75 research outputs found
Perverse Sheaves on affine Grassmannians and Langlands Duality
This is an expanded version of the text ``Perverse Sheaves on Loop
Grassmannians and Langlands Duality'', AG/9703010. The main new result is a
topological realization of algebraic representations of reductive groups over
arbitrary rings.
We outline a proof of a geometric version of the Satake isomorphism. Given a
connected, complex algebraic reductive group G we show that the tensor category
of representations of the Langlands dual group is naturally equivalent to a
certain category of perverse sheaves on the loop Grassmannian of G. The above
result has been announced by Ginsburg in and some of the arguments are borrowed
from his approach. However, we use a more "natural" commutativity constraint
for the convolution product, due to Drinfeld. Secondly, we give a direct proof
that the global cohomology functor is exact and decompose this cohomology
functor into a direct sum of weights. The geometry underlying our arguments
leads to a construction of a canonical basis of Weyl modules given by algebraic
cycles and an explicit construction of the group algebra of the dual group in
terms of the affine Grassmannian. We completely avoid the use of the
decomposition theorem which makes our techniques applicable to perverse sheaves
with coefficients over an arbitrary commutative ring. We deduce the classical
Satake isomorphism using the affine Grassmannian defined over a finite field.
This note contains indications of proofs of some of the results. The details
will appear elsewhere.Comment: Amstex, 12 page
Linear Koszul Duality II - Coherent sheaves on perfect sheaves
In this paper we continue the study (initiated in a previous article) of
linear Koszul duality, a geometric version of the standard duality between
modules over symmetric and exterior algebras. We construct this duality in a
very general setting, and prove its compatibility with morphisms of vector
bundles and base change.Comment: Final version, to appear in JLMS. The numbering differs from the
published version, and is the one used in our papers [MR2] and [MR3] from the
bibliograph
Intersection cohomology of Drinfeld's compactifications
Let be a smooth complete curve, be a reductive group and
a parabolic.
Following Drinfeld, one defines a compactification \widetilde{\on{Bun}}_P
of the moduli stack of -bundles on .
The present paper is concerned with the explicit description of the
Intersection Cohomology sheaf of \widetilde{\on{Bun}}_P. The description is
given in terms of the combinatorics of the Langlands dual Lie algebra
.Comment: An erratum adde
Modules over the small quantum group and semi-infinite flag manifold
We develop a theory of perverse sheaves on the semi-infinite flag manifold
, and show that the subcategory of Iwahori-monodromy
perverse sheaves is equivalent to the regular block of the category of
representations of the small quantum group at an even root of unity
Knot homology via derived categories of coherent sheaves II, sl(m) case
Using derived categories of equivariant coherent sheaves we construct a knot
homology theory which categorifies the quantum sl(m) knot polynomial. Our knot
homology naturally satisfies the categorified MOY relations and is
conjecturally isomorphic to Khovanov-Rozansky homology. Our construction is
motivated by the geometric Satake correspondence and is related to Manolescu's
by homological mirror symmetry.Comment: 51 pages, 9 figure
Categorical geometric skew Howe duality
We categorify the R-matrix isomorphism between tensor products of minuscule
representations of U_q(sl(n)) by constructing an equivalence between the
derived categories of coherent sheaves on the corresponding convolution
products in the affine Grassmannian. The main step in the construction is a
categorification of representations of U_q(sl(2)) which are related to
representations of U_q(sl(n)) by quantum skew Howe duality. The resulting
equivalence is part of the program of algebro-geometric categorification of
Reshitikhin-Turaev tangle invariants developed by the first two authors.Comment: 31 page
The London theory of the crossing-vortex lattice in highly anisotropic layered superconductors
A novel description of Josephson vortices (JVs) crossed by the pancake
vortices (PVs) is proposed on the basis of the anisotropic London theory. The
field distribution of a JV and its energy have been calculated for both dense
() PV lattices with distance
between PVs, and the nonlinear JV core size . It is shown that the
``shifted'' PV lattice (PVs displaced mainly along JVs in the crossing vortex
lattice structure), formed in high out-of-plane magnetic fields transforms into
the PV lattice ``trapped'' by the JV sublattice at a certain field, lower than
, where is the flux quantum, is the
anisotropy parameter and is the distance between CuO planes.
With further decreasing , the free energy of the crossing vortex lattice
structure (PV and JV sublattices coexist separately) can exceed the free energy
of the tilted lattice (common PV-JV vortex structure) in the case of with the in-plane penetration depth if the low
() or high ()
in-plane magnetic field is applied. It means that the crossing vortex structure
is realized in the intermediate field orientations, while the tilted vortex
lattice can exist if the magnetic field is aligned near the -axis and the
-plane as well. In the intermediate in-plane fields
, the
crossing vortex structure with the ``trapped'' PV sublattice seems to settle in
until the lock-in transition occurs since this structure has the lower energy
with respect to the tilted vortex structure in the magnetic field
oriented near the -plane.Comment: 15 pages, 6 figures, accepted for publication in PR
Characterization of red mud/metakaolin-based geopolymers as modified by Ca(OH)2
Geopolymers are an emerging class of materials that offer an alternative to the Portland cement as the binder of structural concrete. One of the advantages is that the primary source of their production is waste alumosilicate materials from different industries. One of the key issues in geopolymer synthesis is the low level of mechanical properties due to porosity as well as the high activity of conductivity carriers. It can often lead to limited application possibilities, so the objective is to obtain an enhanced strength as well as decreased cracking tendency through microstructure modification. The introduction of Ca(OH)2, under certain pH conditions could lead to the filling-the-pores process and improving the mechanical properties. The aim was to understand the role that calcium plays in the geopolymer synthesis, and to define which reaction prevails under the synthesis conditions: formation of geopolymer gel or calcium silicate hydrate that contains aluminum substitution (CASH). The synthesis was performed with different raw materials (with or without red mud) and different alkalinity conditions. Ca(OH)2 was the obligatory supplement to both of the mixtures. Different techniques were performed for the testing of reaction products, as well as to define the microstructural changes as the generator of improved mechanical properties and changed electrical conductivity. The characteristics of the geopolymer's macrostructure were defined by means of an SEM analysis. Compressive strength and electrical conductivity are among the investigated product's properties. X-ray diffraction (XRD) and Fourier transform infra-red spectroscopy (FTIR) were used for the identification of various crystalline phases and an amorphous phase
Azotobacter chroococcum F8/2: a multitasking bacterial strain in sugar beet biopriming
This study assesses the effects of Azotobacter biopriming on the early development of sugar beet. Azotobacter chroococcum F8/2 was screened for plant growth promoting characteristics and biopriming effects were estimated through germination parameters and the structural changes of the root tissues. A. chroococcum F8/2 was characterized as a contributor to nitrogen, iron, and potassium availability, as well as a producer of auxin and 1-aminocyclopropane-1-carboxilic acid deaminase. Applied biopriming had reduced mean germination time by 34.44% and increased vigor I by 90.99% compared to control. Volatile blend comprised 47.67% ethanol, 32.01% 2-methyl-propanol, 17.32% 3-methyl-1-butanol, and a trace of 2,3-butanedione. Root micromorphological analysis of bioprimed sugar beet revealed a considerable increase in primary, secondary xylem area, and vessels size. Obtained results determine A. chroococcum F8/2 as a successful biopriming agent, and active participant in nutrient availability and hormonal status modulation affecting root vascular tissue. © 2022 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group
The hypertoric intersection cohomology ring
We present a functorial computation of the equivariant intersection
cohomology of a hypertoric variety, and endow it with a natural ring structure.
When the hyperplane arrangement associated with the hypertoric variety is
unimodular, we show that this ring structure is induced by a ring structure on
the equivariant intersection cohomology sheaf in the equivariant derived
category. The computation is given in terms of a localization functor which
takes equivariant sheaves on a sufficiently nice stratified space to sheaves on
a poset.Comment: Significant revisions in Section 5, with several corrected proof
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