1,449 research outputs found
A categorification of cyclotomic rings
For any natural number , we construct a triangulated monoidal
category whose Grothendieck ring is isomorphic to the ring of cyclotomic
integers .Comment: 28 pages. Comments welcome! v2, v3: minor correction
On Link Homology Theories from Extended Cobordisms
This paper is devoted to the study of algebraic structures leading to link
homology theories. The originally used structures of Frobenius algebra and/or
TQFT are modified in two directions. First, we refine 2-dimensional cobordisms
by taking into account their embedding into the three space. Secondly, we
extend the underlying cobordism category to a 2-category, where the usual
relations hold up to 2-isomorphisms. The corresponding abelian 2-functor is
called an extended quantum field theory (EQFT). We show that the Khovanov
homology, the nested Khovanov homology, extracted by Stroppel and Webster from
Seidel-Smith construction, and the odd Khovanov homology fit into this setting.
Moreover, we prove that any EQFT based on a Z_2-extension of the embedded
cobordism category which coincides with Khovanov after reducing the
coefficients modulo 2, gives rise to a link invariant homology theory
isomorphic to those of Khovanov.Comment: Lots of figure
A categorification of non-crossing partitions
We present a categorification of the non-crossing partitions given by
crystallographic Coxeter groups. This involves a category of certain bilinear
lattices, which are essentially determined by a symmetrisable generalised
Cartan matrix together with a particular choice of a Coxeter element. Examples
arise from Grothendieck groups of hereditary artin algebras.Comment: 34 pages. Substantially revised and final version, accepted for
publication in Journal of the European Mathematical Societ
Skein Modules from Skew Howe Duality and Affine Extensions
We show that we can release the rigidity of the skew Howe duality process for
knot invariants by rescaling the quantum Weyl group action,
and recover skein modules for web-tangles. This skew Howe duality phenomenon
can be extended to the affine case, corresponding to looking
at tangles embedded in a solid torus. We investigate the relations between the
invariants constructed by evaluation representations (and affinization of them)
and usual skein modules, and give tools for interpretations of annular skein
modules as sub-algebras of intertwiners for particular
representations. The categorification proposed in a joint work with A. Lauda
and D. Rose also admits a direct extension in the affine case
Two-dimensional categorified Hall algebras
In the present paper, we introduce two-dimensional categorified Hall algebras
of smooth curves and smooth surfaces. A categorified Hall algebra is an
associative monoidal structure on the stable -category
of complexes of sheaves with
bounded coherent cohomology on a derived moduli stack .
In the surface case, is a suitable derived enhancement
of the moduli stack of coherent sheaves on the surface. This
construction categorifies the K-theoretical and cohomological Hall algebras of
coherent sheaves on a surface of Zhao and Kapranov-Vasserot. In the curve case,
we define three categorified Hall algebras associated with suitable derived
enhancements of the moduli stack of Higgs sheaves on a curve , the moduli
stack of vector bundles with flat connections on , and the moduli stack of
finite-dimensional local systems on , respectively. In the Higgs sheaves
case we obtain a categorification of the K-theoretical and cohomological Hall
algebras of Higgs sheaves on a curve of Minets and Sala-Schiffmann, while in
the other two cases our construction yields, by passing to , new
K-theoretical Hall algebras, and by passing to ,
new cohomological Hall algebras. Finally, we show that the Riemann-Hilbert and
the non-abelian Hodge correspondences can be lifted to the level of our
categorified Hall algebras of a curve.Comment: 54 page
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