Simple transitive $2$-representations of some $2$-categories of projective functors

TL;DR

This paper proves that all simple transitive 2-representations of a specific 2-category of projective functors are equivalent to cell 2-representations, clarifying their structure in a particular algebraic setting.

Contribution

It establishes the equivalence of simple transitive 2-representations to cell 2-representations for a class of 2-categories related to projective functors.

Findings

All simple transitive 2-representations are equivalent to cell 2-representations.

The result applies to a 2-category associated with a quotient of the quadratic dual of a preprojective algebra.

Provides a classification of simple transitive 2-representations in this context.

Abstract

We show that every simple transitive $2$-representation of the $2$-category of projective functors for a certain quotient of the quadratic dual of the preprojective algebra associated with a tree is equivalent to a cell $2$-representation.

Equations

$$v_1$$

$$v$$

$$1$$

$$3$$

$$\boxed{3}$$

$$a_2{a}_1$$

$$a_1{b}_1$$

$$P^{-1}IP=\left(\begin{array}{ccc}0& AJ& AJB\\ 0& J& JB\\ 0& 0& 0\end{array}\right)\text{ with }J=\left(\begin{array}{cccc}{J}_1& 0& \cdots & 0\\ 0& J_2& & ⋮\\ ⋮& & \ddots & 0\\ 0& \cdots & 0& J_k\end{array}\right).$$

$$k_i=\{\begin{array}{ll}2,& i\notin S;\\ 1,& i\in S.\end{array}$$

$$\frac{1}{k_i}[F_{ii}]=\left(\begin{array}{ccc}0& AJ& AJB\\ 0& J& AB\\ 0& 0& 0\end{array}\right).$$

$$\frac{1}{k_i}[F_{ii}]=\left(\begin{array}{cc}0& 0\\ 0& J\end{array}\right).$$

$$\mathbf{M}(F_{ii})=\mathbf{M}(F_{jj})=\mathbf{M}(F_{ij})=\mathbf{M}(F_{ji})=G_{ss}.$$

$$\dim e_{l}A{e}_k=\{\begin{array}{ll}2,& k=l\notin S,\\ 1,& \{k,l\}\in E\text{ or }k=l\in S,\\ 0,& \text{else}.\end{array}$$

$$G_{ss}^{\oplus 2}$$

$$=\mathbf{M}(F_{ji})\mathbf{M}(F_{ji})=\mathbf{M}(F_{ji}\circ F_{ji})=\mathbf{M}(F_{ji})=G_{ss}.$$

$$F_{si}\circ F_{jt}\cong F_{st}^{\oplus \dim e_{i}A{e}_j}\text{ with }{G}_{si}\circ G_{jt}\cong G_{st}^{\oplus \dim {ϵ}_{i}B{ϵ}_j}$$

Full text

1. Introduction

In [MM1], Mazorchuk and Miemietz started a systematic study of $2$-representations for certain $2$-categories which should be thought of as analogues of finite dimensional algebras. They introduced the notion of cell $2$-representations as a possible $2$-analogue of the notion of simple modules. This was revised in [MM5, MM6] where the notion of a simple transitive $2$-representation was introduced. A weak version of the Jordan-Hölder theory was developed in [MM5] for simple transitive $2$-representations which was a convincing argument that simple transitive $2$-representations are proper $2$-analogues of simple modules. In many important cases, for example for the $2$-category of Soergel bimodules in type $A$, it turns out that every simple transitive $2$-representations is equivalent to a cell $2$-representation.

Another class of natural $2$-categories for which every simple transitive $2$-representations is equivalent to a cell $2$-representation is the class of $2$-categories of projective bimodules for a finite dimensional self-injective associative algebra, see [MM5, MM6]. After [MM5, MM6] there were several attempts to extend this results to other associative algebras. Two particular algebras were considered in [MZ1] and one more in [MMZ]. These two papers have rather different approaches: the approach of [MZ1] is based on existence of a non-zero projective-injective module while [MMZ] treats the smallest algebra which does not have any projective-injective modules. Recently, [MZ2] extended the approach of [MZ1] and completely covered the case of directed algebras which have a non-zero projective-injective module. We refer the reader to [Maz] for a general overview of the problem and related results.

In this note we show that the method developed in [MZ2] can also be extended to some interesting algebras which are not directed (but which have a non-zero projective-injective module). The algebras we consider are certain quotients of quadratic duals of preprojective algebras associated with trees (cf. [Ri]). These kinds of algebras appear naturally in Lie theory (see [St, Mar]), in diagram algebras (see [HK]) and in the theory of Koszul algebras (see [Du]). Our main result is that, for our algebras (which are defined in Subsection 2.1), every simple transitive $2$-representation of the corresponding $2$-category of projective bimodules is equivalent to a cell $2$-representation.

The paper is organized as follows. In the next section we define the type of algebras which we want to study, describe some motivating examples and give all the necessary notions needed to formulate the main result. Section 3 is then devoted to stating and proving the main result.

2. Preliminaries

Throughout the paper we work over an algebraically closed field $\Bbbk$.

2.1. The algebra $A_{T,S}$

Let $n$ be a positive integer. Let $T=(V,E)$ be a tree with vertices labeled by numbers $1,2,\ldots,n$, where $n>1$. We denote by $L\subseteq V$ the set of all leaves of $T$. Denote by $Q=Q_{T}=(V,\hat{E})$ the quiver were we replace every (unoriented) edge $\{i,j\}\in E$, by two arrows (i.e. oriented edges) $(i,j)$ and $(j,i)$. Let $\Bbbk Q$ be the path algebra of $Q$.

Now we define a certain quotient $A_{T,S}$ of $\Bbbk Q$. For this, fix a (possibly empty) subset $S$ of $L$, and consider the ideal $\mathcal{I}$ of $\Bbbk Q$ generated by the following relations:

• •

For all pairwise distinct $v_{1},v_{2},v_{3}\in V$ such that there are arrows $a_{1},a_{2}\in\hat{E}$ with

[TABLE]

we set $a_{2}a_{1}=0$.

• •

For all pairwise distinct vertices $v_{1},v_{2},v_{3}\in V$ such that there exist arrows $a_{1},a_{2},b_{1},b_{2}\in\hat{E}$ with

[TABLE]

we set $a_{1}b_{1}=b_{2}a_{2}$.

• •

For $v\in V$ and $s\in S$ such that there are arrows $a,b\in\hat{E}$ with

[TABLE]

we set $ab=0$.

The algebra $A_{T,S}$, which we will denote simply by $A$, is now defined as the quotient of $\Bbbk Q$ by the ideal $\mathcal{I}$. We denote the idempotents of $A$ by $e_{i}$, for each $i\in V$. For $i\in V$, we set $P_{i}:=Ae_{i}$ and denote by $L_{i}$ the simple top of $P_{i}$.

The structure of projective $A$-modules follows directly from the defining relations:

• •

If $i\in V\setminus S$, then $P_{i}$ is projective-injective of Loewy length three with isomorphic top and socle. The module $\mathrm{Rad}(P_{i})/\mathrm{Soc}(P_{i})$ is multiplicity-free and contains all simple $L_{j}$ such that $\{i,j\}\in E$.

• •

If $i\in S=V$, then $n=2$ and $P_{i}$ is projective-injective of Loewy length two with non-isomorphic top and socle.

• •

If $i\in S\neq V$, then $P_{i}$ is not injective, it has Loewy length two and its socle is isomorphic to $L_{j}$, where $j\in V$ is the unique vertex such that $\{i,j\}\in E$.

From the above description we see that the algebra $A$ is self-injective if and only if $S=\varnothing$ or $S=V$ (in the latter case we have $n=2$).

The motivation for the above definition stems from the following examples.

Example 2.1**.**

Let $T$ be the following Dynkin diagram of type $A_{n}$:

[TABLE]

We have $L=\{1,n\}$. Set $S:=\{n\}$. Then $Q_{T}$ is the following quiver:

[TABLE]

The distinguished leaf $n$ is the one which is in $S$. The relations in $A=A_{T,S}$ are given by {IEEEeqnarray*}r;c;l;r a_i+1a_i & = b_i+1b_i = 0,  i = 1, …, n-2;

a_ib_i = b_i+1a_i+1, i = 1, …, n-2;

a_n-1b_n-1 = 0.

The module category over this algebra is equivalent to the principal block of parabolic category $\mathcal{O}$ associated to the complex Lie algebra $\mathfrak{sl}_{n}$ and a parabolic subalgebra of $\mathfrak{sl}_{n}$ for which the semi-simple part of the Levi quotient is isomorphic to $\mathfrak{sl}_{n-1}$, see e.g. [St].

A second example is:

Example 2.2**.**

Let $T$ be the following tree on $4$ vertices

[TABLE]

We have $L=\{1,3,4\}$. Set $S:=\{3,4\}$. Then $Q_{T}$ looks as follows:

[TABLE]

The relations in $A_{T,S}$ are given by

[TABLE]

These kinds of quivers appear as parts of infinite quivers in e.g. [Mar].

2.2. The $2$-category $\mathscr{C}_{A}$

From now on we fix a tree $T$ and a subset $S$ of its leaves. Let $A=A_{T,S}$. For generalities on finitary $2$-categories, we refer the reader to [MM1].

Following [MM1, Subsection 7.3], we define the finitary $2$-category $\mathscr{C}_{A}$ of projective endofunctors of $A$-mod. Fix a small category $\mathcal{C}$ equivalent to $A$-mod. The $2$-category $\mathscr{C}_{A}$ has one object i, which we identify with $\mathcal{C}$. Indecomposable $1$-morphisms are endofunctors of $\mathcal{C}$ given by tensoring with:

• •

the regular $A$-$A$-bimodule ${}_{A}A_{A}$ (this corresponds to the identity $1$-morphism $\mathbbm{1}_{\texttt{i}}$);

• •

the indecomposable $A$-$A$-bimodule $Ae_{i}\otimes_{\Bbbk}e_{j}A$, for some $i,j\in\{1,2,\dots,n\}$, we denote such a $1$-morphism by $F_{ij}$.

Lastly, $2$-morphisms are homomorphisms of $A$-$A$-bimodules.

A finitary $2$-representation of $\mathscr{C}_{A}$ is a (strict) $2$-functor from $\mathscr{C}_{A}$ to the $2$-category of small finitary additive $\Bbbk$-linear categories. In other words, $\mathbf{M}$ is given by an additive and $\Bbbk$-linear functorial actions on a category $\mathbf{M}(\texttt{i})$ which is equivalent to the category $B$-proj of projective modules over some finite dimensional associative $\Bbbk$-algebra $B$. For $M\in\mathbf{M}(\texttt{i})$, we will often write $FM$ instead of $\mathbf{M}(F)(M)$. All finitary $2$-representation of $\mathscr{C}_{A}$ form a $2$-category $\mathscr{C}_{A}$-afmod where $1$-morphisms are strong $2$-natural transformations and $2$-morphisms are modifications, see [MM3, Section 2.3] for details.

We call a finitary $2$-representation $\mathbf{M}$ of $\mathscr{C}_{A}$ transitive if, for every non-zero object $X\in\mathbf{M}(\texttt{i})$, the additive closure of $\{FX\}$, where $F$ runs through all $1$-morphisms in $\mathscr{C}_{A}$, equals $\mathbf{M}(\texttt{i})$. We call a transitive $2$-representation $\mathbf{M}$ simple if $\mathbf{M}(\texttt{i})$ has no proper $\mathscr{C}_{A}$-invariant ideals. For more details on this, we refer the reader to [MM5, MM6].

One class of examples of simple transitive $2$-representations are so-called cell $2$-representations. For details about these we refer the reader to [MM1, MM2]. The $2$-category $\mathscr{C}_{A}$ has two two-sided cells: the first one consisting of $\mathbbm{1}_{\texttt{i}}$ and the second one containing all $F_{ij}$. The first two-sided cell is a left cell. The second two-sided cell contains $n$ different left cells, namely, $\mathcal{L}_{j}:=\{F_{ij}:i=1,2,\dots,n\}$, where $j=1,2,\dots,n$. Up to equivalence, we have two cell $2$-representations:

• •

The cell $2$-representations $\mathbf{C}_{\mathbbm{1}_{\texttt{i}}}$ which is given as the quotient of the left regular action of $\mathscr{C}_{A}$ on $\mathscr{C}_{A}(\texttt{i},\texttt{i})$ by the unique maximal $\mathscr{C}_{A}$-invariant ideal (cf. [MM2, Section 6]).

• •

Each cell $2$-representations $\mathbf{C}_{\mathcal{L}_{j}}$, where $j=1,2,\dots,n$, is equivalent to the defining $2$-representation (the defining action on $\mathcal{C}$).

2.3. Positive idempotent matrices

One of the ingredients in our proofs is the following classification of non-negative idempotent matrices, see [Fl].

Theorem 2.3**.**

Let $I$ be a non-negative idempotent matrix of rank $k$. Then there exists a permutation matrix $P$ such that

[TABLE]

Here, each $J_{i}$ is a non-negative idempotent matrix of rank one and $A,B$ are non-negative matrices of the appropriate size.

Remark 2.4**.**

This theorem can be applied to quasi-idempotent (but not nilpotent) matrices as well. If $I^{2}=\lambda I$ and $\lambda\neq 0$, then $(\frac{1}{\lambda}I)^{2}=\frac{1}{\lambda^{2}}I^{2}=\frac{1}{\lambda}I$. Hence $\frac{1}{\lambda}I$ is an idempotent and thus can be described by the above theorem.

3. Main result

Fix a tree $T$ and a subset $S$ of its leaves and set $A=A_{T,S}$. Then our main result can be stated as follows:

Theorem 3.1**.**

Let $\mathbf{M}$ be a simple transitive $2$-representation of $\mathscr{C}_{A}$, then $\mathbf{M}$ is equivalent to a cell $2$-representation.

Note that, if $S=\varnothing$ or $S=V$, then the algebra $A$ is self-injective and hence $\mathscr{C}_{A}$ is a weakly fiat $2$-category. In this case the statement follows from [MM5, Theorem 15] and [MM6, Theorem 33]. Therefore, in what follows, we assume that $S\neq\varnothing,V$.

3.1. Some notation

For a simple transitive $2$-representation $\mathbf{M}$ of $\mathscr{C}_{A}$, we denote by $B$ a basic $\Bbbk$-algebra such that $\mathbf{M}(\texttt{i})$ is equivalent to $B$-proj. Moreover, let $1=\epsilon_{1}+\epsilon_{2}+\cdots+\epsilon_{r}$ be a decomposition of the identity in $B$ into a sum of pairwise orthogonal primitive idempotents. Similarly to the situation in $A$, we denote, for $1\leq i,j\leq r$, by $G_{ij}$ the endofunctor of $B$-mod given by tensoring with the indecomposable projective $B$-$B$-bimodule $B\epsilon_{i}\otimes\epsilon_{j}B$. Note that, a priori, there is no reason why we should have $r=n$.

For $i=1,2,\dots,r$, we denote by $Q_{i}$ the projective $B$-module $B\epsilon_{i}$.

We may, without loss of generality, assume that $\mathbf{M}$ is faithful since $\mathscr{C}_{A}$ is simple which was shown in [MMZ, Subsection 3.2]. Indeed, if we assume that $\mathbf{M}$ is not faithful, then $M(F_{ij})=0$, for all $i,j$. However, then the quotient of $\mathscr{C}_{A}$ by the ideal generated by all $F_{ij}$ satisfies all the assumptions of [MM5, Theorem 18] and therefore $\mathbf{M}$ is equivalent to the cell $2$-representation $\mathbf{C}_{\mathbbm{1}_{\texttt{i}}}$ in this case.

So let us from now on assume that $\mathbf{M}$ is faithful and, in particular, that all $\mathbf{M}(F_{ij})$ are non-zero. As we have seen above, $A$ has a non-zero projective-injective module and thus, combining [MZ1, Section 3] and [KMMZ, Theorem 2], we deduce that each $\mathbf{M}(F_{ij})$ is a projective endofunctor of $B$-mod and, as such, is isomorphic to a non-empty direct sum of $G_{st}$, for some $1\leq s,t\leq r$, possibly with multiplicities.

3.2. The sets $X_{i}$ and $Y_{i}$

Following [MZ2], for $1\leq i,j\leq n$, we define

• •

$X_{ij}:=\{s\;|\;$ $G_{st}$ is isomorphic to a direct summand of $\mathbf{M}(F_{ij})$, for some $1\leq t\leq r\}$,

• •

$Y_{ij}:=\{t\;|\;$ $G_{st}$ is isomorphic to a direct summand of $\mathbf{M}(F_{ij})$, for some $1\leq s\leq r\}$.

First of all, note that $X_{ij}$ and $Y_{ij}$ are non-empty as each $\mathbf{M}(F_{ij})$ is non-zero due to faithfulness of $\mathbf{M}$.

In [MZ2, Lemma 20], it is shown that $X_{ij_{1}}=X_{ij_{2}}$, for all $j_{1},j_{2}\in\{1,\dots,n\}$, and thus we may denote by $X_{i}$ the common value of all $X_{ij}$. Similarly, the sets $Y_{ij}$ only depend on $j$, hence we may denote by $Y_{j}$ the common value of the $Y_{ij}$, for all $i$. In [MZ2, Lemma 22, Lemma 21], it is shown that $X_{q}=Y_{q}$, for all $q$ and moreover that $X_{1}\cup X_{2}\cup\dots\cup X_{n}=\{1,2,\dots,r\}$.

3.3. Analysis of the sets $X_{i}$

For a $1$-morphism $H$ in $\mathscr{C}_{A}$, we will denote by $[H]$ the $r\times r$ matrix with coefficients $h_{st}$, where $s,t\in\{1,2,\dots,r\}$, such that $h_{st}$ gives the multiplicity of $Q_{s}$ in $HQ_{t}$.

Lemma 3.2**.**

For each $i\in\{1,\ldots,n\}$, we have $|X_{i}|=1$,

Proof.

First we note that, for all $1\leq i\leq n$, we have $F_{ii}\circ F_{ii}\cong F_{ii}^{\oplus\dim(e_{i}Ae_{i})}$ and hence $[F_{ii}]^{2}=k_{i}[F_{ii}]$, where

[TABLE]

Hence we can apply Theorem 2.3 to $\frac{1}{k_{i}}[F_{ii}]$. This yields that there exists an ordering of the basis vectors such that

[TABLE]

However, we have seen that $X_{i}=Y_{i}$, for any ordering of the basis. This implies that the if the $l$-th row of $\frac{1}{k_{i}}[F_{ii}]$ is zero, then so is the $l$-th column. Thus we get that $A=B=0$ and, in particular, that

[TABLE]

If $i\in S$, we are done as $k_{i}=1$ and thus the trace of the corresponding matrix is $1$ which yields that $[F_{ii}]$ has to contain exactly one non-zero diagonal element and thus $|X_{i}|=1$.

Let now $i\not\in S$. We may restrict the action of $\mathscr{C}_{A}$ to the $2$-full finitary $2$-subcategory $\mathscr{D}$ of $\mathscr{C}_{A}$ whose indecomposable $1$-morphisms are the ones which are isomorphic to either $\mathbbm{1}_{\texttt{i}}$ or $F_{ii}$. This $2$-category, clearly, has only strongly regular two-sided cells. As $i\not\in S$, the projective module $P_{i}$ is also injective and hence $F_{ii}$ is a self-adjoint functor (see [MM1, Subsection 7.3]). Therefore $\mathscr{D}$ satisfies all assumptions of [MM5, Theorem 18] and hence every simple transitive $2$-representation of $\mathscr{D}$ is equivalent to a cell $2$-representation.

The $2$-category $\mathscr{D}$ has two left cells (both are also two-sided cells) and each left cell contains a unique indecomposable $1$-morphism. The matrix of $F_{ii}$ in these $2$-representations is either $(0)$ or $(2)$. This implies that, for $i\not\in S$, all diagonal elements in $[F_{ii}]$ are either equal to [math] or to $2$. As $[F_{ii}]$ has trace $2$, it follows again that $[F_{ii}]$ contains a unique non-zero diagonal element and thus $|X_{i}|=1$. ∎

Next we are going to prove that the $X_{i}$’s are mutually disjoint.

Lemma 3.3**.**

For $i,j\in\{1,\ldots,n\}$ such that $i\neq j$, we have $X_{i}\cap X_{j}=\varnothing$.

Proof.

Let $i,j\in\{1,\ldots,n\}$ be such that $i\neq j$ and assume that $X_{i}\cap X_{j}\neq\varnothing$. This implies, by Lemma 3.2, that $X_{i}=X_{j}$ is a singleton, call it $X_{i}=\{s\}$. By the above, we have that $Y_{i}=X_{i}=X_{j}=Y_{j}=\{s\}$. This implies that

[TABLE]

We have, for any $1\leq k,l\leq n$,

[TABLE]

On the one hand, we know that $\dim{\epsilon_{s}B\epsilon_{s}}\geq 1$ and thus $G_{ss}\circ G_{ss}\neq 0$. On the other hand, we have that $F_{ij}\circ F_{ij}=F_{ij}^{\oplus\dim(e_{j}Ae_{i})}$. This implies that $\dim(e_{j}Ae_{i})\neq 0$ and hence $\{i,j\}\in E$, because of (2). Further, as we assume that $S\neq V$, we also have $\{i,j\}\not\subset S$. Let us assume that $i\not\in S$.

As $i\not\in S$, (2) yields the following:

[TABLE]

As $G_{ss}\neq 0$, this equality is impossible. The obtained contradiction proves our claim. ∎

From the above, we have $n=r$ and, without loss of generality, we may assume $X_{i}=\{i\}$, for all $i=1,2,\dots,n$.

Corollary 3.4**.**

For $i,j=1,2,\ldots,n$, we have $\dim{e_{i}Ae_{j}}=\dim{\epsilon_{i}B\epsilon_{j}}$.

Proof.

This follows immediately since every $F_{st}$ acts via $G_{st}$, by comparing

[TABLE]

∎

3.4. Proof of Theorem 3.1

With the results of Subsection 3.3 at hand, the proof of Theorem 3.1 can now be done using similar arguments as used in [MZ2, Section 5] or [MaMa, Subsection 4.9]. Consider the principal $2$-representation $\mathbf{P}_{\texttt{i}}:=\mathscr{C}_{A}(\texttt{i},\_)$ of $\mathscr{C}_{A}$, that is the regular action of $\mathscr{C}_{A}$ on $\mathscr{C}_{A}(\texttt{i},\texttt{i})$. Set $\mathbf{N}:=\text{add}(F_{i1})$, where $i=1,2,\ldots,k$, be the additive closure of all $F_{i1}$. Now, $\mathbf{N}$ is $\mathscr{C}_{A}$-stable and thus gives rise to a $2$-representation of $\mathscr{C}_{A}$. By [MM2, Subsection 6.5], we have that there exists a unique $\mathscr{C}_{A}$-stable left ideal $\mathbf{I}$ in $\mathbf{N}$ and the corresponding quotient is exactly the cell $2$-representation $\mathbf{C}_{\mathcal{L}_{1}}$.

Now, mapping $\mathbbm{1}_{\texttt{i}}$ to the simple object corresponding to $Q_{1}$ in the abelianization of $\mathbf{M}$, induces a $2$-natural transformation $\Phi:\mathbf{N}\to\mathbf{M}$. Due to the results of the previous subsection, we know that $\Phi$ maps indecomposable $1$-morphisms in $\mathcal{L}_{1}$ to indecomposable objects in $\mathbf{M}$ inducing a bijection on the corresponding isomorphism classes. By uniqueness of the maximal ideal, the kernel of $\Phi$ is contained in $\mathbf{I}$. However, by Corollary 3.4, the Cartan matrices of $A$ and $B$ are the same. This implies that, on the one hand, the kernel of $\Phi$ cannot be smaller than $\mathbf{I}$ and, on the other hand, that $\Phi$ must be full. Therefore $\Phi$ induces an equivalence between $\mathbf{N}/\mathbf{I}\cong\mathbf{C}_{\mathcal{L}_{1}}$ and $\mathbf{M}$. The claim of the theorem follows.

Acknowledgments. The author wants to thank his supervisor Volodymyr Mazorchuk for many helpful discussions.