Extremal weight projectors

@article{Queffelec2018ExtremalWP,
  title={Extremal weight projectors},
  author={Hoel Queffelec and Paul Wedrich},
  journal={Mathematical Research Letters},
  year={2018},
  volume={25},
  pages={1911-1936}
}
We introduce a quotient of the affine Temperley-Lieb category that encodes all weight-preserving linear maps between finite-dimensional sl(2)-representations. We study the diagrammatic idempotents that correspond to projections onto extremal weight spaces and find that they satisfy similar properties as Jones-Wenzl projectors, and that they categorify the Chebyshev polynomials of the first kind. This gives a categorification of the Kauffman bracket skein algebra of the annulus, which is well… 
Extremal weight projectors II.
In previous work, we have constructed diagrammatic idempotents in an affine extension of the Temperley-Lieb category, which describe extremal weight projectors for sl(2), and which categorify
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References

SHOWING 1-10 OF 25 REFERENCES
On Representations of Affine Temperley{lieb Algebras
We study the nite-dimensional simple modules, over an algebraically closed eld, of the aane Temperley{Lieb algebra corresponding to the aane Weyl group of type A. These turn out to be closely related
Hilbert space representations of the annular temperley-lieb algebra
The set of diagrams consisting of an annulus with a finite family of curves connecting some points on the boundary to each other defines a category in which a contractible closed curve counts for a
On representations of affine Temperley–Lieb algebras, II
We study some non-semisimple representations of afine Temperley-Lieb algebras and related cellular algebras. In particular, we classify extensions between simple standard modules. Moreover, we
Triangular decomposition of skein algebras
By introducing a finer version of the Kauffman bracket skein algebra, we show how to decompose the Kauffman bracket skein algebra of a surface into elementary blocks corresponding to the triangles in
Categorification of the Kauffman bracket skein module of I-bundles over surfaces
Khovanov defined graded homology groups for links LR 3 and showed that their polynomial Euler characteristic is the Jones polyno- mial of L. Khovanov's construction does not extend in a
Skein modules and the noncommutative torus
We prove that the Kauffman bracket skein algebra of the cylinder over a torus is a canonical subalgebra of the noncommutative torus. The proof is based on Chebyshev polynomials. As an application, we
Positive basis for surface skein algebras
  • D. Thurston
  • Mathematics
    Proceedings of the National Academy of Sciences
  • 2014
TLDR
This paper proposes a basis for the skein algebra that has positivity properties when q is set to 1, and conjecturally for general values of q as well, and suggests the existence of well-behaved higher-dimensional structures.
Categorification of the polynomial ring
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
Khovanov's homology for tangles and cobordisms
We give a fresh introduction to the Khovanov Homology theory for knots and links, with special emphasis on its extension to tangles, cobordisms and 2-knots. By staying within a world of topological
ON POSITIVITY OF KAUFFMAN BRACKET SKEIN ALGEBRAS OF SURFACES
We show that the Chebyshev polynomials form a basic block of any positive basis of the Kauffman bracket skein algebras of surfaces.
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