## Data

- 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!)