We describe how certain cyclotomic Nazarov–Wenzl algebras occur as endomorphism rings of projective modules in a parabolic version of BGG category O of type D. Furthermore we study a family of ...subalgebras of these endomorphism rings which exhibit similar behaviour to the family of Brauer algebras even when they are not semisimple. The translation functors on this parabolic category O are studied and proven to yield a categorification of a coideal subalgebra of the general linear Lie algebra. Finally this is put into the context of categorifying skew Howe duality for these subalgebras.
A way to construct the natural representation of the quantized affine algebra
U
v
(
s
l
̂
l
)
is via the deformed Fock space by Misra and Miwa. This relates the classes of Weyl modules for
U
q
(
s
l
...N
)
were
q
is a root of unity to the action of
U
v
(
s
l
̂
l
)
as N tends toward infinity. In this paper we investigate the situation outside of type A. In classical types, we construct embeddings of the Grothendieck group of finite dimensional
U
q
(
g
)
-modules into Fock spaces of different charges and define an action of an affine quantum symmetric pair that plays the role of the quantized affine algebra. We describe how the action is related to the linkage principal for quantum groups at a root of unity and tensor product multiplicities.
We give necessary and sufficient conditions for zigzag algebras and certain generalizations of them to be (relative) cellular, quasi-hereditary or Koszul.
We construct a faithful categorical representation of an infinite Temperley-Lieb algebra on the periplectic analogue of Deligne’s universal monoidal category. We use the corresponding combinatorics ...to classify thick tensor ideals in this periplectic Deligne category. This allows us to determine the objects in the kernel of the monoidal functor going to the module category of the periplectic Lie supergroup. We use this to classify indecomposable direct summands in the tensor powers of the natural representation, determine which are projective and determine their simple top.
Functoriality of colored link homologies Ehrig, Michael; Tubbenhauer, Daniel; Wedrich, Paul
Proceedings of the London Mathematical Society,
November 2018, 2018-11-00, Letnik:
117, Številka:
5
Journal Article
Recenzirano
Odprti dostop
We prove that the bigraded, colored Khovanov–Rozansky type A link and tangle invariants are functorial with respect to link and tangle cobordisms.
We study in detail two row Springer fibres of even orthogonal type from an algebraic as well as a topological point of view. We show that the irreducible components and their pairwise intersections ...are iterated
${{\mathbb{P}}^{1}}$
-bundles. Using results of Kumar and Procesi we compute the cohomology ring with its action of the Weyl group. The main tool is a type
$\text{D}$
diagram calculus labelling the irreducible components in a convenient way that relates to a diagrammatical algebra describing the category of perverse sheaves on isotropic Grassmannians based on work of Braden. The diagram calculus generalizes Khovanov's arc algebra to the type
$\text{D}$
setting and should be seen as setting the framework for generalizing well-known connections of these algebras in type
$\text{A}$
to other types.
Koszul Gradings on Brauer Algebras Ehrig, Michael; Stroppel, Catharina
International mathematics research notices,
2016, Letnik:
2016, Številka:
13
Journal Article
For each integer
k
≥
4
, we describe diagrammatically a positively graded Koszul algebra
D
k
such that the category of finite dimensional
D
k
-modules is equivalent to the category of perverse ...sheaves on the isotropic Grassmannian of type
D
k
or
B
k
-
1
, constructible with respect to the Schubert stratification. The algebra is obtained by a (non-trivial) “folding” procedure from a generalized Khovanov arc algebra. Properties such as graded cellularity and explicit closed formulas for graded decomposition numbers are established by elementary tools.
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