1,998 research outputs found
The conjugacy problem and related problems in lattice-ordered groups
We study, from a constructive computational point of view, the techniques
used to solve the conjugacy problem in the "generic" lattice-ordered group
Aut(R) of order automorphisms of the real line. We use these techniques in
order to show that for each choice of parameters f,g in Aut(R), the equation
xfx=g is effectively solvable in Aut(R).Comment: Small update
Local models of Shimura varieties, I. Geometry and combinatorics
We survey the theory of local models of Shimura varieties. In particular, we
discuss their definition and illustrate it by examples. We give an overview of
the results on their geometry and combinatorics obtained in the last 15 years.
We also exhibit their connections to other classes of algebraic varieties such
as nilpotent orbit closures, affine Schubert varieties, quiver Grassmannians
and wonderful completions of symmetric spaces.Comment: 86 pages, small corrections and improvements, to appear in the
"Handbook of Moduli
Group actions on 1-manifolds: a list of very concrete open questions
This text focuses on actions on 1-manifolds. We present a (non exhaustive)
list of very concrete open questions in the field, each of which is discussed
in some detail and complemented with a large list of references, so that a
clear panorama on the subject arises from the lecture.Comment: 21 pages, 2 figure
Kitaev's quantum double model from a local quantum physics point of view
A prominent example of a topologically ordered system is Kitaev's quantum
double model for finite groups (which in particular
includes , the toric code). We will look at these models from
the point of view of local quantum physics. In particular, we will review how
in the abelian case, one can do a Doplicher-Haag-Roberts analysis to study the
different superselection sectors of the model. In this way one finds that the
charges are in one-to-one correspondence with the representations of
, and that they are in fact anyons. Interchanging two of such
anyons gives a non-trivial phase, not just a possible sign change. The case of
non-abelian groups is more complicated. We outline how one could use
amplimorphisms, that is, morphisms to study the superselection
structure in that case. Finally, we give a brief overview of applications of
topologically ordered systems to the field of quantum computation.Comment: Chapter contributed to R. Brunetti, C. Dappiaggi, K. Fredenhagen, J.
Yngvason (eds), Advances in Algebraic Quantum Field Theory (Springer 2015).
Mainly revie
Conjugacy in Garside groups I: Cyclings, powers, and rigidity
In this paper a relation between iterated cyclings and iterated powers of
elements in a Garside group is shown. This yields a characterization of
elements in a Garside group having a rigid power, where 'rigid' means that the
left normal form changes only in the obvious way under cycling and decycling.
It is also shown that, given X in a Garside group, if some power X^m is
conjugate to a rigid element, then m can be bounded above by ||\Delta||^3. In
the particular case of braid groups, this implies that a pseudo-Anosov braid
has a small power whose ultra summit set consists of rigid elements. This
solves one of the problems in the way of a polynomial solution to the conjugacy
decision problem (CDP) and the conjugacy search problem (CSP) in braid groups.
In addition to proving the rigidity theorem, it will be shown how this paper
fits into the authors' program for finding a polynomial algorithm to the
CDP/CSP, and what remains to be done.Comment: 41 page
Towards a dual spin network basis for (3+1)d lattice gauge theories and topological phases
Using a recent strategy to encode the space of flat connections on a
three-manifold with string-like defects into the space of flat connections on a
so-called 2d Heegaard surface, we propose a novel way to define gauge invariant
bases for (3+1)d lattice gauge theories and gauge models of topological phases.
In particular, this method reconstructs the spin network basis and yields a
novel dual spin network basis. While the spin network basis allows to interpret
states in terms of electric excitations, on top of a vacuum sharply peaked on a
vanishing electric field, the dual spin network basis describes magnetic (or
curvature) excitations, on top of a vacuum sharply peaked on a vanishing
magnetic field (or flat connection). This technique is also applicable for
manifolds with boundaries. We distinguish in particular a dual pair of boundary
conditions, namely of electric type and of magnetic type. This can be used to
consider a generalization of Ocneanu's tube algebra in order to reveal the
algebraic structure of the excitations associated with certain 3d manifolds.Comment: 45 page
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