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.     

Bicovariant differential geometry of the quantum group SL h (2)

There are only two quantum group structures on the space of two by two unimodular matrices, these are the SL _q (2) and the SL _h (2) quantum groups. The differential geometry of SL _q (2) is well known. In this Letter, we develop the differential geometry of SL _h (2), and show that the space of left invariant vector fields is three-dimensional.     

A Pseudo-cocycle for the Comultiplication on the Quantum <Emphasis Type="Italic">SU</Emphasis>(<Emphasis Type="Italic">2</Emphasis>) Group

We show that the comultiplication on the quantum group SU _q (2) may be obtained from that on the quantum semigroup SU ₀(2) by twisting with a unitary 2-pseudo-cocycle. Work supported by the ARC Linkage International Fellowship LX0667294, and by the Korea Research Foundation Grant (KRF-2004-041-C00024).     

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.     

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.     

Classification of three-dimensional covariant differential calculi on Podles' quantum spheres and on related spaces

Covariant differential calculi on the quantum space for the quantum group SL _q (2) are classified. Our main assumptions are thatq is not a root of unity and that the differentials de _j of the generators of form a free right module basis for the first-order forms. Our result says, in particular, that apart from the two casesc =c(3), there exists a unique differential calculus with the above properties on the space which corresponds to Podles' quantum sphereS _qc ^/2 .     

Metric on quantum spaces

We introduce the analogue of the metric tensor in the case ofq-deformed differential calculus. We analyse the consequences of the existence of the metric, showing that this enforces severe restrictions on the parameters of the theory. We discuss in detail examples of the Manin plane and theq-deformation of SU(2). Finally we touch the topic of relations with Connes' approach.Partially supported by KBN grant 2P 302 168 4.     

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.     

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.     

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.     

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.     

Noncommutative differential geometry with higher-order derivatives

We build a toy model of differential geometry on the real line, which includes derivatives of the second order. Such construction is possible only within the framework of noncommutative geometry. We introduce the metric and briefly discuss two simple physical models of scalar field theory and gauge theory in this geometry.Partially supported by KBN grant 2 P302 168 4.     

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 borel-weil-bott approach to representations of sl q (2, ℂ)

We use a quite concrete and simple realization of sl _q (2, ) involving finite difference operators. We interpret them as derivations (in the noncommutative sense) on a suitable graded algebra, which gives rise to the noncommutative scheme ¹ II ^1* as the counterpart of the standard ¹ = Sl(2, )/B.     

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.     

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.     

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.     

A counterexample on idempotent states on a compact quantum group

A simple example is given to illustrate that an idempotent state may not be the Haar state of any subgroup in the case of compact quantum groups.     

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.