The quantum de Rham complexes associated with SL h (2)

Quantum de Rham complexes on the quantum plane and the quantum group itself are constructed for the nonstandard deformation of Fun(SL(2)). It is shown that in contrast to the standardq-deformation of SL(2), the above complexes are unique for SL _h (2). Also, as a byproduct, a new deformation of the two-dimensional Heisenberg algebra is obtained which can be used to construct models ofh-deformed quantum mechanics.     

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.     

A *-product on SL(2) and the corresponding nonstandard (\mathfrak{s}\mathfrak{l}({\text{2}}))

We obtain Zakrzewski's deformation of Fun SL(2) through the construction of a *-product on SL(2). We then give the deformation of dual to this, as well as a Poincaré basis for both algebras.Aspirant au Fonds National belge de la Recherche Scientifique. Partially supported by EEC contract SC1-0105-C.     

Singular vectors of quantum group representations for straight Lie algebra roots

We give explicit formulae for singular vectors of Verma modules over U_q(G), where G is any complex simple Lie algebra. The vectors we present correspond exhaustively to a class of positive roots of G which we call straight roots. In some special cases, we give singular vectors corresponding to arbitrary positive roots. For our vectors we use a special basis of U_q(G ^-), where G ^- is the negative roots subalgebra of G, which was introducted in our earlier work in the case q=1. This basis seems more economical than the Poincaré-Birkhoff-Witt type of basis used by Malikov, Feigin, and Fuchs for the construction of singular vectors of Verma modules in the case q=1. Furthermore, this basis turns out to be part of a general basis recently introduced for other reasons by Lusztig for U_q(^-), where ^- is a Borel subalgebra of G.A. v. Humboldt-Stiftung fellow, permanent address and after 22 September 1991: Bulgarian Academy of Sciences, Institute of Nuclear Research and Nuclear Energy, 1784 Sofia, Bulgaria.     

Left regular representation and contraction of sl q (2) to e q (2)

The left regular representation of the quantum algebras sl _q (2) and e _q (2) are discussed and shown to be related by contraction. The reducibility is studied andq-difference intertwining operators are constructed.     

The quantum bialgebra associated with the eight-vertexR-matrix

The quantum bialgebra related to the Baxter's eight-vertexR-matrix is found as a quantum deformation of the Lie algebra of sl(2)-valued automorphic functions on a complex torus.     

Differential calculus on the inhomogeneous quantum groups IGL q (n)

We obtain the inhomogeneousq-groups IGL _q (n) via a projection from GL _q (n + 1). The bicovariant differential calculus of IGL _q (n) is constructed, and the corresponding quantum Lie algebra is given explicitly.     

A superalgebra morphism of U q [osp(1/2N)] onto the deformed oscillator superalgebra W q (N)

We prove that the deformed oscillator superalgebra W _q (n) (which in the Fock representation is generated essentially byn pairs ofq-bosons) is a factor algebra of the quantized universal enveloping algebra U _q [osp(1/2n)]. We write down aq-analog of the Cartan-Weyl basis for the deformed osp(1/2n) and also give an oscillator realization of all Cartan-Weyl generators.     

A quantum-group-like structure on noncommutative 2-tori

We show that on noncommutative 2-tori, there are natural structures which have analogous formal properties as Hopf algebra structures, but where the comultiplication has values in a deformation of the tensor product.Supported by Project P 7724 PHY of Fonds zur Förderung der wissenschaftlichen Forschung.     

On a possible algebra morphism of U q [osp(1/2n)] onto the deformed oscillator algebra W q (n)

We formulate a conjecture stating that the algebra ofn pairs of deformed Bose creation and annihilation operators is a factor algebra of U _q [osp(1/2n)], considered as a Hopf algebra, and prove it for then = 2 case. To this end, we show that for any value ofq, U _q [osp(1/4)] can be viewed as a superalgebra freely generated by two pairsB ₁ ^± ,B ₂ ^± of deformed para-Bose operators. We write down all Hopf algebra relations, an analogue of the Cartan-Weyl basis, the commutation relations between the generators and a basis in U _q [osp(1/2n)] entirely in terms ofB ₁ ^± ,B ₂ ^± .     

A natural and rigid model of quantum groups

We introduce a natural (Fréchet-Hopf) algebra A containing all generic Jimbo algebras U _t (sl(2)) (as dense subalgebras). The Hopf structures on A extend (in a continuous way) the Hopf structures of generic U _t (sl(2)). The Universal R-matrices converge in A A. Using the (topological) dual of A, we recover the formalism of functions of noncommutative arguments. In addition, we show that all these Hopf structures on A are isomorphic (as bialgebras), and rigid in the category of bialgebras.     

A class of bicovariant differential calculi on hopf algebras

We introduce a large class of bicovariant differential calculi on any quantum group A, associated to Ad-invariant elements. For example, the deformed trace element on SL_q (2) recovers Woronowicz's 4D _± calculus. More generally, we obtain a class of differential calculi on each quantum group A(R), based on the theory of the corresponding braided groups B(R). Here R is any regular solution of the QYBE.Supported by St John's College, Cambridge and KBN grant 2 0218 91 01.     

R-matrix formulation of the quantum inhomogeneous groups ISO q,r (N) and ISp q,r (N)

The quantum commutationsRTT=TTR and the orthogonal (symplectic) conditions for the inhomogeneous multiparametricq-groups of theB _n ,C _n ,D _n type are found in terms of theR-matrix ofB _n+1 ,C _n+1 ,D _n+1 .A consistent Hopf structure on these inhomogeneousq-groups is constructed by means of a projection fromB _n+1 ,C _n+1 ,D _n+1 .Real forms are discussed; in particular, we obtain theq-groups ISO _q,r (n+1,n–1), including the quantum Poincaré group. The inhomogeneusq-groups do not contain dilatations when the parameters satisfy certain conditions. For example, we find a dilatation-freeq-Poincaré group depending on one real parameterq.     

Remarks on bicovariant differential calculi and exterior Hopf algebras

We show that every bicovariant differential calculus over the quantum groupA defines a bialgebra structure on its exterior algebra. Conversely, every exterior bialgebra ofA defines bicovariant bimodule overA. We also study a quasitriangular structure on exterior Hopf algebras in some detail.     

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.     

Quantization of U q [so(2n + 1)] with deformed para-Fermi operators

The observation thatn pairs of para-Fermi (pF) operators generate the universal enveloping algebra of the orthogonal Lie algebra so(2n + 1) is used in order to define deformed pF operators. It is shown that these operators are an alternative to the Chevalley generators. With this background U _q [so(2n + 1)] and its Cartan-Weyl generators are written down entirely in terms of deformed para-Fermi operators.     

Differential operators on quantum spaces for GL q (n) and SO q (n)

We prove that the rings of q-differential operators on quantum planes of the GL _q (n) and SO _q (n) types are isomorphic to the rings of classical differential operators. Also, we construct decompositions of the rings of q-differential operators into tensor products of the rings of q-differential operators with less variables.     

Differential calculi on quantum vector spaces with Hecke-type relations

From a vector spaceV equipped with a Yang-Baxter operatorR one may form the r-symmetric algebraS _R V=TV/v w–R(v w), which is a quantum vector space in the sense of Manin, and the associated quantum matrix algebraM _R V=T(End(V))/f g–R(f g)R ^-1. In the case whenR satisfies a Hecke-type identityR ²=(1–q)R+q, we construct a differential calculus _R V forS _R V which agrees with that constructed by Pusz, Woronowicz, Wess, and Zumino whenR is essentially theR-matrix of GL _q (n). Elements of _R V may be regarded as differential forms on the quantum vector spaceS _R V. We show that _R V isM _R V-covariant in the sense that there is a coaction _*: _R V M _R V _R V with _*d=(1 d)_* extending the natural coaction :S _R V M _R V S _R V.     

Fusion structures from quantum groups: II. Why truncation is necessary

It is shown that a finite, reflection positive, and nontruncated fusion structure on an arbitrary Hopf algebra is trivial in the sense thatq-traces coincide with ordinary traces andq-dimensions coincide with ordinary dimensions. Thus, nontruncated fusion structures are ruled out to describe the fusion rules of quantum field theories with noninteger statistical dimensions and a finite number of superselection sectors.Work supported in part by DFG, SFB 288 Differentialgeometrie und Quantenphysik.     

Orbifolds,Hopf algebras,and the Moonshine

We describe the modular properties and fusion rules of holomorphic orbifold models by Hopf algebraic techniques, using the representation theory of the orbifold quantum group. We apply this theory to the construction of generalized Thompson series, and discuss its connections with Moonshine.