297 research outputs found
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
From multiplicative unitaries to quantum groups II
It is shown that all important features of a -algebraic quantum
group defined by a modular multiplicative depend only on the
pair rather than the multiplicative unitary operator . The
proof is based on thorough study of representations of quantum groups. As an
application we present a construction and study properties of the universal
dual of a quantum group defined by a modular multiplicative unitary - without
assuming existence of Haar weights.Comment: 19 pages, LaTe
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
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 -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
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