Corpus ID: 119614446

Categorification and applications in topology and representation theory

@article{Tubbenhauer2013CategorificationAA,
  title={Categorification and applications in topology and representation theory},
  author={D. Tubbenhauer},
  journal={arXiv: Quantum Algebra},
  year={2013}
}
This thesis splits into two major parts. The connection between the two parts is the notion of "categorification" which we shortly explain/recall in the introduction. In the first part of this thesis we extend Bar-Natan's cobordism based categorification of the Jones polynomial to virtual links. Our topological complex allows a direct extension of the classical Khovanov complex (h=t=0), the variant of Lee (h=0,t=1) and other classical link homologies. We show that our construction allows… Expand
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References

SHOWING 1-10 OF 177 REFERENCES
On Khovanov's cobordism theory for SU3 knot homology
We reconsider the link homology theory defined by Knovanov in [9] and generalized by Mackaay and Vaz in [15]. With some slight modifications, we describe the theory as a map from the planar algebraExpand
Fixing the functoriality of Khovanov homology
We describe a modification of Khovanov homology [13], in the spirit of Bar-Natan [2], which makes the theory properly functorial with respect to link cobordisms. This requires introducingExpand
Knot invariants and higher representation theory I: diagrammatic and geometric categorification of tensor products
In this paper, we study 2-representations of 2-quantum groups (in the sense of Rouquier and Khovanov-Lauda) categorifying tensor products of irreducible representations. Our aim is to construct knotExpand
Categorification of the Temperley-Lieb category, tangles, and cobordisms via projective functors
To each generic tangle projection from the three-dimensional real vector space onto the plane, we associate a derived endofunctor on a graded parabolic version of the Bernstein-Gel'fand categoryExpand
A combinatorial approach to functorial quantum slk knot invariants
This paper contains a categorification of the ${\frak sl}(k)$ link invariant using parabolic singular blocks of category ${\cal{O}}$. Our approach is intended to be as elementary as possible,Expand
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 spaceExpand
Parabolic category O, perverse sheaves on Grassmannians, Springer fibres and Khovanov homology
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 theExpand
Frobenius Algebras and 2-D Topological Quantum Field Theories
This 2003 book describes a striking connection between topology and algebra, namely that 2D topological quantum field theories are equivalent to commutative Frobenius algebras. The preciseExpand
Khovanov homology for virtual links using cobordisms
We give a geometric interpretation of the Khovanov complex for virtual links. Geometric interpretation means that we use a cobordism structure like D. Bar-Natan, but we allow non orientableExpand
Semi-abelian categories
Abstract The notion of semi-abelian category as proposed in this paper is designed to capture typical algebraic properties valid for groups, rings and algebras, say, just as abelian categories allowExpand
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