Cellular structures using $\textbf{U}_q$-tilting modules

@article{Andersen2015CellularSU,
  title={Cellular structures using \$\textbf\{U\}\_q\$-tilting modules},
  author={Henning Haahr Andersen and Catharina Stroppel and Daniel Tubbenhauer},
  journal={arXiv: Quantum Algebra},
  year={2015}
}
We use the theory of $\textbf{U}_q$-tilting modules to construct cellular bases for centralizer algebras. Our methods are quite general and work for any quantum group $\textbf{U}_q$ attached to a Cartan matrix and include the non-semisimple cases for $q$ being a root of unity and ground fields of positive characteristic. Our approach also generalizes to certain categories containing infinite-dimensional modules. As applications, we give a new semisimplicty criterion for centralizer algebras… 
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References

SHOWING 1-10 OF 93 REFERENCES
Lectures on quantum groups
Introduction Gaussian binomial coefficients The quantized enveloping algebra $U_q(\mathfrak s \mathfrak {1}_2)$ Representations of $U_q(\mathfrak{sl}_2)$ Tensor products or: $U_q(\mathfrak{sl}_2)$ as
Representations of algebraic groups
Part I. General theory: Schemes Group schemes and representations Induction and injective modules Cohomology Quotients and associated sheaves Factor groups Algebras of distributions Representations
Graded cellular bases for Temperley–Lieb algebras of type A and B
We show that the Temperley–Lieb algebra of type A and the blob algebra (also known as the Temperley–Lieb algebra of type B) at roots of unity are $\mathbb{Z}$-graded algebras. We moreover show that
The strong linkage principle for quantum groups at roots of 1
SEMISIMPLICITY OF HECKE AND (WALLED) BRAUER ALGEBRAS
We show how to use Jantzen’s sum formula for Weyl modules to prove semisimplicity criteria for endomorphism algebras of $\mathbf{U}_{q}$ -tilting modules (for any field $\mathbb{K}$ and any parameter
Character formulas for tilting modules over Kac-Moody algebras
We show how to express the characters of tilting modules in a (possibly parabolic) category O over a Kac-Moody algebra in terms of the characters of simple highest weight modules. This settles, in
Quiver Schur algebras for linear quivers
We define a graded quasi‐hereditary covering of the cyclotomic quiver Hecke algebras RnΛ of type A when e=0 (the linear quiver) or e>n . We prove that these algebras are quasi‐hereditary graded
Highest Weight Categories Arising from Khovanov's Diagram Algebra I: Cellularity
This is the first of four articles studying some slight generalisations Hn m of Khovanov’s diagram algebra, as well as quasi-hereditary covers Kn m of these algebras in the sense of Rouquier, and
Highest weight categories arising from Khovanov's diagram algebra III: category O
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
...
...