1,771 research outputs found
Quantum Principal Bundles
A noncommutative-geometric generalization of the theory of principal bundles
is sketched. A differential calculus over corresponding quantum principal
bundles is analysed. The formalism of connections is presented. In particular,
operators of covariant derivative and horizontal projection are described and
analysed. Quantum counterparts for the Bianchi identity and the Weil's
homomorphism are found. Illustrative examples are considered. (Lecture
presented at the XXII-th Conference on Differential Geometric Methods in
Theoretical Physics, Ixtapa-Zihuatanejo, Mexico, September 1993).Comment: 10 pages, LaTe
Quantum Gauge Transformations and Braided Structure on Quantum Principal Bundles
It is shown that every quantum principal bundle is braided, in the sense that
there exists an intrinsic braid operator twisting the functions on the bundle.
A detailed algebraic analysis of this operator is performed. In particular, it
turns out that the braiding admits a natural extension to the level of
arbitrary differential forms on the bundle. Applications of the formalism to
the study of quantum gauge transformations are presented.Comment: 20 pages, AMS-LaTe
Characteristic Classes of Quantum Principal Bundles
A noncommutative-geometric generalization of classical Weil theory of
characteristic classes is presented, in the conceptual framework of quantum
principal bundles. A particular care is given to the case when the bundle does
not admit regular connections. A cohomological description of the domain of the
Weil homomorphism is given. Relations between universal characteristic classes
for the regular and the general case are analyzed. In analogy with classical
geometry, a natural spectral sequence is introduced and investigated. The
appropriate counterpart of the Chern character is constructed, for structures
admitting regular connections. Illustrative examples and constructions are
presented.Comment: 48 pages, AMS-LaTe
On Differential Structures on Quantum Principal Bundles
A constructive approach to differential calculus on quantum principal bundles
is presented. The calculus on the bundle is built in an intrinsic manner,
starting from given graded (differential) *-algebras representing horizontal
forms on the bundle and differential forms on the base manifold, together with
a family of antiderivations acting on horizontal forms, playing the role of
covariant derivatives of regular connections. In this conceptual framework, a
natural differential calculus on the structure quantum group is described.Comment: 15 pages (AMS-LaTeX
Quantum Classifying Spaces and Universal Quantum Characteristic Classes
A construction of the noncommutative-geometric counterparts of classical
classifying spaces is presented, for general compact matrix quantum structure
groups. A quantum analogue of the classical concept of the classifying map is
introduced and analyzed. Interrelations with the abstract algebraic theory of
quantum characteristic classes are discussed. Various non-equivalent approaches
to defining universal characteristic classes are outlined.Comment: 12 pages, AMS-LaTeX, Lectures, Quantum Groups and Quantum Spaces
Minisemester, Banach Center, Warsaw, Poland, November 199
Classical Spinor Structures on Quantum Spaces
A noncommutative-geometric generalization of the classical concept of spinor
structure is presented. This is done in the framework of the formalism of
quantum principal bundles. In particular, analogs of the Dirac operator and the
Laplacian are introduced and analyzed. A general construction of examples of
quantum spaces with a spinor structure is presented.Comment: 14 pages (AMS-LaTeX
On Braided Quantum Groups
A braided generalization of the concept of Hopf algebra (quantum group) is
presented. The generalization overcomes an inherent geometrical inhomogeneity
of quantum groups, in the sense of allowing completely pointless objects. All
braid-type equations appear as a consequence of initial axioms. Braided
counterparts of basic algebraic relations between fundamental entities of the
standard theory are found.Comment: 14 pages, (AMS-LaTeX
Braided Clifford Algebras as Braided Quantum Groups
The paper deals with braided Clifford algebras, understood as
Chevalley-Kahler deformations of braided exterior algebras. It is shown that
Clifford algebras based on involutive braids can be naturally endowed with a
braided quantum group structure. Basic group entities are constructed
explicitly.Comment: 10 pages, AMS-LaTe
General Frame Structures On Quantum Principal Bundles
A noncommutative-geometric generalization of the classical formalism of frame
bundles is developed, incorporating into the theory of quantum principal
bundles the concept of the Levi-Civita connection. The construction of a
natural differential calculus on quantum principal frame bundles is presented,
including the construction of the associated differential calculus on the
structure group. General torsion operators are defined and analyzed.
Illustrative examples are presented.Comment: 16 pages, AMS-LaTeX, extended versio
First-Order Differential Calculi Over Multi-Braided Quantum Groups
A differential calculus of the first order over multi-braided quantum groups
is developed. In analogy with the standard theory, left/right-covariant and
bicovariant differential structures are introduced and investigated.
Furthermore, antipodally covariant calculi are studied. The concept of the
*-structure on a multi-braided quantum group is formulated, and in particular
the structure of left-covariant *-covariant calculi is analyzed. A special
attention is given to differential calculi covariant with respect to the action
of the associated braid system. In particular it is shown that the left/right
braided-covariance appears as a consequence of the left/right-covariance
relative to the group action. Braided counterparts of all basic results of the
standard theory are found.Comment: 32 pages, AMS-LaTeX/1, this is the revised version of an unpublished
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