4,157 research outputs found
Holomorphic deformation of Hopf algebras and applications to quantum groups
In this article we propose a new and so-called holomorphic deformation scheme
for locally convex algebras and Hopf algebras. Essentially we regard converging
power series expansion of a deformed product on a locally convex algebra, thus
giving the means to actually insert complex values for the deformation
parameter. Moreover we establish a topological duality theory for locally
convex Hopf algebras. Examples coming from the theory of quantum groups are
reconsidered within our holomorphic deformation scheme and topological duality
theory. It is shown that all the standard quantum groups comprise holomorphic
deformations. Furthermore we show that quantizing the function algebra of a
(Poisson) Lie group and quantizing its universal enveloping algebra are
topologically dual procedures indeed. Thus holomorphic deformation theory seems
to be the appropriate language in which to describe quantum groups as deformed
Lie groups or Lie algebras.Comment: 40 page
A Gravity Theory on Noncommutative Spaces
A deformation of the algebra of diffeomorphisms is constructed for
canonically deformed spaces with constant deformation parameter theta. The
algebraic relations remain the same, whereas the comultiplication rule (Leibniz
rule) is different from the undeformed one. Based on this deformed algebra a
covariant tensor calculus is constructed and all the concepts like metric,
covariant derivatives, curvature and torsion can be defined on the deformed
space as well. The construction of these geometric quantities is presented in
detail. This leads to an action invariant under the deformed diffeomorphism
algebra and can be interpreted as a theta-deformed Einstein-Hilbert action. The
metric or the vierbein field will be the dynamical variable as they are in the
undeformed theory. The action and all relevant quantities are expanded up to
second order in theta.Comment: 28 pages, v2: coefficient in equ. (10.15) corrected, references
added, v3: references added, published versio
Mathematical aspects of scattering amplitudes
In these lectures we discuss some of the mathematical structures that appear
when computing multi-loop Feynman integrals. We focus on a specific class of
special functions, the so-called multiple polylogarithms, and discuss introduce
their Hopf algebra structure. We show how these mathematical concepts are
useful in physics by illustrating on several examples how these algebraic
structures are useful to perform analytic computations of loop integrals, in
particular to derive functional equations among polylogarithms.Comment: 58 pages. Lectures presented at TASI 201
Differential calculus and gauge transformations on a deformed space
Deformed gauge transformations on deformed coordinate spaces are considered
for any Lie algebra. The representation theory of this gauge group forces us to
work in a deformed Lie algebra as well. This deformation rests on a twisted
Hopf algebra, thus we can represent a twisted Hopf algebra on deformed spaces.
That leads to the construction of Lagrangian invariant under a twisted Lie
algebra.Comment: 14 pages, to appear in General Relativity and Gravitation Journal,
Obregon's Festschrift 2006, V2: misprints correcte
q-Algebras and Arrangements of Hyperplanes
Varchenko's approach to quantum groups, from the theory of arrangements of
hyperplanes, can be usefully applied to q-algebras in general, of which quantum
groups and quantum (super) Kac-Moody algebras are special cases. New results
are obtained on the classification of q-algebras, and of the Serre ideals of
generalized quantum (super) Kac-Moody algebras.Comment: 21 pages, TeX file, third expanded versio
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