Endomorphisms of cell 2-representations
Volodymyr Mazorchuk, Vanessa Miemietz
TL;DR
The paper extends the 2-representation framework to fiat 2-categories, focusing on endomorphisms of cell 2-representations associated with strongly regular two-sided cells. It proves that, under natural surjectivity and regularity assumptions, the endomorphism category of a cell 2-representation is equivalent to $\Bbbk$-mod, establishing a 2-analogue of Schur’s Lemma; it also derives a 2-fullness property and a $\mathcal{J}$-simple Artin–Wedderburn-type classification for certain 2-categories. The authors develop a graded fiat 2-category theory, describe the passage between representations of a pro-fiat category and its grading quotient, and provide several key examples, including categories $\mathcal{O}$ in types $A$ and $B_2$ and $\mathfrak{sl}_2$-categorification, while addressing corrections via corrigenda. The results have implications for understanding how endomorphisms and center actions control the structure of higher representation theory, with potential applications to categorifications of algebraic structures and their module categories.
Abstract
We determine the endomorphism categories of cell 2-representations of fiat 2-categories associated with strongly regular two-sided cells under some natural assumptions. Along the way, we completely describe J-simple fiat 2-categories which have only one two-sided cell J apart from the identities, under the same conditions as above. For positively graded 2-categories, we show that the additional restrictions are redundant.