122 research outputs found
Categorification of Highest Weight Modules via Khovanov-Lauda-Rouquier Algebras
In this paper, we prove Khovanov-Lauda's cyclotomic categorification
conjecture for all symmetrizable Kac-Moody algebras. Let be the
quantum group associated with a symmetrizable Cartan datum and let
be the irreducible highest weight -module with a dominant integral
highest weight . We prove that the cyclotomic Khovanov-Lauda-Rouquier
algebra gives a categorification of .Comment: Typoes correcte
Nilpotency in type A cyclotomic quotients
We prove a conjecture made by Brundan and Kleshchev on the nilpotency degree
of cyclotomic quotients of rings that categorify one-half of quantum sl(k).Comment: 19 pages, 39 eps files. v3 simplifies antigravity moves and corrects
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The sl_3 web algebra
In this paper we use Kuperbergâs sl3-webs and Khovanovâs sl3-foams to define a new
algebra KS, which we call the sl3-web algebra. It is the sl3 analogue of Khovanovâs arc algebra.
We prove that KS is a graded symmetric Frobenius algebra. Furthermore, we categorify an
instance of q-skew Howe duality, which allows us to prove that KS
is Morita equivalent to a certain cyclotomic KLR-algebra of level 3. This allows us to determine the split Grothendieck group K0
(WS )Q(q) , to show that its center is isomorphic to the cohomology ring of a certain Spaltenstein
variety, and to prove that KS is a graded cellular algebra.info:eu-repo/semantics/publishedVersio
Trace as an alternative decategorification functor
Categorification is a process of lifting structures to a higher categorical
level. The original structure can then be recovered by means of the so-called
"decategorification" functor. Algebras are typically categorified to additive
categories with additional structure and decategorification is usually given by
the (split) Grothendieck group. In this expository article we study an
alternative decategorification functor given by the trace or the zeroth
Hochschild--Mitchell homology. We show that this form of decategorification
endows any 2-representation of the categorified quantum sl(n) with an action of
the current algebra U(sl(n)[t]) on its center.Comment: 47 pages with tikz figures. arXiv admin note: text overlap with
arXiv:1405.5920 by other author
The Symbolic and cancellation-free formulae for Schur elements
In this paper we give a symbolical formula and a cancellation-free formula
for the Schur elements associated to the simple modules of the degenerate
cyclotomic Hecke algebras. As some direct applications, we show that the Schur
elements are symmetric with respect to the natural symmetric group action and
are integral coefficients polynomials and we give a different proof of
Ariki-Mathas-Rui's criterion on the semi-simplicity of degenerate cyclotomic
Hecke algebras.Comment: To appear in Monatshefte fur Mathemati
The degenerate analogue of Ariki's categorification theorem
We explain how to deduce the degenerate analogue of Ariki's categorification
theorem over the ground field C as an application of Schur-Weyl duality for
higher levels and the Kazhdan-Lusztig conjecture in finite type A. We also
discuss some supplementary topics, including Young modules, tensoring with
sign, tilting modules and Ringel duality.Comment: 44 page
Homological algebra for osp(1/2n)
We discuss several topics of homological algebra for the Lie superalgebra
osp(1|2n). First we focus on Bott-Kostant cohomology, which yields classical
results although the cohomology is not given by the kernel of the Kostant
quabla operator. Based on this cohomology we can derive strong
Bernstein-Gelfand-Gelfand resolutions for finite dimensional osp(1|2n)-modules.
Then we state the Bott-Borel-Weil theorem which follows immediately from the
Bott-Kostant cohomology by using the Peter-Weyl theorem for osp(1|2n). Finally
we calculate the projective dimension of irreducible and Verma modules in the
category O
Highest weight categories arising from Khovanov's diagram algebra II: Koszulity
This is the second of a series of four articles studying various
generalisations of Khovanov's diagram algebra. In this article we develop the
general theory of Khovanov's diagrammatically defined "projective functors" in
our setting. As an application, we give a direct proof of the fact that the
quasi-hereditary covers of generalised Khovanov algebras are Koszul.Comment: Minor changes, extra sections on Kostant modules and rigidity of cell
modules adde
A finite analog of the AGT relation I: finite W-algebras and quasimaps' spaces
Recently Alday, Gaiotto and Tachikawa proposed a conjecture relating
4-dimensional super-symmetric gauge theory for a gauge group G with certain
2-dimensional conformal field theory. This conjecture implies the existence of
certain structures on the (equivariant) intersection cohomology of the
Uhlenbeck partial compactification of the moduli space of framed G-bundles on
P^2. More precisely, it predicts the existence of an action of the
corresponding W-algebra on the above cohomology, satisfying certain properties.
We propose a "finite analog" of the (above corollary of the) AGT conjecture.
Namely, we replace the Uhlenbeck space with the space of based quasi-maps from
P^1 to any partial flag variety G/P of G and conjecture that its equivariant
intersection cohomology carries an action of the finite W-algebra U(g,e)
associated with the principal nilpotent element in the Lie algebra of the Levi
subgroup of P; this action is expected to satisfy some list of natural
properties. This conjecture generalizes the main result of arXiv:math/0401409
when P is the Borel subgroup. We prove our conjecture for G=GL(N), using the
works of Brundan and Kleshchev interpreting the algebra U(g,e) in terms of
certain shifted Yangians.Comment: minor change
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