study what we call -local subsets and quasi-local subsets of for any vector space W, and we prove that any -local subset generates a (weak) quantum vertex algebra and that any quasi-local subset generates a vertex algebra with W as a (left) quasi module-at-infinity. Using this result we associate the Lie algebra of pseudo-differential operators on the circle with vertex algebras in terms of quasi modules-at-infinity.     

q-deformed W-algebras and elliptic algebras

The elliptic algebra A _q,p {sl}(N _c) at the critical level c = –N has an extended center containing trace-like operators t(z). Families of Poisson structures, defining q-deformations of the W _N algebra, are constructed. The operators t(z) also close an exchange algebra when (–p ^1/2) ^NM = q ^–c–N for M . It becomes Abelian when in addition p = q ^Nh where h is a non-zero integer. The Poisson structures obtained in these classical limits contain different q-deformed W _N algebras depending on the parity of h, characterizing the exchange structures at p q ^Nh as new W _q,p(sl(N)) algebras.     

Vertex operators of level-one U q (B n (1) )-modules

We construct explicitly the level-one vertex operators for the fundamental modulesV(₁) (i=0, 1,n) of the quantum affine algebra of typeB using free boson and fermion fields.     

Algebra of Deformed Differential Operators and Induced Integrable Toda Field Theory

We build in this paper the algebra of q-deformed pseudo-differential operators, shown to be an essential step toward setting a q-deformed integrability program. In fact, using the results of this q-deformed algebra, we derive the q-analogues of the generalized KdV hierarchy. We focus in particular on the first leading orders of this q-deformed hierarchy, namely the q-KdV and q-Boussinesq integrable systems. We also present the q-generalization of the conformal transformations of the currents u _n ,n 2, and discuss the primary condition of the fields W _n , n 2, by using the Volterra gauge group transformations for the q-covariant Lax operators. An induced su(n)-Toda(su(2)-Liouville) field theory construction is discussed and other important features are presented.     

Some remarks on producing Hopf algebras

We report some observations concerning two well-known approaches to construction of quantum groups. Thus, starting from a bialgebra of inhomogeneous type and imposing quadratic, cubic or quartic commutation relations on a subset of its generators we come, in each case, to aq-deformed universal enveloping algebra of a certain simple Lie algebra. An interesting correlation between the order of initial commutation relations and the Cartan matrix of the resulting algebra is observed. Another example demonstrates that the bialgebra structure ofsl _q (2) can be completely determined by requiring theq-oscillator algebra to be its covariant comodule, in analogy with Manin's approach to defineSL _q (2) as a symmetry algebra of the bosonic and fermionic quantum planes.Presented at the 4th Colloquium Quantum Groups and Integrable Systems, Prague, 22–24 June 1995.This work was supported in part by International Sciences Foundation (grant RFF-300) and by Russian Basic Research Foundation (grant 95-02-05679).I acknowledge helpful discussions with A. Isaev, P. Kulish, V. Lyakhovsky, O. Ogievetsky, P. Pyatov, and V. Tolstoy.     

Characters of U q (gl(n))-Reflection Equation Algebra

We list characters (one-dimensional representations) of the reflection equation algebra associated with the fundamental vector representation of the Drinfeld–Jimbo quantum group _q (gl(n)).     

Classification of bicovariant differential calculi on quantum groups of type A,B, C and D

Under the assumptions thatq is not a root of unity and that the differentialsdu _j ⁱ of the matrix entries span the left module of first order forms, we classify bicovariant differential calculi on quantum groupsA _n–1 ,B _n ,C _n andD _n . We prove that apart one dimensional differential calculi and from finitely many values ofq, there are precisely2n such calculi on the quantum groupA _n–1 =SL _q (n) forn3. All these calculi have the dimensionn ². For the quantum groupsB _n ,C _n andD _n we show that except for finitely manyq there exist precisely twoN ²-dimensional bicovariant calculi forN3, whereN=2n+1 forB _n andN=2n forC _n ,D _n . The structure of these calculi is explicitly described and the corresponding ad-invariant right ideals of ker are determined. In the limitq1 two of the 2n calculi forA _n–1 and one of the two calculi forB _n ,C _n andD _n contain the ordinary classical differential calculus on the corresponding Lie group as a quotient.     

Quantum algebra structure of certain Jackson integrals

Theq-difference system satisfied by Jackson integrals with a configuration ofA-type root system is studied. We explicitly construct some linear combination of Jackson integrals, which satisfies the quantum Knizhnik-Zamolodchikov equation for the 2-point correlation function ofq-vertex operators, introduced by Frenkel and Reshetikhin, for the quantum affine algebra . The expression of integrands for then-point case is conjectured, and a set of linear relations for the corresponding Jackson integrals is proved.     

Vecteurs totalisateurs d'une algèbre de von Neumann

We prove that the set of cyclic vectors for a von Neumann algebra in a Hilbert spaceH is aG set, which is empty or dense. We obtain some corollaries, for instance: if (A ₁,A ₂ ...) is a sequence of von Neumann algebras inH, and if eachA _n has a cyclic vector and a separating vector, then there exists a vector inH which is cyclic and separating for eachA _n. For algebras of local observables, we improve the known results connecting the infinite type of the algebras and the existence of cyclic and separating vectors.     

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.     

Roots of unity: Representations of quantum groups

Representations of Quantum GroupsU (g _n ), g _n any semi-simple Lie algebra of rankn, are constructed from arbitrary representations of rankn–1 quantum groups for a root of unity. Representations which have the maximal dimension and number of free parameters for irreducible representations arise as special cases.Supported by the Japan Society for the Promotion of ScienceDeceased     

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.     

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.