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

Differential calculus on the quantum Heisenberg group

The differential calculus on the quantum Heisenberg group is conlinebreak structed. The duality between quantum Heisenberg group and algebra is proved.Comment: AMSTeX, Pages

On the structure of inhomogeneous quantum groups

We investigate inhomogeneous quantum groups G built from a quantum group H and translations. The corresponding commutation relations contain inhomogeneous terms. Under certain conditions (which are satisfied in our study of quantum Poincare groups [12]) we prove that our construction has correct `size', find the R-matrices and the analogues of Minkowski space for G.Comment: LaTeX file, 47 pages, existence of invertible coinverse assumed, will appear in Commun. Math. Phy

Landstad-Vaes theory for locally compact quantum groups

Landstad-Vaes theory deals with the structure of the crossed product of a C$^*$-algebra by an action of locally compact (quantum) group. In particular it describes the position of original algebra inside crossed product. The problem was solved in 1979 by Landstad for locally compact groups and in 2005 by Vaes for regular locally compact quantum groups. To extend the result to non-regular groups we modify the notion of $G$-dynamical system introducing the concept of weak action of quantum groups on C$^*$-algebras. It is still possible to define crossed product (by weak action) and characterise the position of original algebra inside the crossed product. The crossed product is unique up to an isomorphism. At the end we discuss a few applications

Representations of SU(1,1) in Non-commutative Space Generated by the Heisenberg Algebra

SU(1,1) is considered as the automorphism group of the Heisenberg algebra H. The basis in the Hilbert space K of functions on H on which the irreducible representations of the group are realized is explicitly constructed. The addition theorems are derived.Comment: Latex, 8 page

Quantum Semigroups from Synchronous Games

We show that the C*-algebras associated with synchronous games give rise to certain quantum families of maps between the input and output sets of the game. In particular situations (e.g. for graph endomorphism games) these quantum families have a natural quantum semigroup structure and if the condition of preservation of a natural state is added, they are in fact compact quantum groups.Comment: 9 page

CQG algebras: a direct algebraic approach to compact quantum groups

The purely algebraic notion of CQG algebra (algebra of functions on a compact quantum group) is defined. In a straightforward algebraic manner, the Peter-Weyl theorem for CQG algebras and the existence of a unique positive definite Haar functional on any CQG algebra are established. It is shown that a CQG algebra can be naturally completed to a $C^\ast$-algebra. The relations between our approach and several other approaches to compact quantum groups are discussed.Comment: 14 pp., Plain TeX, accepted by Lett. Math. Phy