2,896 research outputs found
Branching laws for Verma modules and applications in parabolic geometry. I
We initiate a new study of differential operators with symmetries and combine
this with the study of branching laws for Verma modules of reductive Lie
algebras. By the criterion for discretely decomposable and multiplicity-free
restrictions of generalized Verma modules [T. Kobayashi,
http://dx.doi.org/10.1007/s00031-012-9180-y {Transf. Groups (2012)}], we are
brought to natural settings of parabolic geometries for which there exist
unique equivariant differential operators to submanifolds. Then we apply a new
method (F-method) relying on the Fourier transform to find singular vectors in
generalized Verma modules, which significantly simplifies and generalizes many
preceding works. In certain cases, it also determines the Jordan--H\"older
series of the restriction for singular parameters. The F-method yields an
explicit formula of such unique operators, for example, giving an intrinsic and
new proof of Juhl's conformally invariant differential operators [Juhl,
http://dx.doi.org/10.1007/978-3-7643-9900-9 {Progr. Math. 2009}] and its
generalizations. This article is the first in the series, and the next ones
include their extension to curved cases together with more applications of the
F-method to various settings in parabolic geometries
Chiral Observables and Modular Invariants
Various definitions of chiral observables in a given Moebius covariant
two-dimensional theory are shown to be equivalent. Their representation theory
in the vacuum Hilbert space of the 2D theory is studied. It shares the general
characteristics of modular invariant partition functions, although SL(2,Z)
transformation properties are not assumed. First steps towards classification
are made.Comment: 28 pages, 1 figur
Quantum Principal Bundles and Corresponding Gauge Theories
A generalization of classical gauge theory is presented, in the framework of
a noncommutative-geometric formalism of quantum principal bundles over smooth
manifolds. Quantum counterparts of classical gauge bundles, and classical gauge
transformations, are introduced and investigated. A natural differential
calculus on quantum gauge bundles is constructed and analyzed. Kinematical and
dynamical properties of corresponding gauge theories are discussed.Comment: 28 pages, AMS-LaTe
The H-Covariant Strong Picard Groupoid
The notion of H-covariant strong Morita equivalence is introduced for
*-algebras over C = R(i) with an ordered ring R which are equipped with a
*-action of a Hopf *-algebra H. This defines a corresponding H-covariant strong
Picard groupoid which encodes the entire Morita theory. Dropping the positivity
conditions one obtains H-covariant *-Morita equivalence with its H-covariant
*-Picard groupoid. We discuss various groupoid morphisms between the
corresponding notions of the Picard groupoids. Moreover, we realize several
Morita invariants in this context as arising from actions of the H-covariant
strong Picard groupoid. Crossed products and their Morita theory are
investigated using a groupoid morphism from the H-covariant strong Picard
groupoid into the strong Picard groupoid of the crossed products.Comment: LaTeX 2e, 50 pages. Revised version with additional examples and
references. To appear in JPA
Homomorphisms between fuzzy information systems revisited
Recently, Wang et al. discussed the properties of fuzzy information systems
under homomorphisms in the paper [C. Wang, D. Chen, L. Zhu, Homomorphisms
between fuzzy information systems, Applied Mathematics Letters 22 (2009)
1045-1050], where homomorphisms are based upon the concepts of consistent
functions and fuzzy relation mappings. In this paper, we classify consistent
functions as predecessor-consistent and successor-consistent, and then proceed
to present more properties of consistent functions. In addition, we improve
some characterizations of fuzzy relation mappings provided by Wang et al.Comment: 10 page
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