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1
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Link homology and Frobenius extensions

100%
Khovanov M.
Fundamenta Mathematicae
|
2006
|
tom 190
|
nr 1
179-190
EN
We explain how rank two Frobenius extensions of commutative rings lead to link homology theories and discuss relations between these theories, Bar-Natan theories, equivariant cohomology and the Rasmussen invariant.
2
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How to categorify one-half of quantum 𝔤𝔩(1|2)

100%
Khovanov M.
Banach Center Publications
|
2014
|
tom 103
|
nr 1
211-232
EN
We describe a collection of differential graded rings that categorify weight spaces of the positive half of the quantized universal enveloping algebra of the Lie superalgebra 𝔤𝔩(1|2).
3
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Heisenberg algebra and a graphical calculus

100%
Khovanov M.
Fundamenta Mathematicae
|
2014
|
tom 225
|
nr 1
169-210
EN
A new calculus of planar diagrams involving diagrammatics for biadjoint functors and degenerate affine Hecke algebras is introduced. The calculus leads to an additive monoidal category whose Grothendieck ring contains an integral form of the Heisenberg algebra in infinitely many variables. We construct bases of the vector spaces of morphisms between products of generating objects in this category.
4
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An invariant of tangle cobordisms via subquotients of arc rings

64%
Chen Y., Khovanov M.
Fundamenta Mathematicae
|
2014
|
tom 225
|
nr 1
23-44
EN
We construct an explicit categorification of the action of tangles on tensor powers of the fundamental representation of quantum sl(2).
5
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Categorifications of the polynomial ring

64%
Khovanov M., Sazdanovic R.
Fundamenta Mathematicae
|
2015
|
tom 230
|
nr 3
251-280
EN
We develop a diagrammatic categorification of the polynomial ring ℤ[x]. Our categorification satisfies a version of Bernstein-Gelfand-Gelfand reciprocity property with the indecomposable projective modules corresponding to xⁿ and standard modules to (x-1)ⁿ in the Grothendieck ring.
6
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Matrix factorizations and link homology

64%
Khovanov M., Rozansky L.
Fundamenta Mathematicae
|
2008
|
tom 199
|
nr 1
1-91
EN
For each positive integer n the HOMFLYPT polynomial of links specializes to a one-variable polynomial that can be recovered from the representation theory of quantum sl(n). For each such n we build a doubly-graded homology theory of links with this polynomial as the Euler characteristic. The core of our construction utilizes the theory of matrix factorizations, which provide a linear algebra description of maximal Cohen-Macaulay modules on isolated hypersurface singularities.
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