Quantum double and differential calculi

We show that bicovariant bimodules as defined by Woronowicz are in one-to-one correspondence with the Drinfeld quantum double representations. We then prove that a differential calculus associated to a bicovariant bimodule of dimension n is connected to the existence of a particular (n+1)-dimensional representation of the double. An example of bicovariant differential calculus on the nonquasitriangular quantum group E _q (2) is developed. The construction is studied in terms of Hochschild cohomology and a correspondence between differential calculi and 1-cocycles is proved. Some differences of calculi on quantum and finite groups with respect to Lie groups are stressed.     

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.     

Differential calculus on quantum Euclidean spheres

We study covariant differential calculus on the quantum Euclidean spheres S _q ^N−1 which are quantum homogeneous spaces with coactions of the quantum groups O _q (N). First order differential calculi on the quantum Euclidean spheres satisfying a dimension constraint are found and classified: ForN≥6, there exist exactly two such calculi one of which is closely related to the classical differential calculus in the commutative case. Higher order differential forms and symmetry are discussed. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000.     

Differential calculus on quantized simple lie groups

Differential calculi, generalizations of Woronowicz's four-dimensional calculus on SU _q (2), are introduced for quantized classical simple Lie groups in a constructive way. For this purpose, the approach of Faddeev and his collaborators to quantum groups was used. An equivalence of Woronowicz's enveloping algebra generated by the dual space to the left-invariant differential forms and the corresponding quantized universal enveloping algebra, is obtained for our differential calculi. Real forms for q are also discussed.     

Left-covariant differential calculi on SLq(2) and SLq(3)

We study (N²−1)-dimensional left-covariant differential calculi on the quantum group SL_q(N) for which the generators of the quantum Lie algebras annihilate the quantum trace. In this way we obtain one distinguished calculus on SL_q(2) (which corresponds to Woronowicz' 3D-calculus on SU_q(2)) and two distinguished calculi on SL_q(3) such that the higher-order calculi give the ordinary differential calculus on SL(2) and SL(3), respectively, in the limit q → 1. Two new differential calculi on SL_q(3) are introduced and developed in detail.     

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.     

Riemannian Geometry of Quantum Groups¶and Finite Groups with Nonuniversal Differentials

We construct noncommutative “Riemannian manifold” structures on dual quasitriangular Hopf algebras such as ℂ _q [SU ₂] with its standard bicovariant differential calculus, using the quantum frame bundle approach introduced previously. The metric is provided by the braided-Killing form on the braided-Lie algebra on the tangent space and the n-bein by the Maurer–Cartan form. We also apply the theory to finite sets and in particular to finite group function algebras ℂ[G] with differential calculi and Killing forms determined by a conjugacy class. The case of the permutation group ℂ[S ₃] is worked out in full detail and a unique torsion free and cotorsion free or “Levi–Civita” connection is obtained with noncommutative Ricci curvature essentially proportional to the metric (an Einstein space). We also construct Dirac operators in the metric background, including on finite groups such as S ₃. In the process we clarify the construction of connections from gauge fields with nonuniversal calculi on quantum principal bundles of tensor product form. Received: 22 June 2000 / Accepted: 26 August 2001     

Complex quantum group,dual algebra and bicovariant differential calculus

The method used to construct the bicovariant bimodule in ref. [CSWW] is applied to examine the structure of the dual algebra and the bicovariant differential calculus of the complex quantum group. The complex quantum group Fun _q (SL(N, C)) is defined by requiring that it contains Fun _q (SU(N)) as a subalgebra analogously to the quantum Lorentz group. Analyzing the properties of the fundamental bimodule, we show that the dual algebra has the structure of the twisted product Fun _q (SU(N))Fun _q (SU(N)) _reg ^* . Then the bicovariant differential calculi on the complex quantum group are constructed.     

Inhomogeneous quantum groups and their quantized universal enveloping algebras

The quantum group IGL _q (N), the inhomogenization of GL _q (N), is formulated with -matrices. Theq-deformed universal enveloping algebra is constructed as the algebra of regular functionals in this formulation and contains the partial derivatives of the covariant differential calculus on the quantum space.     

Exterior algebras related to the quantum group O(O q(3))

For the 9-dimensional bicovariant differential calculi on the quantum group O(O _q(3)) several kinds of exterior algebras are examined. The corresponding dimensions, bicovariant subbimodules and eigenvalues of the antisymmetrizer are given. Exactly one of the exterior algebras studied by the authors has a unique left invariant form with maximal degree.     

Differential calculus on quantum projective spaces

Covariant first order differential calculus on the quantum projective spaces CP _q ^N-1 is studied by two approaches. First, the embedding of CP _q ^N-1 into the quantum spheres S _q ^2N-1 is used to obtain differential calculi on CP _q ^N-1 by restriction; second, classification results for differential calculi on CP _q ^N-1 under three different constraint settings are proved directly. The main results are that under each of the constraints considered, there exists a differential calculus which is uniquely determined if N 6, and that (essentially) all of the differential *-calculi on S _q ^2N-1 known from a previous classification paper admit restriction to CP _q ^N-1 .     

Differential calculi on the quantum superplane

Using some natural conditions less restrictive than theGL _ql/s(m/n) invariance, we present two possible multiparametric differential calculi on the quantum superplane. We show that there exists a new differential calculus which is different from the known one, generalizing the Wess-Zumino formalism to the superspace case. We discuss some^*-algebra structures leaving invariant this differential calculus. The (1 + 1)-dimensional case is analyzed and a realization of the super-Virasoro algebra on this particular quantum superspace is given.     

Levi-Civita Connections on the Quantum Groups SLq(N), Oq(N) and Spq(N)

For bicovariant differential calculi on quantum groups various notions on connections and metrics (bicovariant connections, invariant metrics, the compatibility of a connection with a metric, Levi-Civita connections) are introduced and studied. It is proved that for the bicovariant differential calculi on SL _q (N), O _q (N) and Sp _q (N) from the classification in [18] there exist unique Levi-Civita connections. Received: Received: 28 February 1996 / Accepted: 1 October 1996     

Covering and gluing of algebras and differential algebras

Extending work of Budzyński and Kondracki, we investigate coverings and gluings of algebras and differential algebras. We describe in detail the gluing of two quantum discs along their classical subspace, giving a C^*-algebra isomorphic to a certain Podleś sphere, as well as the gluing of U_q^1/2(sl₂)-covariant differential calculi on the discs.     

The problem of differential calculus on quantum groups

The bicovariant differential calculi on quantum groups of Woronowicz have the drawback that their dimensions do not agree with that of the corresponding classical calculus. In this paper we discuss the first-order differential calculus which arises from a simple quantum Lie algebra l _h (g) This calculus has the correct dimension and is shown to be bicovariant and complete. But it doesnot satisfy the Leibniz rule. Forsl _n this approach leads to a differential calculus which satisfies a simple generalization of the Leibniz rule.Presented at the 5th International Colloquium on Quantum Groups: Quantum Groups and Integrable Systems, Prague, 20–22 June 1996.     

Geometrical Tools for Quantum Euclidean Spaces

We apply one of the formalisms of noncommutative geometry to ℝ ^N _q , the quantum space covariant under the quantum group SO _q (N). Over ℝ ^N _q there are two SO _q (N)-covariant differential calculi. For each we find a frame, a metric and two torsion-free covariant derivatives which are metric compatible up to a conformal factor and which have a vanishing linear curvature. This generalizes results found in a previous article for the case of ℝ³ _q . As in the case N=3, one has to slightly enlarge the algebra ℝ ^N _q ; for N odd one needs only one new generator whereas for N even one needs two. As in the particular case N=3 there is a conformal ambiguity in the natural metrics on the differential calculi over ℝ ^N _q . While in our previous article the frame was found “by hand”, here we disclose the crucial role of the quantum group covariance and exploit it in the construction. As an intermediate step, we find a homomorphism from the cross product of ℝ ^N _q with U _q so(N) into ℝ ^N _q , an interesting result in itself. Received: 4 March 2000 / Accepted: 11 October 2000     

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.     

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).     

Differential calculus on the quantum superspace and deformation of phase space

We investigate non-commutative differential calculus on the supersymmetric version of quantum space in which quantum supergroups are realized. Multiparametric quantum deformation of the general linear super-group,GL _q(m|n), is studied and the explicit form for the \(\hat R - matrix\) is presented. We apply these results to the quantum phase-space construction ofOSp _q(2n|2m) and calculate their \(\hat R - matrices\) .