1,273 research outputs found
Hopf algebras and subfactors associated to vertex models
If H is a Hopf algebra whose square of the antipode is the identity, v\in\l
(V)\otimes H is a corepresentation, and \pi :H\to\l (W) is a representation,
then satisfies the equation of the vertex models for subfactors. A universal construction shows
that any solution of this equatio n arises in this way. A more elaborate
construction shows that there exists a ``minimal'' triple
satisfying . This paper is devoted to the study of this
latter construction of Hopf algebras. If is unitary we construct a
\c^*-norm on and we find a new description of the standard invariant of
the subfactor associated to . We discuss also the ``twisted'' (i.e. ) case.Comment: 25 pages, Late
Construction of Curtis-Phan-Tits system in black box classical groups
We present a polynomial time Monte-Carlo algorithm for finite simple black
box classical groups of odd characteristic which constructs all root
-subgroups associated with the nodes of the extended Dynkin
diagram of the corresponding algebraic group.Comment: 35 page
Automorphisms of the bipartite graph planar algebra
For any abstract subfactor planar algebra , there exists a finite index
extremal subfactor with as its standard invariant. In
this paper, we classify the automorphism group of a bipartite graph planar
algebra, and obtain subfactor planar subalgebras by taking fixed points under
groups of automorphisms. This construction provides both new examples of
subfactors and new descriptions of the planar algebras of previously known
examples.Comment: 22 pp
The non-commuting, non-generating graph of a finite simple group
Let be a group such that is finite and simple. The
non-commuting, non-generating graph of has vertex set , with edges corresponding to pairs of elements that do not commute and do
not generate . We show that is connected with diameter at most ,
with smaller upper bounds for certain families of groups. When itself is
simple, we prove that the diameter of the complement of the generating graph of
has a tight upper bound of . In the companion paper arXiv:2211.08869, we
consider when is not simple.Comment: 20 page
Quantitative K-Theory Related to Spin Chern Numbers
We examine the various indices defined on pairs of almost commuting unitary
matrices that can detect pairs that are far from commuting pairs. We do this in
two symmetry classes, that of general unitary matrices and that of self-dual
matrices, with an emphasis on quantitative results. We determine which values
of the norm of the commutator guarantee that the indices are defined, where
they are equal, and what quantitative results on the distance to a pair with a
different index are possible. We validate a method of computing spin Chern
numbers that was developed with Hastings and only conjectured to be correct.
Specifically, the Pfaffian-Bott index can be computed by the "log method" for
commutator norms up to a specific constant
-algebras from planar algebras I: canonical -algebras associated to a planar algebra
From a planar algebra, we give a functorial construction to produce numerous
associated -algebras. Our main construction is a Hilbert -bimodule
with a canonical real subspace which produces Pimsner-Toeplitz, Cuntz-Pimsner,
and generalized free semicircular -algebras. By compressing this system,
we obtain various canonical -algebras, including Doplicher-Roberts
algebras, Guionnet-Jones-Shlyakhtenko algebras, universal
(Toeplitz-)Cuntz-Krieger algebras, and the newly introduced free graph
algebras. This is the first article in a series studying canonical
-algebras associated to a planar algebra.Comment: 47 pages, many figure
Isotropic quantum walks on lattices and the Weyl equation
We present a thorough classification of the isotropic quantum walks on
lattices of dimension for cell dimension . For there exist
two isotropic walks, namely the Weyl quantum walks presented in Ref. [G. M.
D'Ariano and P. Perinotti, Phys. Rev. A 90, 062106 (2014)], resulting in the
derivation of the Weyl equation from informational principles. The present
analysis, via a crucial use of isotropy, is significantly shorter and avoids a
superfluous technical assumption, making the result completely general.Comment: 16 pages, 1 figur
- …