703 research outputs found
Twisted geometries: A geometric parametrisation of SU(2) phase space
A cornerstone of the loop quantum gravity program is the fact that the phase
space of general relativity on a fixed graph can be described by a product of
SU(2) cotangent bundles per edge. In this paper we show how to parametrize this
phase space in terms of quantities describing the intrinsic and extrinsic
geometry of the triangulation dual to the graph. These are defined by the
assignment to each triangle of its area, the two unit normals as seen from the
two polyhedra sharing it, and an additional angle related to the extrinsic
curvature. These quantities do not define a Regge geometry, since they include
extrinsic data, but a looser notion of discrete geometry which is twisted in
the sense that it is locally well-defined, but the local patches lack a
consistent gluing among each other. We give the Poisson brackets among the new
variables, and exhibit a symplectomorphism which maps them into the Poisson
brackets of loop gravity. The new parametrization has the advantage of a simple
description of the gauge-invariant reduced phase space, which is given by a
product of phase spaces associated to edges and vertices, and it also provides
an abelianisation of the SU(2) connection. The results are relevant for the
construction of coherent states, and as a byproduct, contribute to clarify the
connection between loop gravity and its subset corresponding to Regge
geometries.Comment: 28 pages. v2 and v3 minor change
Bicrossed products with the Taft algebra
Let be a group which admits a generating set consisting of finite order
elements. We prove that any Hopf algebra which factorizes through the Taft
algebra and the group Hopf algebra (equivalently, any bicrossed product
between the aforementioned Hopf algebras) is isomorphic to a smash product
between the same two Hopf algebras. The classification of these smash products
is shown to be strongly linked to the problem of describing the group
automorphisms of . As an application, we completely describe by generators
and relations and classify all bicrossed products between the Taft algebra and
the group Hopf algebra , where denotes the dihedral group.Comment: Continues arXiv:1205.6110, arXiv:1205.6564, arXiv:1603.01854,
arXiv:1611.05674; restates preliminaries and definitions for sake of clarit
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