Tilting modules for quantum groups

introduction

modules for $U_q(\mathfrak{g})$
Verma modules $M_{\lambda}=U_q(\mathfrak{g})\otimes_{U_q(\mathfrak{b})}\mathbb{C}_{\lambda}$
Weyl modules : quotients of Verma modules

$$ W_{\lambda}=M_{\lambda}/\operatorname{span}(M_{s_i\cdot \lambda}) $$

a tilting module is a module $T$ that admies a filtration whose associated graded pieces are Weyl modules and that admits another filtration whose associated graded are dual Weyl modules

Hazi, Amit. “Balanced Semisimple Filtrations for Tilting Modules.” arXiv:1510.02596 [math], October 9, 2015. http://arxiv.org/abs/1510.02596.
Andersen, Henning Haahr, Catharina Stroppel, and Daniel Tubbenhauer. “Cellular Structures Using $\textbf{U}_q$-Tilting Modules.” arXiv:1503.00224 [math], March 1, 2015. http://arxiv.org/abs/1503.00224.
Andersen, Henning Haahr, and Masaharu Kaneda. 2009. “Rigidity of Tilting Modules.” arXiv:0909.2935 [math] (September 16). http://arxiv.org/abs/0909.2935.
Andersen, Henning Haahr, and Jan Paradowski. 1995. “Fusion Categories Arising from Semisimple Lie Algebras.” Communications in Mathematical Physics 169 (3) (May 1): 563–588. doi:10.1007/BF02099312.