# Highest weight categories arising from Khovanov's diagram algebra III: category O

@article{Brundan2008HighestWC, title={Highest weight categories arising from Khovanov's diagram algebra III: category O}, author={Jonathan Brundan and Catharina Stroppel}, journal={arXiv: Representation Theory}, year={2008} }

We prove that integral blocks of parabolic category O associated to the subalgebra gl(m) x gl(n) of gl(m+n) are Morita equivalent to quasi-hereditary covers of generalised Khovanov algebras. Although this result is in principle known, the existing proof is quite indirect, going via perverse sheaves on Grassmannians. Our new approach is completely algebraic, exploiting Schur-Weyl duality for higher levels. As a by-product we get a concrete combinatorial construction of 2-Kac-Moody…

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## References

SHOWING 1-10 OF 102 REFERENCES

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

- Mathematics
- 2010

This is the second of a series of four papers studying various generalisations of Khovanov's diagram algebra. In this paper we develop the general theory of Khovanov's diagrammatically defined…

A categorification of finite-dimensional irreducible representations of quantum sl(2) and their tensor products

- Mathematics
- 2005

The purpose of this paper is to study categorifications of tensor products of finite dimensional modules for the quantum group for sl(2). The main categorification is obtained using certain…

Parabolic category O, perverse sheaves on Grassmannians, Springer fibres and Khovanov homology

- MathematicsCompositio Mathematica
- 2009

Abstract For a fixed parabolic subalgebra 𝔭 of $\mathfrak {gl}(n,\mathbb {C})$ we prove that the centre of the principal block 𝒪0𝔭 of the parabolic category 𝒪 is naturally isomorphic to the…

Duality between sln(C) and the Degenerate Affine Hecke Algebra

- Mathematics
- 1998

Abstract We construct a family of exact functors from the Bernstein–Gelfand–Gelfand category O of s l n-modules to the category of finite-dimensional representations of the degenerate affine Hecke…

Projective modules in the category _{}: self-duality

- Mathematics
- 1985

Given a parabolic subalgebra ps of a complex, semisimple Lie algebra , there is a naturally defined category °s of g-modules which includes all the g-modules induced from finite-dimensional…

Representations of Semisimple Lie Algebras in the BGG Category O

- Mathematics
- 2008

Review of semisimple Lie algebras Highest weight modules: Category $\mathcal{O}$: Basics Characters of finite dimensional modules Category $\mathcal{O}$: Methods Highest weight modules I Highest…

Category O and slk link invariants

- Mathematics
- 2007

We construct a functor valued invariant of oriented tangles on certain singular blocks of category O. Parabolic subcategories of these blocks categorify tensor products of various fundamental sl(k)…

KOSZUL DUALITY FOR PARABOLIC AND SINGULAR CATEGORY O

- Mathematics
- 1999

This paper deals with a generalization of the “Koszul duality theorem” for the Bernstein-Gelfand-Gelfand category O over a complex semisimple Lie-algebra, established by Beilinson, Ginzburg and…

Centers of degenerate cyclotomic Hecke algebras and parabolic category

- Mathematics
- 2006

We prove that the center of each degenerate cyclotomic Hecke algebra associated to the complex reflection group of type B_d(l) consists of symmetric polynomials in its commuting generators. The…