Cell 2-representations of finitary 2-categories

Abstract

We study 2-representations of finitary 2-categories with involution and adjunctions by functors on module categories over finite-dimensional algebras. In particular, we define, construct and describe in detail (right) cell 2-representations inspired by KazhdanLusztig cell modules for Hecke algebras. Under some natural assumptions we show that cell 2-representations are strongly simple and do not depend on the choice of a right cell inside a two-sided cell. This reproves and extends the uniqueness result on categorification of Kazhdan-Lusztig cell modules for Hecke algebras of type A from [V. Mazorchuk and C. Stroppel, Categorification of (induced) cell modules and the rough structure of generalised Verma modules, Adv. Math. 219 (2008Math. 219 ( ), 1363Math. 219 ( -1426.

Classification problems in 2-representation theory

“…Here we also note that many results in [40]- [44] assume that a certain numerical condition is satisfied. This assumption was rendered superfluous by [45,Proposition 1].…”

Transitive $2$-representations of finitary $2$-categories

“…More specifically, we study finitary 2-categories over an algebraically closed field which include the 2-category of Soergel bimodules associated to a finite Coxeter system (see [BG, So, EW]), an exhaustive family of quotients of 2-Kac-Moody algebras (see [BFK,KL,Ro1,CL,We]), quiver 2-categories constructed in [Xa] and the 2-category of projective functors on the module category of a finite dimensional algebra (see [MM1]). We define a new class of 2-representations for such 2-categories which we call simple transitive 2-representations and which we believe serves as the correct 2-analogue for the class of irreducible representations of an algebra.…”

Multisemigroups with multiplicities and complete ordered semi-rings

“…Consider the 2-category S 3 of Soergel bimodules over the coinvariant algebra of S 3 as detailed in, e.g., [Mazorchuk and Miemietz (2011) …”

“…We refer the reader to Mazorchuk and Miemietz (2016b) for details. The combinatorics of this multisemigroup reflects and encodes various structural properties of the underlying additive k-linear 2-category and controls major parts of the 2-representation theory of the latter, see Mazorchuk and Miemietz (2011, 2016a for details.…”

“…Let W be a finite Coxeter group with a fixed geometric representation. To these data, one associates the so-called 2-category S W of Soergel bimodules over the coinvariant algebra of the geometric representation, see Soergel (2007) and [Mazorchuk and Miemietz (2011), Sect. 7.1].…”