92 research outputs found
Neveu-Schwarz and operators algebras I: Vertex operators superalgebras
This paper is the first of a series giving a self-contained way from the
Neveu-Schwarz algebra to a new series of irreducible subfactors. Here we
present an elementary, progressive and self-contained approch to vertex
operator superalgebra. We then build such a structure from the loop algebra
of any simple finite dimensional Lie algebra . The Neveu-Schwarz
algebra emerges naturally on. As application, we obtain a unitary
action of on the unitary discrete series of .Comment: 40 page
Neveu-Schwarz and operators algebras II: Unitary series and characters
This paper is the second of a series giving a self-contained way from the
Neveu-Schwarz algebra to a new series of irreducible subfactors. Here we give a
unitary complete proof of the classification of the unitary series of the
Neveu-Schwarz algebra, by the way of GKO construction, Kac determinant and FQS
criterion. We then obtain the characters directly, without Feigin-Fuchs
resolutions.Comment: 30 page
Ore's theorem on subfactor planar algebras
This article proves that an irreducible subfactor planar algebra with a
distributive biprojection lattice admits a minimal 2-box projection generating
the identity biprojection. It is a generalization (conjectured in 2013) of a
theorem of Oystein Ore on distributive intervals of finite groups (1938), and a
corollary of a natural subfactor extension of a conjecture of Kenneth S. Brown
in algebraic combinatorics (2000). We deduce a link between combinatorics and
representations in finite group theory.Comment: 14 pages. It reproduces some preliminaries of arXiv:1702.02124 and
arXiv:1703.04486, for being self-containe
Neveu-Schwarz and operators algebras III: Subfactors and Connes fusion
This paper is the third of a series giving a self-contained way from the
Neveu-Schwarz algebra to a new series of irreducible subfactors. Here we
introduce the local von Neumann algebra of the Neveu-Schwarz algebra, to obtain
Jones-Wassermann subfactors for each representation of the discrete series.
Then using primary fields we prove the irreducibility of these subfactors; to
next compute the Connes fusion ring and obtain the explicit formula of the
subfactors indices.Comment: 54 page
Fusion Bialgebras and Fourier Analysis
We introduce fusion bialgebras and their duals and systematically study their
Fourier analysis. As an application, we discover new efficient analytic
obstructions on the unitary categorification of fusion rings. We prove the
Hausdorff-Young inequality, uncertainty principles for fusion bialgebras and
their duals. We show that the Schur product property, Young's inequality and
the sum-set estimate hold for fusion bialgebras, but not always on their duals.
If the fusion ring is the Grothendieck ring of a unitary fusion category, then
these inequalities hold on the duals. Therefore, these inequalities are
analytic obstructions of categorification. We classify simple integral fusion
rings of Frobenius type up to rank 8 and of Frobenius-Perron dimension less
than 4080. We find 34 ones, 4 of which are group-like and 28 of which can be
eliminated by applying the Schur product property on the dual. In general,
these inequalities are obstructions to subfactorize fusion bialgebras.Comment: 39 pages; 8 figures; addition of a classification in Subsection 9.2;
the long lists in Subsection 9.3 are now more pleasant to read; addition of
Section 7 providing a categorical proof of Schur Product Theore
Spectral triples for finitely generated groups, index 0.
This paper of 7 pages is just a first draft, it contains very few proofs. Â It is possible that some propositions are false, or that some proofs are incomplete or trivially false.Using Cayley graphs and Clifford algebras, we are able to give, for every finitely generated groups, a uniform construction of spectral triples with a generically non-trivial phase for the Dirac operator. Unfortunatly is index , but we are naturally led to an interesting classification of finitely generated groups into three types
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