7,959 research outputs found
On Complex Manifolds and Observable Schemes
We work out the construction of a Stein manifold from a commutative
Arens-Michael algebra, under assumptions that are mild enough for the process
to be useful in practice. Then, we do the passage to arbitrary complex
manifolds by proposing a suitable notion of scheme. We do this in the abstract
language of spectral functors, in view of its potential usefulness in
non-commutative geometry.Comment: 15 pages. Supported by Fondecyt Postdoctoral Grant No. 311004
On the geometry underlying a real Lie algebra representation
Let be a real Lie group with Lie algebra . Given a unitary
representation of , one obtains by differentiation a representation
of by unbounded, skew-adjoint operators. Representations
of admitting such a description are called \emph{integrable,} and
they can be geometrically seen as the action of by derivations on
the algebra of representative functions , which are
naturally defined on the homogeneous space . In other words,
integrable representations of a real Lie algebra can always be seen as
realizations of that algebra by vector fields on a homogeneous manifold. Here
we show how to use the coproduct of the universal enveloping algebra of
to generalize this to representations which are not necessarily
integrable. The geometry now playing the role of is a locally homogeneous
space. This provides the basis for a geometric approach to integrability
questions regarding Lie algebra representations.Comment: 12 pages. Author supported by Fondecyt Postdoctoral Grant N{\deg}
311004
Pre-freezing of multifractal exponents in Random Energy Models with logarithmically correlated potential
Boltzmann-Gibbs measures generated by logarithmically correlated random
potentials are multifractal. We investigate the abrupt change ("pre-freezing")
of multifractality exponents extracted from the averaged moments of the measure
- the so-called inverse participation ratios. The pre-freezing can be
identified with termination of the disorder-averaged multifractality spectrum.
Naive replica limit employed to study a one-dimensional variant of the model is
shown to break down at the pre-freezing point. Further insights are possible
when employing zero-dimensional and infinite-dimensional versions of the
problem. In particular, the latter version allows one to identify the pattern
of the replica symmetry breaking responsible for the pre-freezing phenomenon.Comment: This is published version, 11 pages, 1 figur
- …