Catharina Stroppel

Highest weight categories arising from Khovanov's diagram algebra II: Koszulity

This is the second of a series of four papers studying various generalisations of Khovanovs diagram algebra. In this paper we develop the general theory of Khovanovs 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.

HIGHEST WEIGHT CATEGORIES ARISING FROM KHOVANOV'S DIAGRAM ALGEBRA IV: THE GENERAL LINEAR SUPERGROUP

We prove that blocks of the general linear supergroup are Morita equivalent to a limiting version of Khovanovs diagram algebra. We deduce that blocks of the general linear supergroup are Koszul.

Projective-injective modules, Serre functors and symmetric algebras

Abstract We describe Serre functors for (generalisations of) the category associated with a semisimple complex Lie algebra. In our approach, projective-injective modules, that is modules which are both, projective and injective, play an important role. They control the Serre functor in the case of a quasi-hereditary algebra having a double centraliser with respect to a projective-injective module whose endomorphism ring is a symmetric algebra. As an application of the double centraliser property together with our description of Serre functors, we prove three conjectures of Khovanov about the projective-injective modules in the parabolic category .

The slˆ(n)k-WZNW fusion ring: A combinatorial construction and a realisation as quotient of quantum cohomology

Abstract A simple, combinatorial construction of the sl ˆ ( n ) k -WZNW fusion ring, also known as Verlinde algebra, is given. As a byproduct of the construction one obtains an isomorphism between the fusion ring and a particular quotient of the small quantum cohomology ring of the Grassmannian Gr k , k + n . We explain how our approach naturally fits into known combinatorial descriptions of the quantum cohomology ring, by establishing what one could call a ‘Boson–Fermion-correspondence’ between the two rings. We also present new recursion formulae for the structure constants of both rings, the fusion coefficients and the Gromov–Witten invariants.

Categorifying fractional Euler characteristics, Jones–Wenzl projectors and 3j-symbols

We study the representation theory of the smallest quantum group and its categori- fication. The first part of the paper contains an easy visualization of the3j -symbols in terms of weighted signed line arrangements in a fixed triangle and new binomial expressions for the 3j -symbols. All these formulas are realized as graded Euler characteristics. The3j -symbols appear as new generalizations of Kazhdan-Lusztig polynomials. A crucial result of the paper is that complete intersection rings can be employed to obtain rational Euler characteristics, hence to categorify rational quantum numbers. This is the main tool for our categorification of the Jones-Wenzl projector, ‚-networks and tetrahedron net- works. Networks and their evaluations play an important role in the Turaev-Viro construction of 3-manifold invariants. We categorify these evaluations by Ext-algebras of certain simple Harish-Chandra bimodules. The relevance of this construction to categorified colored Jones invariants and invariants of3-manifolds will be studied in detail in subsequent papers.

2-block Springer fibers: convolution algebras and coherent sheaves

For a fixed 2-block Springer fiber, we describe the structure of its irreducible components and their relation to the Bialynicki-Birula paving, following work of Fung. That is, we consider the space of complete flags in C^n preserved by a fixed nilpotent matrix with 2 Jordan blocks, and study the action of diagonal matrices commuting with our fixed nilpotent. In particular, we describe the structure of each component, its set of torus fixed points, and prove a conjecture of Fung describing the intersection of any pair. Then we define a convolution algebra structure on the direct sum of the cohomologies of pairwise intersections of irreducible components and closures of C^*-attracting sets (that is, Bialynicki-Birula cells), and show this is isomorphic to a generalization of the arc algebra of Khovanov defined by the first author. We investigate the connection of this algebra to Cautis & Kamnitzers recent work on link homology via coherent sheaves and suggest directions for future research.

2-row Springer Fibres and Khovanov Diagram Algebras for Type D

We study in detail two row Springer fibres of even orthogonal type from an algebraic as well as topological point of view. We show that the irreducible components and their pairwise intersections are iterated 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 D diagram calculus labelling the irreducible components in an convenient way which relates to a diagrammatical algebra describing the category of perverse sheaves on isotropic Grassmannians based on work of Braden. The diagram calculus generalizes Khovanovs arc algebra to the type D setting and should be seen as setting the framework for generalizing well-known connections of these algebras in type A to other types. The results will be connected to Brauer algebras at non-generic parameters in a subsequent paper.

Schur–Weyl duality for the Brauer algebra and the ortho-symplectic Lie superalgebra

We give a proof of a Schur–Weyl duality statement between the Brauer algebra and the ortho-symplectic Lie superalgebra

Cellular structures using TEXTBACKSLASHmathboldTEXTBACKSLASHUq-tilting modules

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