66 research outputs found
Note On Endomorphism Algebras Of Separable Monoidal Functors
We recall the Tannaka construction for certain types of split monoidal
functor into Vect_{k}, and remove the compactness restriction on the domain
Some Exceptional Cases in Mathematics: Euler Characteristic, Division Algebras, Cross Vector Product and Fano Matroid
We review remarkable results in several mathematical scenarios, including
graph theory, division algebras, cross product formalism and matroid theory.
Specifically, we mention the following subjects: (1) the Euler relation in
graph theory, and its higher-dimensional generalization, (2) the dimensional
theorem for division algebras and in particular the Hurwitz theorem for normed
division algebras, (3) the vector cross product dimensional possibilities, (4)
some theorems for graphs and matroids. Our main goal is to motivate a possible
research work in these four topics, putting special interest in their possible
links.Comment: 14 pages, Late
Aspects of (0,2) Orbifolds and Mirror Symmetry
We study orbifolds of (0,2) models and their relation to (0,2) mirror
symmetry. In the Landau-Ginzburg phase of a (0,2) model the superpotential
features a whole bunch of discrete symmetries, which by quotient action lead to
a variety of consistent (0,2) vacua. We study a few examples in very much
detail. Furthermore, we comment on the application of (0,2) mirror symmetry to
the calculation of Yukawa couplings in the space-time superpotential.Comment: 13 pages, TeX (harvmac, big) with 4 enclosed PostScript figures, one
reference adde
Spin and Statistics and First Principles
It was shown in the early Seventies that, in Local Quantum Theory (that is
the most general formulation of Quantum Field Theory, if we leave out only the
unknown scenario of Quantum Gravity) the notion of Statistics can be grounded
solely on the local observable quantities (without assuming neither the
commutation relations nor even the existence of unobservable charged field
operators); one finds that only the well known (para)statistics of Bose/Fermi
type are allowed by the key principle of local commutativity of observables. In
this frame it was possible to formulate and prove the Spin and Statistics
Theorem purely on the basis of First Principles.
In a subsequent stage it has been possible to prove the existence of a
unique, canonical algebra of local field operators obeying ordinary Bose/Fermi
commutation relations at spacelike separations. In this general guise the Spin
- Statistics Theorem applies to Theories (on the four dimensional Minkowski
space) where only massive particles with finite mass degeneracy can occur. Here
we describe the underlying simple basic ideas, and briefly mention the
subsequent generalisations; eventually we comment on the possible validity of
the Spin - Statistics Theorem in presence of massless particles, or of
violations of locality as expected in Quantum Gravity.Comment: Survey based on a talk given at the Meeting on "Theoretical and
experimental aspects of the spin - statistics connection and related
symmetries", Trieste, Italy - October 21-25, 200
A note on geometric duality in matroid theory and knot theory
We observe that for planar graphs, the geometric duality relation generates
both 2-isomorphism and abstract duality. This observation has the surprising
consequence that for links, the equivalence relation defined by isomorphisms of
checkerboard graphs is the same as the equivalence relation defined by
2-isomorphisms of checkerboard graphs.Comment: v1: 10 pages, 6 figures. v2: minor edits. v3: 11 pages, 6 figures.
final prepublication versio
On the super replication price of unbounded claims
In an incomplete market the price of a claim f in general cannot be uniquely
identified by no arbitrage arguments. However, the ``classical'' super
replication price is a sensible indicator of the (maximum selling) value of the
claim. When f satisfies certain pointwise conditions (e.g., f is bounded from
below), the super replication price is equal to sup_QE_Q[f], where Q varies on
the whole set of pricing measures. Unfortunately, this price is often too high:
a typical situation is here discussed in the examples. We thus define the less
expensive weak super replication price and we relax the requirements on f by
asking just for ``enough'' integrability conditions. By building up a proper
duality theory, we show its economic meaning and its relation with the
investor's preferences. Indeed, it turns out that the weak super replication
price of f coincides with sup_{Q\in M_{\Phi}}E_Q[f], where M_{\Phi} is the
class of pricing measures with finite generalized entropy (i.e., E[\Phi
(\frac{dQ}{dP})]<\infty) and where \Phi is the convex conjugate of the utility
function of the investor.Comment: Published at http://dx.doi.org/10.1214/105051604000000459 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
The C*-algebra of a Hilbert Bimodule
We regard a right Hilbert C*-module X over a C*-algebra A endowed with an
isometric *-homomorphism \phi: A\to L_A(X) as an object X_A of the C*-category
of right Hilbert A-modules. Following a construction by the first author and
Roberts, we associate to it a C*-algebra O_{X_A} containing X as a ``Hilbert
A-bimodule in O_{X_A}''. If X is full and finite projective O_{X_A} is the
C*-algebra C*(X), the generalization of the Cuntz-Krieger algebras introduced
by Pimsner. More generally, C*(X) is canonically embedded in O_{X_A} as the
C*-subalgebra generated by X. Conversely, if X is full, O_{X_A} is canonically
embedded in the bidual of C*(X). Moreover, regarding X as an object A_X_A of
the C*-category of Hilbert A-bimodules, we associate to it a C*-subalgebra
O_{A_X_A} of O_{X_A} commuting with A, on which X induces a canonical
endomorphism \rho. We discuss conditions under which A and O_{A_X_A} are the
relative commutant of each other and X is precisely the subspace of
intertwiners in O_{X_A} between the identity and \rho on O_{A_X_A}. We also
discuss conditions which imply the simplicity of C*(X) or of O_{X_A}; in
particular, if X is finite projective and full, C*(X) will be simple if A is
X-simple and the ``Connes spectrum'' of X is the circle.Comment: 22 pages, LaTe
A unified interpretation of several combinatorial dualities
AbstractSeveral combinatorial structures exhibit a duality relation that yields interesting theorems, and, sometimes, useful explanations or interpretations of results that do not concern duality explicitly. We present a common characterization of the duality relations associated with matroids, clutters (Sperner families), oriented matroids, and weakly oriented matroids. The same conditions characterize the orthogonality relation on certain families of vector spaces. This leads to a notion of abstract duality
- …