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Title: Simple transitive $2$-representations via (co)algebra $1$-morphisms
Authors: Marco Mackaay, Volodymyr Mazorchuk, Vanessa Miemietz and Daniel Tubbenhauer
Status: Indiana Univ. Math. J. 68 (2019), no. 1, 1-33. Last update: Wed, 6 Feb 2019 09:54:48 UTC
ArXiv link: https://arxiv.org/abs/1612.06325
ArXiv version = 0.99 published version, but please use the arXiv version
LaTex Beamer presentation: Slides1, Slides2, Slides3, Slides4, Slides5, Slides6

Abstract

For any fiat $2$-category $\mathscr{C}$, we show how its simple transitive $2$-representations can be constructed using coalgebra $1$-morphisms in the injective abelianization of $\mathscr{C}$. Dually, we show that these can also be constructed using algebra $1$-morphisms in the projective abelianization of $\mathscr{C}$. We also extend Morita-Takeuchi theory to our setup and work out several examples explicitly.

A few extra words

The subject of $2$-representation theory is the higher categorical analogue of the classical representation theory of algebras. A systematic study of “finite” $2$-representation was initiated by Mazorchuk and Miemietz and is concerned about studying $2$-representations of so-called finitary $2$-categories.
In particular, for a given finitary $2$-category $\mathscr{C}$, there is an appropriate $2$-analog of simple representations, the so-called simple transitive $2$-representations. These $2$-representations have a Jordan-Hölder $2$-theory very much in the spirit of the classical Jordan-Hölder theory, i.e. one can see them as “atomic” $2$-representations.
This motivates the problem of classifying the simple transitive $2$-representations of a given finitary $2$-category $\mathscr{C}$. Sadly, this is a very hard problem in general.
In the present paper, we show that, for a given fiat $2$-category $\mathscr{C}$ (a fiat $2$-category is a finitary $2$-category with additional structure making classifications easier), there is a bijection between the equivalence classes of simple transitive $2$-representations of $\mathscr{C}$ and certain equivalence classes of so-called simple algebra $1$-morphisms. (This is a generalization of a similar story for tensor categories where so-called algebra objects correspond to our algebra $1$-morphisms in $2$-categories with one object.)
It is worth emphasizing that the approach using algebra $1$-morphism does not seem to be very helpful for the classification of $2$-representations, because the classification of algebra $1$-morphisms looks like a very hard problem in general. However, this method is quite helpful if one would like to check existence of some $2$-representations, since this can be reformulated into the problem of checking that certain $1$-morphisms have an additional structure of an algebra $1$-morphism. This is sometimes quite easy, e.g. if algebra $1$-morphisms decategorify to idempotents in the Grothendieck group, then this basically fixes the algebra structure.
Here an explicit example. Let $D_n$ be the dihedral group with $2n$ elements for $n>2$. Fix its usual Coxeter presentation \[ D_n=\langle s,t\mid s^2=t^2=1, \underbrace{\dots sts}_n=\underbrace{\dots tst}_n\rangle. \]
Then the associated Hecke algebra $H_n$ is categorified by the monoidal category of Soergel bimodules $\mathscr{S}_n$, which we see as a $2$-category with one object. Take its small quotient $\mathscr{s}_n$ (which is basically the same $2$-category, but one kills the $2$-ideal spanned by the longest element $w_0$ of $D_n$).
$\mathscr{s}_n$ has one indecomposable $1$-morphism $B_w$ for each element $w\neq w_0$ in $D_n$. These decategorify to the corresponding Kazhdan-Lusztig basis elements of $H_n$.
Now, the $2$-representation theory of $\mathscr{s}_n$ is by our results controlled by algebra $1$-morphisms in $\mathscr{s}_n$. One can work out explicitly the following list (which, of course, exists in a “$t$-copy” as well). Hereby we write $s_j=\dots sts$ with $j$-symbols in total, “diagram” means the underlying quiver diagram of the module category of the algebra $1$-morphism in question, “dimension” is its dimension, $n$ is the $n$ from $\mathscr{s}_n$, and $k\geq 4$ for $\mathbf{D}_k$. \begin{array}{c|c|c|c} \mbox{Algebra }1\mbox{-morphism} & \mbox{Diagram} & \mathscr{s}_n & \mbox{Dimension} \\ \hline B_s & \mathbf{A}_k & n=k & k-1 \\ B_s\oplus B_{s_{n-1}} & \mathbf{D}_k & n=2k-2 & k \\ B_s\oplus B_{s_7} & \mathbf{E}_6 & n=12 & 6 \\ B_s\oplus B_{s_9} \oplus B_{s_{17}} & \mathbf{E}_7 & n=18 & 7 \\ B_s\oplus B_{s_{11}} \oplus B_{s_{19}} \oplus B_{s_{29}} & \mathbf{E}_8 & n=30 & 8 \\ \end{array} Hereby one easily checks that the type $\mathbf{A}$ and $\mathbf{D}$ algebra $1$-morphisms decent to idempotents in the Grothendieck group. This basically immediately shows that these are indeed algebra $1$-morphisms. In contrast, that the type $\mathbf{E}$ algebra $1$-morphisms are actually algebra $1$-morphisms is way more complicated to prove, and to see that the list above “is complete” requires also extra work (for both we refer to the last section of the paper).
Thus, all of these give non-equivalent (check the dimensions!) $2$-representations of $\mathscr{s}_n$. This gives us an $\mathbf{A}\mathbf{D}\mathbf{E}$ classification for the simple transitive $2$-representations of $\mathscr{s}_n$.
Bonus observation: It is no coincidence that the table above looks very similar to the table in the work of Kirillov-Ostrik https://arxiv.org/abs/math/0101219 on the classification of “finite subgroups of $\mathrm{U}_q(\mathfrak{sl}_2)$” for $q$ being a primitive, complex, even root of unity. (Check the last section of our paper!)