17,955 research outputs found
Kostka systems and exotic t-structures for reflection groups
Let W be a complex reflection group, acting on a complex vector space H. Kato
has recently introduced the notion of a "Kostka system," which is a certain
collection of finite-dimensional W-equivariant modules for the symmetric
algebra on H. In this paper, we show that Kostka systems can be used to
construct "exotic" t-structures on the derived category of finite-dimensional
modules, and we prove a derived-equivalence result for these t-structures.Comment: 21 pages. v2: minor corrections; simplified proof in Section
Heisenberg Idempotents on Unipotent Groups
Let G be an algebraic group over an algebraically closed field of positive
characteristic such that its neutral connected component is a unipotent group.
We consider a certain class of closed idempotents in the braided monoidal
category (under convolution of complexes) D_G(G) known as Heisenberg
idempotents. For such an idempotent e, we will prove certain results about the
Hecke subcategory eD_G(G) conjectured by V. Drinfeld. In particular, we will
see that it is the bounded derived category of a modular category.Comment: 20 pages, added some definition
Affine hom-complexes
For two general polytopal complexes the set of face-wise affine maps between
them is shown to be a polytopal complex in an algorithmic way. The resulting
algorithm for the affine hom-complex is analyzed in detail. There is also a
natural tensor product of polytopal complexes, which is the left adjoint
functor for Hom. This extends the corresponding facts from single polytopes,
systematic study of which was initiated in [6,12]. Explicit examples of
computations of the resulting structures are included. In the special case of
simplicial complexes, the affine hom-complex is a functorial subcomplex of
Kozlov's combinatorial hom-complex [14], which generalizes Lovasz' well-known
construction [15] for graphs.Comment: final version, to appear in Portugaliae Mathematic
Morita cohomology
We consider two categorifications of the cohomology of a topological space X
by taking coefficients in the category of differential graded categories. We
consider both derived global sections of a constant presheaf and singular
cohomology and find the resulting dg-categories are quasi-equivalent and
moreover quasi-equivalent to representations in perfect complexes of chains on
the loop space of X.Comment: 33 page
On global deformation quantization in the algebraic case
We give a proof of Yekutieli's global algebraic deformation quantization
result which does not rely on the choice of local sections of the bundle of
affine coordinate systems. Instead we use an argument inspired by algebraic De
Rham cohomology.Comment: 60 pages; references added; relation to Hinich's work explaine
Cluster combinatorics of d-cluster categories
We study the cluster combinatorics of cluster tilting objects in
cluster categories. By using mutations of maximal rigid objects in
cluster categories which are defined similarly for cluster tilting
objects, we prove the equivalences between cluster tilting objects, maximal
rigid objects and complete rigid objects. Using the chain of triangles of
cluster tilting objects in [IY], we prove that any almost complete
cluster tilting object has exactly complements, compute the extension
groups between these complements, and study the middle terms of these
triangles. All results are the extensions of corresponding results on cluster
tilting objects in cluster categories established in [BMRRT] to cluster
categories. They are applied to the Fomin-Reading's generalized cluster
complexes of finite root systems defined and studied in [FR2] [Th] [BaM1-2],
and to that of infinite root systems [Zh3].Comment: correted many typos according to the referee's comments, final
version to appear in J. Algebr
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