Functional form of unitary representations of the quantum "az+b" group

The formula for all unitary representations of the quantum "az+b" group for a real deformation parameter is given. The description involves the quantum exponential function introduced by Woronowicz

Quantum families of maps and quantum semigroups on finite quantum spaces

Quantum families of maps between quantum spaces are defined and studied. We prove that quantum semigroup (and sometimes quantum group) structures arise naturally on such objects out of more fundamental properties. As particular cases we study quantum semigroups of maps preserving a fixed state and quantum commutants of given quantum families of maps.Comment: update: last section generalized to non-tracial state

Gauge theories of quantum groups

We find two different q-generalizations of Yang-Mills theories. The corresponding lagrangians are invariant under the q-analogue of infinitesimal gauge transformations. We explicitly give the lagrangian and the transformation rules for the bicovariant q-deformation of $SU(2) \times U(1)$. The gauge potentials satisfy q-commutations, as one expects from the differential geometry of quantum groups. However, in one of the two schemes we present, the field strengths do commute.Comment: 12 pages, DFTT-19/9

Rieffel deformation via crossed products

We start from Rieffel data (A,f,X) where A is a C*-algebra, X is an action of an abelian group H on A and f is a 2-cocycle on the dual group. Using Landstad theory of crossed product we get a deformed C*-algebra A(f). In the case of H being the n-th Cartesian product of the real numbers we obtain a very simple proof of invariance of K-groups under the deformation. In the general case we also get a very simple proof that nuclearity is preserved under the deformation. We show how our approach leads to quantum groups and investigate their duality. The general theory is illustrated by an example of the deformation of SL(2,C). A description of it, in terms of noncommutative coordinates is given.Comment: 39 page

Symmetries of quantum spaces. Subgroups and quotient spaces of quantum SU(2)SU(2) and SO(3)SO(3) groups

We prove that each action of a compact matrix quantum group on a compact quantum space can be decomposed into irreducible representations of the group. We give the formula for the corresponding multiplicities in the case of the quotient quantum spaces. We describe the subgroups and the quotient spaces of quantum SU(2) and SO(3) groups.Comment: 30 pages (with very slight changes

Geometry of Quantum Principal Bundles I

A theory of principal bundles possessing quantum structure groups and classical base manifolds is presented. Structural analysis of such quantum principal bundles is performed. A differential calculus is constructed, combining differential forms on the base manifold with an appropriate differential calculus on the structure quantum group. Relations between the calculus on the group and the calculus on the bundle are investigated. A concept of (pseudo)tensoriality is formulated. The formalism of connections is developed. In particular, operators of horizontal projection, covariant derivative and curvature are constructed and analyzed. Generalizations of the first structure equation and of the Bianchi identity are found. Illustrative examples are presented.Comment: 64 pages, AMS-LaTeX, To appear in CM

R matrix and bicovariant calculus for the inhomogeneous quantum groups IGL_q(n)

We find the R matrix for the inhomogeneous quantum groups whose homogeneous part is $GL_q(n)$, or its restrictions to $SL_q(n)$,$U_q(n)$ and $SU_q(n)$. The quantum Yang-Baxter equation for R holds because of the Hecke relation for the braiding matrix of the homogeneous subgroup. A bicovariant differential calculus on $IGL_q(n)$ is constructed, and its application to the $D=4$ Poincar\'e group ISL_q(2,\Cb) is discussed.Comment: 8 pp., LaTeX, DFTT-59/9

Bicovariant Differential Calculus on the Quantum D=2 Poincare Group

We present a bicovariant differential calculus on the quantum Poincare group in two dimensions. Gravity theories on quantum groups are discussed.Comment: 17 page

The Lagrangian of q-Poincare' Gravity

The gauging of the q-Poincar\'e algebra of ref. hep-th 9312179 yields a non-commutative generalization of the Einstein-Cartan lagrangian. We prove its invariance under local q-Lorentz rotations and, up to a total derivative, under q-diffeomorphisms. The variations of the fields are given by their q-Lie derivative, in analogy with the q=1 case. The algebra of q-Lie derivatives is shown to close with field dependent structure functions. The equations of motion are found, generalizing the Einstein equations and the zero-torsion condition.Comment: 12 pp., LaTeX, DFTT-01/94 (extra blank lines introduced by mailer, corrupting LaTeX syntax, have been hopefully eliminated