79 research outputs found
Differential Calculus on Quantum Spaces and Quantum Groups
A review of recent developments in the quantum differential calculus. The
quantum group is treated by considering it as a particular quantum
space. Functions on are defined as a subclass of functions on
. The case of is also briefly considered. These notes cover
part of a lecture given at the XIX International Conference on Group Theoretic
Methods in Physics, Salamanca, Spain 1992.Comment: 23 page
Cartan Calculus on Quantum Lie Algebras
A generalization of the differential geometry of forms and vector fields to
the case of quantum Lie algebras is given. In an abstract formulation that
incorporates many existing examples of differential geometry on quantum spaces
we combine an exterior derivative, inner derivations, Lie derivatives, forms
and functions all into one big algebra, the ``Cartan Calculus''. (This is an
extended version of a talk presented by P. Schupp at the XXII
International Conference on Differential Geometric Methods in Theoretical
Physics, Ixtapa, Mexico, September 1993)Comment: 15 pages in LaTeX, LBL-34833 and UCB-PTH-93/3
Realization of Vector fields for Quantum Groups as Pseudodifferential Operators on Quantum Spaces
The vector fields of the quantum Lie algebra are described for the quantum
groups and as pseudodifferential operators on the
linear quantum spaces covariant under the corresponding quantum group. Their
expressions are simple and compact. It is pointed out that these vector fields
satisfy certain characteristic polynomial identities. The real forms
and are discussed in detail.Comment: 16 pages, Latex, no figures, to appear in the Proceedings of the XX
International Conference on Group Theory Methods in Physics, Toyonaka, Japan
(1994
Duality Rotations in Nonlinear Electrodynamics and in Extended Supergravity
We review the general theory of duality rotations which, in four dimensions,
exchange electric with magnetic fields. Necessary and sufficient conditions in
order for a theory to have duality symmetry are established. A nontrivial
example is Born-Infeld theory with n abelian gauge fields and with Sp(2n,R)
self-duality. We then review duality symmetry in supergravity theories. In the
case of N=2 supergravity duality rotations are in general not a symmetry of the
theory but a key ingredient in order to formulate the theory itself. This is
due to the beautiful relation between the geometry of special Kaehler manifolds
and duality rotations.Comment: Invited contribution to Rivista del Nuovo Cimento in occasion of the
2005 Enrico Fermi Prize of the Italian Physical Society. 96 pages, corrected
typo
Cartan Calculus for Hopf Algebras and Quantum Groups
A generalization of the differential geometry of forms and vector fields to
the case of quantum Lie algebras is given. In an abstract formulation that
incorporates many existing examples of differential geometry on quantum groups,
we combine an exterior derivative, inner derivations, Lie derivatives, forms
and functions all into one big algebra. In particular we find a generalized
Cartan identity that holds on the whole quantum universal enveloping algebra of
the left-invariant vector fields and implicit commutation relations for a
left-invariant basis of 1-forms.Comment: 15 pages (submitted to Comm. Math. Phys.
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