Bimodules over uniformly oriented A_n quivers with radical square zero

TL;DR

This paper classifies bimodule categories over certain finite-dimensional algebras derived from uniformly oriented A_n quivers with radical square zero, analyzing their structure and representation theory.

Contribution

It provides a detailed description of the cell structure, adjunctions, and simple transitive 2-representations of the tensor category of bimodules over these algebras.

Findings

Classification of connected finite-dimensional algebras with finitely many bimodule classes

Description of the cell structure of the bimodule tensor category

Proof that all simple transitive 2-representations are cell 2-representations

Abstract

We start with observing that the only connected finite dimensional algebras with finitely many isomorphism classes of indecomposable bimodules are the quotients of the path algebras of uniformly oriented $A_n$-quivers modulo the radical square zero relations. For such algebras we study the (finitary) tensor category of bimodules. We describe the cell structure of this tensor category, determine existing adjunctions between its $1$-morphisms and find a minimal generating set with respect to the tensor structure. We also prove that, for the algebras mentioned above, every simple transitive $2$-representation of the $2$-category of projective bimodules is equivalent to a cell $2$-representation.

Equations

$$1\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\left(\begin{array}[]{cc}0&1\\ 0&0\end{array}\right)\quad\text{ and }\quad\left(\begin{array}[]{cc}0&\lambda\\ 0&0\end{array}\right),\quad\text{ where }\lambda\in\Bbbk,$$

$$i\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\bullet \text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\bullet \text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\bullet \text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$1|1\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$1|\mathtt{n}\text{-}1$$

$$\frac{4n^3+3n^2-7n+3}{3}$$

$$\mathtt{i}\mid \mathtt{j}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 13.4167pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr\\\crcr}}}\ignorespaces{\hbox{\kern-8.41669pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathtt{i\mid j,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 0.0pt\raise-20.9445pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern-13.4167pt\raise-30.6231pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathtt{i\text{+}1\mid j}}$}}}}}}}\ignorespaces}}}}\ignorespaces\quad i\in\mathbb{N}_{n}^{\ast},j\in\mathbb{N}_{n+1}^{\ast};\qquad\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 7.02779pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-7.02779pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathtt{i\mid j}}$}}}}}}}{\hbox{\kern 23.02786pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathtt{i\mid j\text{+}1,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 7.0278pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces\quad i\in\mathbb{N}_{n+1}^{\ast},j\in\mathbb{N}_{n}^{\ast}.$$

$$\left(K_1{\otimes }_{\mathbb{k}}A\right){\otimes }_A\left(K_2{\otimes }_{\mathbb{k}}X\right)\cong {\left(K_1{\otimes }_{\mathbb{k}}X\right)}^{\oplus \dim K_2}.$$

$$\bullet \text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\mathbf{v}(N):=\text{max}\{\mathbf{v}(M):M\text{ is a string module and is as a direct summand of }N\}.$$

$$\mathtt{i}\mid \mathtt{j}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\mathtt{i}\mid \mathtt{j}$$

$$\mathbf{w}(X):=1+\max \{i\in \mathrm{Rsupp}(X)\}-\min \{i\in \mathrm{Rsupp}(X)\},$$

$$\mathbf{h}(X):=1+\max \{i\in \mathrm{Lsupp}(X)\}-\min \{i\in \mathrm{Lsupp}(X)\}.$$

$$\mathrm{Lsupp}(X{\otimes }_{A}Y)\subset \mathrm{Lsupp}(X)\text{ and }\mathrm{Rsupp}(X{\otimes }_{A}Y)\subset \mathrm{Rsupp}(Y).$$

$${\mathcal{J}}_{\mathbb{k}}{\ge }_J{\mathcal{J}}_0{\ge }_J{\mathcal{J}}_1{\ge }_J\cdots {\ge }_J{\mathcal{J}}_{n-1}.$$

$$e_i\mapsto \{\begin{array}{ll}{e}_{i+1},& i+1\le n;\\ 0,& \text{ else;}\end{array}{\alpha }_i\mapsto \{\begin{array}{ll}{\alpha }_{i+1},& i+1<n;\\ 0,& \text{ else.}\end{array}$$

$${}^{\phi }\tilde{e}A{\otimes }_{A}A{\tilde{e}}^{\phi }\to {}^{\phi }\tilde{e}A{\tilde{e}}^{\phi }$$

$$\mathrm{Lsupp}(A{\tilde{e}}^{\phi }{\otimes }_{A}X)=(\mathrm{Lsupp}(X))[1]\text{and}\mathrm{Rsupp}(A{\tilde{e}}^{\phi }{\otimes }_{A}X)=\mathrm{Rsupp}(X).$$

$$0\to Q\to A\to \mathrm{Coker}\to 0$$

$${\mathrm{Hom}}_{A\text{-}A}(Q{\otimes }_{A}X,Y)\cong {\mathrm{Hom}}_{A\text{-}A}(X,{\mathrm{Hom}}_{A\text{-}}(Q,Y)).$$

$${\mathrm{Hom}}_{A\text{-}A}(Q^{\prime }{\otimes }_{A}Y,Y)\cong {\mathrm{Hom}}_{A\text{-}A}(Y,{\mathrm{Hom}}_{A\text{-}}(Q^{\prime },Y)).$$

$${\mathrm{Hom}}_{A\text{-}A}(Q^{\prime }{\otimes }_{A}Y,Y)=0.$$

Full text

1. Introduction and description of the results

Finitary $2$-categories were introduced in [MM1] as “finite dimensional” counterparts of, in general, “infinite dimensional” $2$-categories which were studied in the categorification literature on the borderline between algebra and topology during the last twenty years, see [BFK, CR, Kh, KL, Ro, St]. The series [MM1, MM2, MM3, MM4, MM5, MM6] of papers develops basics of abstract representation theory for finitary $2$-categories. Classical examples of finitary $2$-categories are: the $2$-category of Soergel bimodules over the coinvariant algebra of a finite Weyl group, see [MM1, Subsection 7.1], and the $2$-category of projective functors over finite dimensional associative algebra, see [MM1, Subsection 7.3]. Further examples of finitary $2$-categories were constructed and studied in [GM1, GM2, Xa, Zh1, Zh2], see also the above mentioned series [MM1]–[MM6].

In the present paper we consider a new natural class of examples of finitary $2$-categories. The paper started from the following question:

For which finite-dimensional algebras, the corresponding tensor category of bimodules is finitary?

Although we suspect that the answer to this question could be known to specialists in representation type, we did not manage to find any explicit answer in the literature. In Theorem 1 we show that the only finite dimensional algebras over an algebraically closed field, for which the number of isomorphism classes of bimodules is finite, are those algebras whose connected components are radical square zero quotients of uniformly oriented path algebras of type $A$ Dynkin quivers. This result motivates the rest of the paper where we take a closer look at the tensor category of bimodules over such algebras.

We start with an attempt to understand the combinatorics of left-, right- and two-sided cells of this tensor category. These cells are natural generalizations of Green’s relations for semigroups from [Gr] to the setup of tensor categories, see [MM2, Section 3]. Apart from $\Bbbk$-split bimodules, that is bimodules of the form $X\otimes_{\Bbbk}Y$, where $X$ is a left module and $Y$ is a right module, see [MMZ], the remaining bimodules can be split into four families, which we call $W$, $M$, $N$ and $S$, motivated by the shape of the diagram of a bimodule. All $\Bbbk$-split bimodules always form the maximum two-sided cell which is easy to understand.

To describe the remaining structure, we introduce several combinatorial invariants of bimodules, called left support and right support and also the number of valleys in the diagram of a bimodule. We show that these invariants, in combination with bimodule types, classify left, right and two-sided cells. For example, two-sided cells are classified, in the case of non $\Bbbk$-split bimodules, by the number of valleys in bimodule diagrams. This result is a first step in understanding combinatorial structure for bimodule categories over arbitrary finite dimensional algebras, the latter question forming the core of our motivation. We also give an explicit description for all adjoint pairs of functors formed by our bimodules. This description covers only some bimodules as, in general, the right adjoint of tensoring with a bimodule is not exact and hence is not isomorphic to tensoring with some (possibly different) bimodule.

Furthermore, we find a minimal generating set for our tensor category, with respect to the tensor structure. It consists of the identity bimodule and three additional bimodules in the two-sided cell closest to the one formed by the identity bimodule, with respect to the two-sided order. To prove the statement, we give explicit formulae for tensor products of each of these three bimodules with all other indecomposable bimodules.

Finally, we study simple transitive of the $2$-category of projective bimodules over our algebras. Classification of such $2$-representations is a natural problem which was considered, for various classes of $2$-categories in [MM5, MM6, Zh1, Zi, MZ, MaMa, KMMZ, MT, MMMT, MMZ]. It also has interesting applications, see [KiM1]. A natural class of simple transitive $2$-representations is given by the so-called cell $2$-representations constructed in [MM1, MM2]. For the $2$-category of projective bimodules over a finite dimensional algebra $A$, it is known that cell $2$-representations exhaust all simple transitive $2$-representations if $A$ is self-injective (see [MM5, MM6]), if $A=\Bbbk[x,y]/(x^{2},y^{2},xy)$ (see [MMZ]) and if $A$ is the radical square zero quotient of the path algebra of a uniformly oriented quiver of type $A_{2}$ or $A_{3}$ (see [MZ]). In the present paper we extend this result to all directed algebras admitting a non-zero projective-injective module, see Theorem 19. We recover, with a much shorter and much more elegant proof, the main results of [MZ], however, our approach is strongly inspired by [MZ, Section 3]. Our approach to the proof of Theorem 19 contains some new general ideas which could help to attack similar problems for other finitary $2$-categories. We note that there are natural examples of $2$-categories which have simple transitive $2$-representations that are not equivalent to cell $2$-representations, see [MaMa, KMMZ, MT, MMMT].

The paper is organized as follows: Section 2 contains the material related to the formulation and proof of Theorem 1. Section 3 studies combinatorics of bimodules. The main results of this section which provide a combinatorial description of the cell structure are collected in Subsection 3.5. In Theorem 15 of Section 4, we describe a minimal generating set of our tensor category with respect to the tensor structure. Finally, Section 5 contains the material related to classification of simple transitive $2$-representations.

Acknowledgements: This research was partially supported by the Swedish Research Council, Knut and Alice Wallenberg Stiftelse and Göran Gustafsson Stiftelse.

2. Characterization via representation type

2.1. Main object of study

Throughout the paper we fix an algebraically closed field $\Bbbk$. For $n\in\{1,2,3,\dots\}$, we denote by $A_{n}$ the quotient of the path algebra of the quiver

[TABLE]

modulo the relations that the product of any two arrows is zero. In particular, we have $A_{1}\cong\Bbbk$, $A_{2}$ is isomorphic to the algebra of upper triangular $2\times 2$ matrices with coefficients in $\Bbbk$ and $\mathrm{Rad}(A_{n})^{2}=0$, for any $n$. If $n$ is fixed or clear from the context, we will simply write $A$ for $A_{n}$.

We denote by

• •

$A$-mod the category of finite dimensional left $A$-modules;

• •

mod-$A$ the category of finite dimensional right $A$-modules;

• •

$A$-mod-$A$ the category of finite dimensional $A$-$A$-bimodules.

For $i=1,2,\dots,n$, we denote by $e_{i}$ the trivial path at the vertex $\mathtt{i}$. Thus we have a primitive decomposition $1=e_{1}+e_{2}+\dots+e_{n}$ of the identity $1\in A$. Then $P_{i}=Ae_{i}$ is an indecomposable projective in $A$-mod and we denote by $L_{i}$ the simple top of $P_{i}$. Further, we denote by $I_{i}$ the indecomposable injective envelope of $L_{i}$. Note that $P_{i}$ has dimension $2$, for $i=1,2,\dots,n-1$, and $P_{n}=L_{n}$. Similarly, $I_{i}$ has dimension $2$, for $i=2,3,\dots,n$, and $I_{1}=L_{1}$. Moreover, $P_{i}\cong I_{i+1}$, for $i=1,2,\dots,n-1$.

2.2. Bimodule representation type

The main result of this subsection is the following statement. We suspect that this claim should be known to experts, but we failed to find any explicit reference in the literature.

Theorem 1**.**

Let $B$ be a finite dimensional associative $\Bbbk$-algebra. Then the following conditions are equivalent:

1. $($a$)$

The category $B$-mod-$B$* has finitely many isomorphism classes of indecomposable objects.* 2. $($b$)$

Each connected component of $B$ is Morita equivalent to some $A_{n}$.

As $B$-mod-$B$ is equivalent to $B\otimes_{\Bbbk}B^{\mathrm{op}}$-mod, condition (a) is equivalent to saying that $B\otimes_{\Bbbk}B^{\mathrm{op}}$ is of finite representation type.

2.3. Proof of the implication (a)$\Rightarrow$(b)

Let $B$ be a basic finite dimensional associative $\Bbbk$-algebra such that the category $B$-mod-$B$ has finitely many isomorphism classes of indecomposable objects. Note that this, in particular, implies that $B$ has finite representation type. Consider the Gabriel quiver $Q=(Q_{0},Q_{1})$ of $B$, where $Q_{0}$ is the set of vertices and $Q_{1}$ the set of arrows. Then, for any $i,j\in Q_{0}$, we have at most one arrow from $i$ to $j$ for otherwise $B$ would surject onto the Kronecker algebra and thus have infinite representation type.

Next we claim that $Q$ has no loops. Indeed, if $Q$ would have a loop, $B$ would have a quotient isomorphic to the algebra $D:=\Bbbk(x)/(x^{2})$ of dual numbers. However, the algebra $D\otimes_{\Bbbk}D^{\mathrm{op}}$ has a quotient isomorphic to $\Bbbk(x,y)/(x^{2},y^{2},xy)$ and the latter has infinite representation type since we have an infinite family of pairwise non-isomorphic indecomposable $2$-dimensional modules of this algebra on which $x$ and $y$ act via

[TABLE]

respectively (note that $\Bbbk$, being algebraically closed, is infinite). This is a contradiction.

Next we claim that each vertex of $Q$ has indegree at most $1$ and outdegree at most $1$. We prove the claim for indegrees and, for outdegrees, the proof is similar. If $Q$ has a vertex with indegree at least two, then, taking the above into account, $Q$ has a subgraph of the form:

[TABLE]

Then $B$ has a quotient isomorphic to the path algebra $C$ of (2). We claim that $C\otimes_{\Bbbk}C^{\mathrm{op}}$ has infinite representation type thus giving us a contradiction. Indeed, $C\otimes_{\Bbbk}C^{\mathrm{op}}$ is isomorphic to the quotient of the path algebra of the solid part of the following quiver:

[TABLE]

modulo the commutativity relations indicated by dotted lines. In the middle of this picture we see an orientation of the affine Dynkin diagram of type $\tilde{D}_{4}$. The corresponding centralizer subalgebra thus has infinite representation type. It follows that $C\otimes_{\Bbbk}C^{\mathrm{op}}$ has infinite representation type.

Next we claim that $Q$ does not have any components of the form

\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces} . If the latter were the case, then $B\otimes_{\Bbbk}B^{\mathrm{op}}$ would have a quotient isomorphic to the path algebra of

[TABLE]

modulo the relations that all squares commute. The latter has a quotient isomorphic to the path algebra corresponding to the following orientation

[TABLE]

of an affine Dynkin quiver of type $\tilde{A}_{4}$ and thus has infinite representation type, a contradiction.

The above shows that $Q$ is a disjoint union of graphs of the form (1). We now only need to show that $\mathrm{Rad}(B)^{2}=0$. If $\mathrm{Rad}(B)^{2}\neq 0$, then $B$ has a quotient which is isomorphic to the path algebra $F$ of (1), for $n=3$. Then $F\otimes_{\Bbbk}F^{\mathrm{op}}$ is isomorphic to the quotient of the path algebra of the solid part of the following quiver:

[TABLE]

modulo the commutativity relations indicated by dotted lines. Similarly to the previous paragraph, this algebra has a centralizer subalgebra which is the path algebra of a type $\tilde{D}_{4}$ quiver. Therefore it has infinite representation type.

2.4. Proof of the implication (b)$\Rightarrow$(a)

Note that $A_{n}/(e_{n})\cong A_{n-1}$, for any $n$. Therefore, for all $m,n$, there is a full embedding of the category of $A_{n}$-$A_{m}$-bimodules into the category of $A_{k}$-$A_{k}$-bimodules, where $k=\max(m,n)$. Using additivity and the fact that Morita equivalence, by definition, does not affect representation type, we obtain that it is sufficient to prove that the category $A_{n}$-mod-$A_{n}$ has finitely many isomorphism classes of indecomposable objects, for every $n$.

The algebra $A_{n}\otimes_{\Bbbk}A_{n}^{\mathrm{op}}$ is the quotient of the path algebra of the following quiver:

[TABLE]

modulo the relations that all squares commute and the product of any two horizontal or any two vertical arrows is zero.

The algebra $A_{n}\otimes_{\Bbbk}A_{n}^{\mathrm{op}}$ is thus a special biserial algebra in the sense of [BR, WW]. According to these two references, each indecomposable module over a special biserial algebra is either a string modules or a band module or a non-uniserial projective-injective module. In our case, there are certainly finitely many indecomposable non-uniserial projective-injective modules (they correspond to commutative squares in our quiver).

We claim that, in the case of $A_{n}\otimes_{\Bbbk}A_{n}^{\mathrm{op}}$, there are only finitely many string modules (up to isomorphism) and that there are no band modules. This follows from the form of the relations. Indeed, the maximal (with respect to inclusion) strings avoiding zero relations are:

[TABLE]

and so on (in total, there are $2n-2$ such maximal strings). We see that edges of these strings never intersect and that the strings never close into bands (corresponding to primitive cyclic words in some references). Consequently, there are no band modules and only finitely many string modules. The claim follows.

An exact enumeration of isomorphism classes of indecomposable objects in the category $A_{n}$-mod-$A_{n}$ is given in the next subsection.

2.5. Enumeration of indecomposable $A_{n}$-$A_{n}$-bimodules

Proposition 2**.**

For $n=1,2,\dots$, the category $A_{n}$-mod-$A_{n}$ contains exactly

[TABLE]

isomorphism classes of indecomposable objects.

Proof.

We have $(n-1)^{2}$ indecomposable projective-injective objects in $A_{n}$-mod-$A_{n}$. From Subsection 2.4, we know that the remaining indecomposable objects correspond to string $A_{n}\otimes_{\Bbbk}A_{n}^{\mathrm{op}}$-module. A string module is uniquely identified by the string the module is supported at, that is by a substring of one of the maximal string as shown in (4).

There are two maximal strings with three vertices, two maximal strings with $5$ vertices and so on. The maximal number of vertices on a maximal string is $2n-1$. A string with $k$ vertices supports $k(k+1)/2$ string modules (which correspond to connected substrings). Note that simple modules are supported just by vertices and there are $n^{2}-2$ vertices, that is all but the left upper and right lower corners, which belong to two different maximal substrings and hence are counted twice above. Putting all this together and simplifying gives the necessary formula. ∎

3. Combinatorics of $A$-$A$-bimodules

3.1. Cells

The main aim of the section is to describe the cell combinatorics of $A$-$A$-bimodules in the sense of [MM2, Section 3]. Denote by $\mathcal{S}$ the set of isomorphism classes of indecomposable $A$-$A$-bimodules. Recall that the left preorder $\leq_{L}$ is defined as follows: for $X,Y\in\mathcal{S}$ we have $X\leq_{L}Y$ provided that $Y$ is isomorphic to a direct summand of $Z\otimes_{A}X$, for some $A$-$A$-bimodule $X$. An equivalence class with respect to $\leq_{L}$ is called a left cell and the corresponding equivalence relation is denoted by $\sim_{L}$. The right preorder $\leq_{R}$ and the corresponding right cells and $\sim_{R}$ are defined similarly using tensoring over $A$ from the right. The two-sided preorder $\leq_{J}$ and the corresponding two-sided cells and $\sim_{J}$ are defined similarly using tensoring over $A$ from both sides.

3.2. $\Bbbk$-split $A$-$A$-bimodules

For simplicity, in this section we will denote by $\mathbb{N}_{n}$ the set $\{0,1,2,\dots,n-1\}$ and $\mathbb{N}_{n}^{\ast}$ the set $\{1,2,\dots,n-1\}$.

An $A$-$A$-bimodule $X$ is called $\Bbbk$-split, cf. [MMZ], provided that $X$ is isomorphic to a direct sum of $A$-$A$-bimodules of the form $M\otimes_{\Bbbk}N$, where $M\in A$-mod and $N\in\mathrm{mod}\text{-}A$.

We will often argue using bimodule action graphs. We will depict the left action using vertical arrows and the right action using horizontal arrows. Following the proof of Proposition 2, we can now list all non-zero indecomposable $\Bbbk$-split $A$-$A$-bimodules (up to isomorphism) and describe their action graphs as follows:

• •

the projective-injective bimodules $Ae_{i}\otimes_{\Bbbk}e_{j+1}A$, where $i,j\in\mathbb{N}_{n}^{\ast}$:

[TABLE]

• •

the simple bimodules $\mathtt{i\mid j}$, where $i,j\in\mathbb{N}_{n+1}^{\ast}$, that is, string $A\otimes_{\Bbbk}A^{\mathrm{op}}$-modules of dimension $1$ (we will denote the bimodule $\mathtt{i\mid j}$ by $L_{ij}$);

• •

string $A\otimes_{\Bbbk}A^{\mathrm{op}}$-modules of dimension $2$:

[TABLE]

Note that the projective $A$-$A$-bimodules $Ae_{n}\otimes_{\Bbbk}e_{j+1}A$, where $j\in\mathbb{N}_{n}^{\ast}$, and also $Ae_{i}\otimes_{\Bbbk}e_{1}A$, where $i\in\mathbb{N}_{n}^{\ast}$, belong to the last type of indecomposables (and they are not injective).

We denote by $\mathcal{J}_{\Bbbk}$ the set of all $\Bbbk$-split elements in $\mathcal{S}$.

Proposition 3**.**

1. $($i$)$

The set $\mathcal{J}_{\Bbbk}$ is the unique maximal two-sided cell in $\mathcal{S}$. 2. $($ii$)$

For each indecomposable right $A$-module $N$, the set of elements in $\mathcal{J}_{\Bbbk}$ of the form $X\otimes_{\Bbbk}N$, for some $X\in A$-mod, forms a left cell. Moreover, each left cell in $\mathcal{J}_{\Bbbk}$ is of such form. 3. $($iii$)$

For each indecomposable left $A$-module $K$, the set of elements in $\mathcal{J}_{\Bbbk}$ of the form $K\otimes_{\Bbbk}Y$, for some $Y\in\mathrm{mod}\text{-}A$, forms a right cell. Moreover, each right cell in $\mathcal{J}_{\Bbbk}$ is of such form.

Proof.

For $X\in\mathcal{J}_{\Bbbk}$ and $Y$ any $A$-$A$-bimodule, both $X\otimes_{A}Y$ and $Y\otimes_{A}X$ are obviously $\Bbbk$-split, so $\Bbbk$-split bimodules form a tensor ideal in $A$-mod-$A$. For any indecomposable $K_{1}$ and $K_{2}$ in $A$-mod and any indecomposable $X\in\mathrm{mod}\text{-}A$, we have

[TABLE]

This implies claim (ii) and claim (iii) is proved similarly. This, combined with the fact that $\Bbbk$-split bimodules form a tensor ideal in $A$-mod-$A$, also implies claim (i), completing the proof. ∎

3.3. $A$-$A$-bimodules which are not $\Bbbk$-split

Consider the action graph $\Gamma_{M}$ for a string $A\otimes_{\Bbbk}A^{\mathrm{op}}$-module $M$. Then, directly from the construction of string modules, we can make the following easy observations:

• •

the indegree of each vertex in $\Gamma_{M}$ is at most two;

• •

the outdegree of each vertex in $\Gamma_{M}$ is at most two;

• •

each vertex of $\Gamma_{M}$ is either a source or a sink (or both);

• •

$M$ is simple if and only if both the indegree and the outdegree of each vertex in $\Gamma_{M}$ is zero.

A vertex of $\Gamma_{M}$ of indegree exactly two will be called a valley. A typical example of a valley in an action graph is the white vertex of the following graph:

[TABLE]

We denote by $\mathbf{v}(M)$ the number of valleys in $\Gamma_{M}$. Clearly, $0\leq\mathbf{v}(M)\leq n-1$, moreover, the only $M$ for which $\mathbf{v}(M)=n-1$ is the regular bimodule $M\cong{}_{A}A_{A}$. For any $A\otimes_{\Bbbk}A^{\mathrm{op}}$-module $N$ which is not projective-injective, define

[TABLE]

Indecomposable $A$-$A$-bimodules that are not $\Bbbk$-split correspond to string $A\otimes_{\Bbbk}A^{\mathrm{op}}$-modules whose action graphs have $k$ vertices, where $3\leq k\leq 2n-1$. Below we list all such bimodules, fix notation for them and describe the corresponding action graphs. We use the number of valleys in the action graph as a parameter, denoted by $t$. Our choice for notation is motivated by the shape of the action graph.

Bimodules $W_{ij}^{t}$. For any $t\in\mathbb{N}_{n}^{\ast}$ and $i,j\in\mathbb{N}_{n-t+1}^{\ast}$, we denote by $W_{ij}^{t}$ the following $A$-$A$-bimodule:

[TABLE]

In particular, we have ${}_{A}A_{A}\cong W_{11}^{n-1}$.

Bimodules $S_{ij}^{t}$. For any $t\in\mathbb{N}_{n-1}^{\ast}$, $i\in\mathbb{N}_{n-t}^{\ast}$, and $j\in\mathbb{N}_{n-t+1}^{\ast}$, we denote by $S_{ij}^{t}$ the following $A$-$A$-bimodule:

[TABLE]

Bimodules $N_{ij}^{t}$. For any $t\in\mathbb{N}_{n-1}^{\ast}$, $i\in\mathbb{N}_{n-t+1}^{\ast}$ and $j\in\mathbb{N}_{n-t}^{\ast}$, we denote by $N_{ij}^{t}$ the following $A$-$A$-bimodule:

[TABLE]

Bimodules $M_{ij}^{t}$. For any $t\in\mathbb{N}_{n-1}$ and $i,j\in\mathbb{N}_{n-t}^{\ast}$, we denote by $M_{ij}^{t}$ the following $A$-$A$-bimodule:

[TABLE]

In particular, $\mathrm{Hom}_{\Bbbk}({}_{A}A_{A},\Bbbk)\cong M_{11}^{n-2}$. As a concrete example, the action graph of the bimodule $M_{ij}^{0}$, where $i,j\in\mathbb{N}_{n}^{\ast}$, is:

[TABLE]

Let $M$ be an indecomposable $A$-$A$-bimodule listed in Subsection 3.2-3.3. Then vertices in $\Gamma_{M}$ correspond to the standard basis of $M$ denoted $\mathbf{B}(M)$. We will often identify vertices in $\Gamma_{M}$ with this standard basis. Note the following:

• •

every vertex of non-zero indegree generates a simple sub-bimodule;

• •

every vertex of outdegree two generates a subbimodule which is isomorphic to some $M_{ij}^{0}$.

3.4. Supports

For an $A$-$A$-bimodule $X$, we define the left support of $X$ as the set of all $i\in\{1,2,\dots,n\}$ such that $e_{i}X\neq 0$. The left support of $X$ will be denoted $\mathrm{Lsupp}(X)$. Similarly, the right support of $X$, denoted $\mathrm{Rsupp}(X)$, is the set of all $i\in\{1,2,\dots,n\}$ such that $Xe_{i}\neq 0$. Note that, for any indecomposable $A$-$A$-bimodule $X$, both $\mathrm{Lsupp}(X)$ and $\mathrm{Rsupp}(X)$ are convex subset of $\{1,2,\dots,n\}$. Here $X\subset\{1,2,\dots,n\}$ is convex if, for any $x,y,z\in\{1,2,\dots,n\}$, the properties $x,z\in X$ and $x<y<z$ imply $y\in X$.

We define the width $\mathbf{w}(X)$ and the height $\mathbf{h}(X)$ of $X$ to be, respectively,

[TABLE]

Note that, for an indecomposable $X$, we have $|\mathbf{w}(X)-\mathbf{h}(X)|\leq 1$.

It is worth noting that indecomposable bimodules in the families $N$ and $S$ are uniquely determined (inside the set of isomorphism classes of indecomposable $A$-$A$-bimodules) by their left and right supports and that for bimodules in these families we always have $|\mathbf{w}(X)-\mathbf{h}(X)|=1$. Further, each indecomposable bimodule in the families $M$ and $W$ is uniquely determined by its left and right support inside its family. Moreover, for each bimodule $X$ in the families $M$ and $W$, there is a unique bimodule in the other family (i.e. in $W$ if $X$ is in $M$ and vice versa) which shares the left and right supports with $X$. Note that, if $X$ of the $M$-family and $Y$ of the $W$-family share the left and right supports, then $\mathbf{v}(Y)>\mathbf{v}(X)$.

The following lemma contains a crucial observation for our combinatorial description. The claim follows directly from the definitions.

Lemma 4**.**

For any $A$-$A$-bimodules $X$ and $Y$, we have

[TABLE]

3.5. Description of cells

For $v\in\mathbb{N}_{n}$, denote by $\mathcal{J}_{v}$ the set of all non $\Bbbk$-split elements in $\mathcal{S}$ having exactly $v$ valleys. Note that $\mathcal{J}_{n-1}$ consists just of the identity $A$-$A$-bimodule ${}_{A}A_{A}$. Recall from [CM, Subsection 2.3] that a two-sided cell is called idempotent provided that it contains three elements $\mathrm{F}$, $\mathrm{G}$ and $\mathrm{H}$ (not necessarily different), such that $\mathrm{F}$ is isomorphic to a direct summand of $\mathrm{G}\circ\mathrm{H}$.

In this subsection we present our main results. We start by describing two-sided cells in $\mathcal{S}$.

Theorem 5**.**

1. $($i$)$

Each $\mathcal{J}_{v}$ is a two-sided cell in $\mathcal{S}$, moreover, each two-sided cell in $\mathcal{S}$ coincides with either $\mathcal{J}_{\Bbbk}$ or with one of the $\mathcal{J}_{v}$’s. 2. $($ii$)$

The two-sided cells of $\mathcal{S}$ are linearly ordered as follows:

[TABLE] 3. $($iii$)$

All two-sided cells but $\mathcal{J}_{0}$ are idempotent.

As left and right cells in $\mathcal{J}_{\Bbbk}$ are already described in Proposition 3, it is left to consider left and right cells formed by non $\Bbbk$-split bimodules.

Theorem 6**.**

Let $v\in\mathbb{N}_{n}$.

1. $($i$)$

If $v>0$, then, for each $j\in\mathbb{N}_{n-v+1}^{*}$, the set of all bimodules of type $W$ and $S$ in $\mathcal{J}_{v}$ with right support $\{j,j+1,\dots,j+v\}$ forms a left cell in $\mathcal{J}_{v}$. 2. $($ii$)$

For each $j\in\mathbb{N}_{n-v}^{*}$, the set of all bimodules of type $M$ and $N$ in $\mathcal{J}_{v}$ with right support $\{j,j+1,\dots,j+v+1\}$ forms a left cell in $\mathcal{J}_{v}$. 3. $($iii$)$

Each left cell in $\mathcal{J}_{v}$ is of the form given by (i) or (ii).

Theorem 7**.**

Let $v\in\mathbb{N}_{n}$.

1. $($i$)$

If $v>0$, then, for each $j\in\mathbb{N}_{n-v+1}^{*}$, the set of all bimodules of type $W$ and $N$ in $\mathcal{J}_{v}$ with left support $\{j,j+1,\dots,j+v\}$ forms a right cell in $\mathcal{J}_{v}$. 2. $($ii$)$

For each $j\in\mathbb{N}_{n-v}^{*}$, the set of all bimodules of type $M$ and $S$ in $\mathcal{J}_{v}$ with left support $\{j,j+1,\dots,j+v+1\}$ forms a right cell in $\mathcal{J}_{v}$. 3. $($iii$)$

Each right cell in $\mathcal{J}_{v}$ is of the form given by (i) or (ii).

As an immediate consequence from Theorems 5, 6 and 7, we have:

Corollary 8**.**

All two-sided cells in $\mathcal{J}$ are strongly regular in the sense that we have $|\mathcal{L}\cap\mathcal{R}|=1$, for any left cell $\mathcal{L}$ in $\mathcal{J}$ and any right cell $\mathcal{R}$ in $\mathcal{J}$.

Regular two-sided cells play an important role in the theory developed in [MM1]–[MM6], see also [KiM2].

The algebra $A$ has an involutive anti-automorphism $\imath$ which swaps $e_{1}$ with $e_{n}$, $e_{2}$ with $e_{n-1}$, $\alpha_{1}$ with $\alpha_{n-1}$ and so on. Using this anti-automorphism, we can define an anti-involution on the tensor category $A$-mod-$A$ which we, abusing notation, also denote by $\imath$. This anti-involution swaps the sides of bimodules and twists both action by $\imath$. Consequently, the multisemigroup $\mathcal{S}$ has an involutive anti-automorphism. This anti-automorphism makes the statements of Theorem 6 and Theorem 7 equivalent. Remainder of the section is devoted to the proof of Theorems 5 and 6.

3.6. Proof of Theorem 6

We start by introducing the following notation: for a subset $U\subset\mathbb{Z}$ and for $r\in\mathbb{Z}$, we will denote by $U[r]$ the set $\{i+r\,:\,i\in U\}$.

Lemma 9**.**

Let us fixed a type, $W$, $M$, $S$ or $N$, of non-$\Bbbk$-split $A$-$A$-bimodules. Then all bimodules of this type and with fixed right support belong to the same left cell.

Proof.

Denote by $\varphi$ the endomorphism of $A$ which sends

[TABLE]

Set $\tilde{e}=e_{2}+e_{3}+\dots+e_{n}=\varphi(1)$. Then $\tilde{e}A$ has the natural structure of an $A$-$A$-bimodule where the right action of $a$ is given by multiplication with $a$ and the left action of $a$ is given by multiplication with $\varphi(a)$. We will denote this bimodule by ${}^{\varphi}\tilde{e}A$. Similarly, $A\tilde{e}$ has the natural structure of an $A$-$A$-bimodule where the right action of $a$ is given by multiplication with $\varphi(a)$ and the left action of $a$ is given by multiplication with $a$. We will denote this bimodule by $A\tilde{e}^{\varphi}$.

We have $A\tilde{e}=\tilde{e}A\tilde{e}$. Consequently, the multiplication map

[TABLE]

is an isomorphism of $A$-$A$-bimodules. Set $I:=Ae_{n}A$. Then, mapping $1+I\mapsto\tilde{e}$, gives rise to an isomorphism of $A$-$A$-bimodules between $A/I$ and ${}^{\varphi}\tilde{e}A\tilde{e}^{\varphi}$. In particular, it follows that ${}^{\varphi}\tilde{e}A\otimes_{A}A\tilde{e}^{\varphi}\cong{}_{A}(A/I)_{A}$.

This means that, if $X\in A$-mod is annihilated by $I$, then, tensoring $X$ (from the left) first with $A\tilde{e}^{\varphi}$ and then with ${}^{\varphi}\tilde{e}A$, gives $X$ back. If $X$ has, additionally, the structure of an indecomposable $A$-$A$-bimodule, then tensoring with $A\tilde{e}^{\varphi}$ just twists the left action of $A$ on $X$ by $\varphi$ and hence does not change the type ($W$, $M$, $S$ or $N$) of $X$. In particular, we have

[TABLE]

Starting now with $X$ such that $1\in\mathrm{Lsupp}(X)$ and applying this procedure inductively, we will obtain that all bimodules of the same type as $X$ and with the same right support as $X$ belong to the left cell of $X$. This proves the claim. ∎

Lemma 10**.**

Bimodules of types $W$ and $S$ with the same right support belong to the same left cell.

Proof.

After Lemma 9, it is enough to prove this claim for any two particular bimodules of types $W$ and $S$ with the same right support. Let $X$ be the unique indecomposable subquotient of ${}_{A}A_{A}$ of type $W$ with right support $\{j,j+1,\dots,k\}$. Assume that $k<n$ and let $Y$ be the unique indecomposable subquotient of ${}_{A}A_{A}$ of type $S$ with right support $\{j,j+1,\dots,k\}$. Then $Y$ surjects onto $X$ with one-dimensional kernel.

Let $Q=A(e_{j}+e_{j+1}+\dots+e_{k})A$ be a subbimodule of the regular bimodule ${}_{A}A_{A}$. Consider the short exact sequence

[TABLE]

of bimodules. By construction, $\mathrm{Coker}\otimes_{A}X=0$ and hence $Q\otimes_{A}X$ surjects onto $X$. Consequently, we either have $Q\otimes_{A}X\cong X$ or $Q\otimes_{A}X\cong Y$. To determine which of these cases takes place, we use adjunction:

[TABLE]

It is now easy to check that $\mathrm{Hom}_{A\text{-}}(Q,Y)\cong X$ and thus the right hand side of the above isomorphism is non-zero. As $\mathrm{Hom}_{A\text{-}A}(X,Y)=0$, it follows that $Q\otimes_{A}X=Y$.

If we denote by $Q^{\prime}$ the quotient of the above bimodule $Q$ by the subbimodule $e_{k+1}Q$, then we again have either $Q^{\prime}\otimes_{A}Y\cong X$ or $Q^{\prime}\otimes_{A}Y\cong Y$. By adjunction,

[TABLE]

However, now $Q^{\prime}$ is a proper quotient of $Q$ and one sees that the dimension of $\mathrm{Hom}_{A\text{-}}(Q^{\prime},Y)$ is strictly smaller than that of $\mathrm{Hom}_{A\text{-}}(Q,Y)$. Therefore $\mathrm{Hom}_{A\text{-}}(Q^{\prime},Y)$ is a proper subbimodule of $X$ which yields

[TABLE]

This implies $Q^{\prime}\otimes_{A}Y\cong X$ and hence $X$ and $Y$ do belong to the same left cell.

In the case $k=n$ the arguments are similar with the only difference that $X$ has to be changed to ${}^{\varphi}\tilde{e}A\otimes_{A}X$. This works only if $j>1$. If $j=1$ and $k=n$, then there are no $A$-$A$-bimodules of type $S$ with such right support. ∎

Lemma 11**.**

Bimodules of types $M$ and $N$ with the same right support belong to the same left cell.

Proof.

This is similar to the proof of Lemma 10 and is left to the reader. ∎

Lemma 12**.**

Bimodules of types $W$ and $M$ cannot belong to the same left cell.

Proof.

Let $X$ be an $A$-$A$-bimodule of type $W$ and $Y$ an $A$-$A$-bimodule of type $M$. Taking Lemma 4 into account, suppose that $\mathrm{Rsupp}(X)=\mathrm{Rsupp}(Y)$ and consider the right $A$-modules $X_{A}$ and $Y_{A}$. Then from our explicit description of bimodules we can see that $X_{A}$ is not a quotient of any module in $\mathrm{add}(Y_{A})$. Therefore no $Z\otimes_{A}Y_{A}$ can have $X_{A}$ as a quotient, let alone direct summand. The claim follows. ∎

Claims (i) and (ii) of Theorem 6 follow from Lemmata 9–12. Claim (iii) follows from claims (i) and (ii) and classification of indecomposable $A$-$A$-bimodules.

3.7. Proof of Theorem 5

Claim (i) follows from Theorems 6 and Theorem 7. That fact that $\mathcal{J}_{v}$, for $v>1$, are idempotent follows from the proof of Lemma 10.

To prove that two-sided cells are linearly ordered, for $j=1,2,\dots,n-1$, consider the quotient $A$-$A$-bimodule $Q_{j}:=A/A(e_{j+2}+\dots+e_{n})A$ of $A$ (in particular, $Q_{n-1}=A$). We have $Q_{j}\in\mathcal{J}_{j}$, for all $j$, in fact, $Q_{j}\cong W_{11}^{j}$. Consider also the $A$-$A$-bimodule $\mathrm{Hom}_{\Bbbk}(Q_{j},\Bbbk)\cong M_{11}^{j-1}\in\mathcal{J}_{j-1}$. We can, in fact, interpret $Q_{j}$ as the identity bimodule for the algebra $B:=A/A(e_{j+2}+\dots+e_{n})A$. After this interpretation it is clear that

[TABLE]

Claim (ii) follows.

To complete the proof of Theorem 5, it remains to show that $\mathcal{J}_{0}$ is not idempotent. Note that $\mathcal{J}_{0}$ only consists of bimodules of type $M$. Let $X\otimes_{A}Y$ be the tensor product of two bimodules from $\mathcal{J}_{0}$. We need to check that $X\otimes_{A}Y$ is $\Bbbk$-split. This can be done by a direct computation or, alternatively, argued theoretically as follows.

If $\mathrm{Rsupp}(X)\cap\mathrm{Lsupp}(Y)=\varnothing$, then $X\otimes_{A}Y=0$. If $\mathrm{Rsupp}(X)=\{i,i+1\}$ and $\mathrm{Lsupp}(Y)=\{i+1,i+2\}$, then, tensoring $Y$ with the short exact sequence

[TABLE]

where Ẍ́^{\prime} is simple with $\mathrm{Rsupp}(X^{\prime})=\{i\}$ and using $X^{\prime}\otimes_{A}Y=0$, implies $X\otimes_{A}Y\cong\mathrm{Coker}\otimes_{A}Y$ is $\Bbbk$-split as $\mathrm{Coker}$ is $\Bbbk$-split.

If $\mathrm{Rsupp}(X)=\{i+1,i+2\}$ and $\mathrm{Lsupp}(Y)=\{i,i+1\}$, then, tensoring $Y$ with the short exact sequence

[TABLE]

where $X^{\prime}$ is simple with $\mathrm{Rsupp}(X^{\prime})=\{i+1\}$, we obtain $\mathrm{Coker}\otimes_{A}Y=0$ and $X^{\prime}\otimes Y=0$ implying $X\otimes_{A}Y=0$ which is certainly $\Bbbk$-split.

If $\mathrm{Rsupp}(X)=\mathrm{Lsupp}(Y)=\{i,i+1\}$, then we tensor $Y$ with the short exact sequence

[TABLE]

where $X^{\prime}$ is simple with $\mathrm{Rsupp}(X^{\prime})=\{i\}$. Then $\mathrm{Coker}\otimes_{A}Y=0$ and thus the two-dimensional bimodule $X^{\prime}\otimes_{A}Y$ surjects onto $X\otimes_{A}Y$. Therefore $X\otimes_{A}Y$ has dimension at most two and thus is $\Bbbk$-split. The proof of Theorem 5 is complete.

3.8. Adjunctions between indecomposable $A$-$A$-bimodules

In this subsection we will describe all pairs of bimodules which correspond to adjoint pairs of functors.

Lemma 13**.**

For any $X,Y\in A$-mod-$A$, the pair $(X\otimes_{A}{}_{-},Y\otimes_{A}{}_{-})$ is an adjoint pair of endofunctors of $A$-mod if and only if $X$ is projective as a left $A$-module and $\mathrm{Hom}_{A}(X,A)\cong Y$ as $A$-$A$-bimodules.

Proof.

As $(X\otimes_{A}{}_{-},\mathrm{Hom}_{A}(X,{}_{-}))$ is an adjoint pair of functors, we have

[TABLE]

in particular, the latter functor must be exact and hence $X$ must be left $A$-projective. As exact functors are uniquely determined by their action on the category of projective $A$-modules, we get an isomorphism $\mathrm{Hom}_{A}(X,A)\cong Y$ of a $A$-$A$-bimodule. The converse implication is straightforward. ∎

An indecomposable $A$-$A$-bimodule $X$ which is not $\Bbbk$-split is left projective if and only if it belongs to the following set:

[TABLE]

As $W_{11}^{n-1}\cong{}_{A}A_{A}$, the functor $\mathrm{W}_{11}^{n-1}$ is self adjoint. Considering the right adjoint of the functors given by tensoring with bimodules in (5), we obtain the following proposition.

Proposition 14**.**

*For any $t\in\mathbb{N}_{n-1}^{\ast}$, we have the following:     *

1. $($a$)$

for any $j\in\mathbb{N}_{n-t+1}^{\ast}$, the pair $(\mathrm{W}_{n-t|j}^{t},\mathrm{N}_{j|n-t-1}^{t})$ is an adjoint pair of functors; 2. $($b$)$

for any $j\in\mathbb{N}_{n-t+1}^{\ast}$, the pair $(\mathrm{S}_{1j}^{t},\mathrm{W}_{j1}^{t})$ is an adjoint pair of functors; 3. $($c$)$

for any $i\in\mathbb{N}_{n-t}^{\ast}\setminus\{1\}$ and $j\in\mathbb{N}_{n-t+1}^{\ast}$, the pair $(\mathrm{S}_{ij}^{t},\mathrm{N}_{j|i-1}^{t})$ is an adjoint pair of functors.

Proof.

We start by proving claim (a) any claim (c) is proved similarly. As $\mathrm{W}_{n-t|j}^{t}$ is left projective, its right adjoint is automatically exact. We need to prove that $\mathrm{Hom}_{A}(\mathrm{W}_{n-t|j}^{t},A)\cong\mathrm{N}_{j|n-t-1}^{t}$.

As $\mathrm{W}_{n-t|j}^{t}$ is indecomposable, as a bimodule, so is $\mathrm{Hom}_{A}(\mathrm{W}_{n-t|j}^{t},A)$. Now, recall that indecomposable $A$-$A$-bimodules of type $N$ are uniquely determined by their left and right supports. Therefore it is enough to check that the bimodules $\mathrm{Hom}_{A}(\mathrm{W}_{n-t|j}^{t},A)$ and $\mathrm{N}_{j|n-t-1}^{t}$ have the same left support and the same right support.

The left $A$-action on $X:=\mathrm{Hom}_{A}(\mathrm{W}_{n-t|j}^{t},A)$ comes from the right $A$-action on $\mathrm{W}_{n-t|j}$. Because of our notation, the minimum $s$ for which $e_{i}$ does not annihilate $\mathrm{W}_{n-t|j}$ on the right is $s=j$. Furthermore, as a left $A$-module, $\mathrm{W}_{n-t|j}$ has $t+1$ direct summands. This implies that

[TABLE]

The direct summand $Ae_{n-t+s}$, for $s=0,1,\dots,t$, of $\mathrm{W}_{n-t|j}^{t}$ maps only to the direct summands $Ae_{n-t+s}$ and $Ae_{n-t+s-1}$ of $A$. This implies that

[TABLE]

The claim follows.

Claim (b) is also proved analogously to claim (a) with one additional remark: we determine the right adjoint using its support. For claim (b), the support argument implies that the right adjoint might be either of type $W$ or of type $M$. However, exactness of this right adjoint determines its type uniquely as $W$. The rest is completely analogous to claim (a). ∎

4. A minimal generating set

4.1. The main result of the section

The main result of this subsection is the following:

Theorem 15**.**

Assume $n\geq 3$.

1. $($i$)$

The category $A$-mod-$A$* coincides with the minimal subcategory of $A$-mod-$A$ containing*

[TABLE]

and closed under isomorphisms, tensor products and taking direct sums and direct summands. 2. $($ii$)$

The set (6) is minimal in the sense that no proper subset of (6) has the property described in (i).

The cases $n=1,2$ are too small to be interesting.

4.2. Comments on the proof of Lemma 9

Note that some of the bimodules listed in (6) appeared already in Lemma 9. Namely, we have $A\tilde{e}^{\varphi}\cong W_{21}^{n-2}$ and ${}^{\varphi}\tilde{e}A\cong N_{11}^{n-2}$ in $A$-mod-$A$. We can similarly describe the $A$-$A$-bimodule $S_{12}^{n-2}$ as being isomorphic to $A\tilde{e^{\prime}}^{\psi}$, where $\tilde{e^{\prime}}=e_{1}+e_{2}+\dots+e_{n-1}$ and $\psi:=\imath\circ\varphi\circ\imath$ (see Subsection 3.5 for the definition of $\imath$). Note that $\tilde{e^{\prime}}=\psi(1)$ and that $\psi$ sends

[TABLE]

Making a parallel with the proof of Lemma 9, we also have the $A$-$A$-bimodule ${}^{\psi}\tilde{e^{\prime}}A$. This bimodule is isomorphic to $W_{21}^{n-2}$. In fact, the anti-involution $\imath$ on $A$-mod-$A$ fixes both $W_{11}^{n-1}$ and $W_{21}^{n-2}$ but swaps $N_{11}^{n-2}$ with $S_{12}^{n-2}$. Similarly to the proof of Lemma 9, we have

[TABLE]

where $I^{\prime}=Ae_{1}A=Ae_{1}$. This means that, if $X\in$ mod-$A$ is annihilated by $I^{\prime}$, then, tensoring $X$ first with ${}^{\psi}\tilde{e^{\prime}}A$ and then with $A\tilde{e^{\prime}}^{\psi}$ (both from the right), gives $X$ back. If $X$ has the structure of an indecomposable $A$-$A$-bimodule, then tensoring with ${}^{\psi}\tilde{e^{\prime}}A$ just twists the right action of $A$ on $X$ by $\psi$ and hence does not change the type ($W$, $M$, $S$ or $N$) of $X$. Moreover, we have

[TABLE]

(here we use the notation $U[-1]$ from Subsection 3.6). This fact can be used to prove the “other side” version of Lemma 9, which consequently contributes to the proof of Theorem 7.

4.3. Auxiliary lemmata

By construction, the action graph $\Gamma_{X}$ of an $A$-$A$-bimodule $X$ is a subgraph of the graph (3). In what follows, for a fixed $A$-$A$-bimodule $X$, we will describe the outcome of tensoring of $X$ with $W_{21}^{n-2}$, $N_{11}^{n-2}$, and $S_{12}^{n-2}$, both from the left and from the right, using combinatorial manipulations with the graph $\Gamma_{X}$, considering the latter as a subgraph of (3).

Lemma 16**.**

Let $X$ be an indecomposable $A$-$A$-bimodule.

1. $($i$)$

The action graph of the $A$-$A$-bimodule $W_{21}^{n-2}\otimes_{A}X$ is obtained from $\Gamma_{X}$ by moving the latter vertically one step down and then cutting off all vertices and arrows which fall outside the graph in figure (3). 2. $($ii$)$

The action graph of the $A$-$A$-bimodule $X\otimes_{A}W_{21}^{n-2}$ is obtained from $\Gamma_{X}$ by moving the latter horizontally one step to the left and then cutting off all vertices and arrows which fall outside the graph in figure (3).

Proof.

From the proof of Lemma 9, it follows that claim (i) is true as soon as $IX=0$ (recall that $I=Ae_{n}A=e_{n}A$). Note that in this case, no “cutting off” is necessary.

In general, we have $W_{21}^{n-2}\cong A\tilde{e}^{\varphi}$ and $IX=e_{n}X$ is an $A$-$A$-subbimodule of $X$. If $e_{n}X\neq 0$, then we have $A\tilde{e}^{\varphi}\otimes_{A}e_{n}X=0$ and hence $A\tilde{e}^{\varphi}\otimes_{A}X\cong A\tilde{e}^{\varphi}\otimes_{A}(X/e_{n}X)$. As $I(X/e_{n}X)=0$, we can apply the argument from the previous paragraph. Factoring $e_{n}X$ out corresponds precisely to “cutting off” those vertices and edges which fall outside the graph in figure (3) after the move. This completes the proof of claim (i).

From Subsection 4.2 it follows that claim (ii) is true as soon as $XI^{\prime}=0$. Note that in this case, no “cutting off” is necessary.

In general, we have $W_{21}^{n-2}\cong{}^{\psi}\tilde{e^{\prime}}A$ and $XI^{\prime}=Xe_{1}$ is an $A$-$A$-subbimodule of $X$. If $Xe_{1}\neq 0$, then we have $Xe_{1}\otimes_{A}{}^{\psi}\tilde{e^{\prime}}A=0$ and hence

[TABLE]

As $(X/Xe_{1})I^{\prime}=0$, we can apply the argument from the previous paragraph. Factoring $Xe_{1}$ out corresponds precisely to “cutting off” those vertices and edges which fall outside the graph in figure (3) after the move. This completes the proof of claim (ii). ∎

Let $X$ be an indecomposable $A$-$A$-bimodule. A full subgraph $\Gamma$ of $\Gamma_{X}$ will be called thick provided that, for any arrow $x\to y$ in $\Gamma_{X}$, the condition $x\in\Gamma$ implies $y\in\Gamma$.

Lemma 17**.**

Let $X$ be an indecomposable and not $\Bbbk$-split $A$-$A$-bimodule.

1. $($i$)$

The action graph of the $A$-$A$-bimodule $N_{11}^{n-2}\otimes_{A}X$ is obtained from $\Gamma_{X}$ by moving the latter vertically one step up and then cutting off all vertices and edges which fall outside the graph in figure (3). 2. $($ii$)$

If $\dim_{\Bbbk}Xe_{1}=2$, then the action graph of the $A$-$A$-bimodule $X\otimes_{A}N_{11}^{n-2}$ is obtained from $\Gamma_{X}$ in the following three steps:

• •

first we move $\Gamma_{X}$ one step to the right;

• •

then we cut off the thick subgraph generated by all vertices which fall outside the graph in figure (3), we denote the resulting graph $\Gamma$;

• •

finally, we add to $\Gamma$ one new vertex and one new arrow as follows: let $v$ be the north-west corner of $\Gamma$, then we add to $\Gamma$ the immediate west neighbor $w$ of $v$ and the arrow connecting $v$ to $w$. 3. $($iii$)$

If $\dim_{\Bbbk}Xe_{1}\neq 2$, then the action graph of the $A$-$A$-bimodule $X\otimes_{A}N_{11}^{n-2}$ is obtained from $\Gamma_{X}$ by moving the latter horizontally one step to the right and then cutting off the thick subgraph generated by all vertices which fall outside the graph in figure (3).

A good example to illustrate the procedure described in Lemma 17(ii) is the obvious isomorphism $W_{11}^{n-1}\otimes_{A}N_{11}^{n-2}\cong N_{11}^{n-2}$ based on the fact that $W_{11}^{n-1}\cong{}_{A}A_{A}$. In particular, this example shows that the second step of the procedure described in Lemma 17(ii) can lead to elimination of some vertices which do not fall outside the graph in figure (3). For $n=4$, the transformation in this example can be depicted explicitly as follows:

[TABLE]

Here dashed lines indicate the boundaries of the graph in figure (3), dotted arrows and white vertices are the ones which are deleted during the second step of the procedure described in Lemma 17 (ii). Finally, the triangle vertex and the double arrow are the ones which are added during the last step of the procedure described in Lemma 17 (ii).

Proof.

Assume that $e_{1}X=0$ and let $Y$ be an $A$-$A$-bimodule such that $\Gamma_{Y}$ is obtained by moving $\Gamma_{X}$ one step up (note that this is well-defined as $e_{1}X=0$). Then $Y$ has the same type as $X$ and

[TABLE]

From Lemma 16 (i) it follows that $X\cong W_{21}^{n-2}\otimes_{A}Y$. This, together with the discussion in Subsection 4.2, implies $N_{11}^{n-2}\otimes_{A}X\cong Y$ in $A$-mod-$A$. Claim (i) in the case $e_{1}X=0$ follows.

Assume now that $e_{1}X\neq 0$. Consider the following short exact sequence of $A$-$A$-bimodules:

[TABLE]

where $\tilde{e}X\to X$ is the natural inclusion. In this sequence we have that ${}_{A}\mathrm{Coker}$ is semisimple, moreover, $e_{i}\mathrm{Coker}=0$, for all $i>1$. Recall that $N_{11}^{n-2}\cong{}^{\varphi}\tilde{e}A$. Then ${}^{\varphi}\tilde{e}A\otimes_{A}\mathrm{Coker}=0$. The functor $N_{11}^{n-2}\otimes_{A}{}_{-}$ is exact as $N_{11}^{n-2}$ is right projective (or due to Proposition 14 (a)). Therefore, applying $N_{11}^{n-2}\otimes_{A}{}_{-}$ to (8), gives $N_{11}^{n-2}\otimes_{A}\tilde{e}X\cong N_{11}^{n-2}\otimes_{A}X$. We can now apply the previous paragraph to $N_{11}^{n-2}\otimes_{A}\tilde{e}X$ and thus complete the proof of claim (i).

We have $\mathrm{Lsupp}(X\otimes_{A}{}^{\varphi}\tilde{e}A)\subset\mathrm{Lsupp}(X)$ by Lemma 4. Similarly to the proof of Lemma 9, if $XI=0$, then, tensoring $X$ (from the right) with ${}^{\varphi}\tilde{e}A$ and then with $A\tilde{e}^{\varphi}$, gives $X$ back. Consequently, in the case $XI=0$ we have

[TABLE]

Consider the short exact sequence

[TABLE]

of $A$-$A$-bimodules where ${}^{\varphi}\tilde{e}Ae_{1}\to{}^{\varphi}\tilde{e}A$ is the natural inclusion. Then we have $\mathrm{Coker^{\prime}}\cong W_{12}^{n-2}$. Using Lemma 16 (ii) or the fact that $e_{1}A\tilde{e}^{\varphi}=0$, we have

[TABLE]

Therefore, tensoring $X$ (from the right) with $\mathrm{Coker^{\prime}}$ and then with $A\tilde{e}^{\varphi}$ also gives $X$ back. Applying the functor $X\otimes_{A}{}_{-}$ to (9), we thus obtain

[TABLE]

As tensor functors are right exact, we also have

[TABLE]

Just like in the proof of Lemma 9, tensoring with $W_{12}^{n-2}$ twists the right $A$-action on $X$ by $\varphi$ and hence does not change the type ($W$, $M$, $S$, or $N$) of $X$. In particular, we have

[TABLE]

Therefore, considering the action graph of $X\otimes_{A}\mathrm{Coker^{\prime}}$ corresponds to the first step of claim (ii).

Note that $\mathrm{Rsupp}(X\otimes_{A}{}^{\varphi}\tilde{e}Ae_{1})\subset\{1\}$. Hence, the above arguments imply

[TABLE]

Now, if $\dim_{\Bbbk}Xe_{1}=2$, then $\mathbf{B}(X)$ contains the vertices $\mathtt{j\mid 1}$ and $\mathtt{j\text{+}1\mid 1}$, for some $j$. Moreover, we obtain that the $A$-$A$-bimodule $X\otimes_{A}{}^{\varphi}\tilde{e}Ae_{1}$ is of dimension one with basis $\mathtt{j\mid 1}\otimes\alpha_{1}$ (here it is important that $X$ is not $\Bbbk$-split). The image of this element in $X\otimes_{A}{}^{\varphi}\tilde{e}A$ is also non-zero. This implies that

[TABLE]

Therefore the sequence

[TABLE]

is exact, which completes the proof of claim (ii) in the case $XI=0$.

If $\dim_{\Bbbk}Xe_{1}\neq 2$, then we have $X\otimes_{A}{}^{\varphi}\tilde{e}Ae_{1}=0$ as ${}_{A}({}^{\varphi}\tilde{e}Ae_{1})$ is simple and $X$ is not $\Bbbk$-split. This implies that $X\otimes_{A}{}^{\varphi}\tilde{e}A\cong X\otimes_{A}\mathrm{Coker^{\prime}}$, which establishes claim (iii) in the case $XI=0$.

If $XI\neq 0$, then we have $XI\otimes_{A}{}^{\varphi}\tilde{e}A=0$ as the left action of $e_{n}$ annihilates ${}^{\varphi}\tilde{e}A$. Hence we obtain $X\otimes_{A}{}^{\varphi}\tilde{e}A\cong(X/XI)\otimes_{A}{}^{\varphi}\tilde{e}A$ so that we can reduce our consideration to the previous case $XI=0$. This proves claims (ii) and (iii) in full generality and we are done. ∎

Lemma 18**.**

Let $X$ be an indecomposable and not $\Bbbk$-split $A$-$A$-bimodule.

1. $($i$)$

If $\dim e_{n}X=2$, then the action graph of the $A$-$A$-bimodule $S_{12}^{n-2}\otimes_{A}X$ is obtained from $\Gamma_{X}$ in the following three steps:

• •

first we move $\Gamma_{X}$ vertically one step up;

• •

then we cut off the thick subgraph generated by all vertices which fall outside the graph in figure (3), we denote the resulting graph $\Gamma$;

• •

finally, we add to $\Gamma$ one new vertex and one new arrow as follows: let $v$ be the south-east corner of $\Gamma$, then we add to $\Gamma$ the immediate south neighbor $w$ of $v$ and the arrow connecting $v$ to $w$. 2. $($ii$)$

If $\dim e_{n}X\neq 2$, then the action graph of the $A$-$A$-bimodule $S_{12}^{n-2}\otimes_{A}X$ is obtained from $\Gamma_{X}$ by moving the latter vertically one step up and then cutting off the thick subgraph generated by all vertices which fall outside the graph in figure (3). 3. $($iii$)$

The action graph of the $A$-$A$-bimodule $X\otimes_{A}S_{12}^{n-2}$ is obtained from $\Gamma_{X}$ by moving the latter one step to the right and then cutting off all vertices and edges which fall outside the graph in figure (3).

Proof.

Observing that we have an isomorphism $S_{12}^{n-2}\cong A\tilde{e^{\prime}}^{\psi}$ of $A$-$A$-bimodules, the proof is similar to that of Lemma 17 and is left to the reader. ∎

Again, a good example to illustrate the procedure described in Lemma 18 (i) is the obvious isomorphism $S_{12}^{n-2}\otimes_{A}W_{11}^{n-1}\cong S_{12}^{n-2}$ based on the fact that $W_{11}^{n-1}\cong{}_{A}A_{A}$. In particular, this example shows that the second step of the procedure described in Lemma 18 (i) can lead to elimination of some vertices which do not fall outside the graph in figure (3). For $n=4$, the transformation in this example can be depicted (using the same conventions as in (7)) explicitly as follows:

[TABLE]

4.4. Proof of Theorem 15

Let $\mathcal{X}$ denote the minimal subcategory of $A$-mod-$A$ containing (6) and closed under isomorphisms, tensor products and taking direct sums and direct summands. To prove Theorem 15 (i), we need to show that all indecomposable $A$-$A$-bimodules belong to $\mathcal{X}$.

Clearly, $W_{11}^{n-1}\in\mathcal{X}$ as it appears in (6). Further, $M_{11}^{n-2}\in\mathcal{X}$ as, for example,

[TABLE]

where we used Lemma 17 (ii) for the first isomorphism and Lemma 18 (i) for the second one.

Starting from $W_{11}^{n-1}$, $M_{11}^{n-2}$, $S_{12}^{n-2}$ and $N_{11}^{n-2}$, and applying the manipulations described in Lemma 16 (i)-(ii), Lemma 17 (i) and Lemma 18 (iii), it is clear that we can obtain action graphs of all indecomposable $A$-$A$-bimodules which are not isomorphic to $Ae_{i}\otimes_{\Bbbk}e_{j+1}A$, where $i,j\in\mathbb{N}_{n}^{\ast}$. Hence all such bimodules must belong to $\mathcal{X}$, in particular, all indecomposable string $A$-$A$-bimodules of dimension two are in $\mathcal{X}$.

Finally, tensoring indecomposable string $A$-$A$-bimodules of dimension two with each other, we can get all bimodules of the form $Ae_{i}\otimes_{\Bbbk}e_{j+1}A$, where $i,j\in\mathbb{N}_{n}^{\ast}$. Claim (i) of Theorem 15 follows.

Now, let us prove claim (ii). Let $T$ be a proper subset of (6) and $\mathcal{X}_{T}$ the minimal subcategory of $A$-mod-$A$ containing $T$ and closed under isomorphisms, tensor products and taking direct sums and direct summands. We have to show that $\mathcal{X}_{T}$ is a proper subcategory of $A$-mod-$A$. If $W_{11}^{n-1}\not\in T$, then the latter claim follows directly from Theorem 5 (ii).

If $W_{21}^{n-2}\not\in T$, then, from Lemmata 17 and 18 it follows that the only possible manipulations with actions graphs are to move them up or to the right. As we can only start with action graphs of bimodules from $T$, it follows that, in particular, $W_{21}^{n-2}\not\in\mathcal{X}_{T}$. A similar argument also shows that, necessarily, either $S_{12}^{n-2}\in T$ or $N_{11}^{n-2}\in T$.

Assume that $T$ is (6) without $S_{12}^{n-2}$. Then, from Lemmata 16 and 17, it follows that all bimodules in $\mathcal{X}_{T}$ are either $\Bbbk$-split or of type $W$ or $N$. Assume that $T$ is (6) without $N_{11}^{n-2}$. Then, from Lemmata 16 and 18, it follows that all bimodules in $\mathcal{X}_{T}$ are either $\Bbbk$-split or of type $W$ or $S$. Claim (ii) follows and the proof of Theorem 15 is complete.

5. Simple transitive $2$-representations of projective $A$-$A$-bimodules

5.1. Finitary $2$-categories and their $2$-representations

In this section we switch from the concrete algebras $A_{n}$ considered above to general finite dimensional algebras $A$.

For a finite dimensional $\Bbbk$-algebra $A$, consider the $2$-category $\sc\mbox{C}\hskip 1.0pt_{A}$ of projective endofunctors of $A$-mod, see [MM1, Subsection 7.3] (we note that this $2$-category depends on the choice of a small category equivalent to $A$-mod). We assume that $A$ is basic and connected and let $\varepsilon_{1}+\varepsilon_{2}+\dots+\varepsilon_{k}=1$ be a primitive decomposition of the identity in $A$. The $2$-category $\sc\mbox{C}\hskip 1.0pt_{A}$ has one object $\mathtt{i}$. A complete list of representatives of isomorphism classes of indecomposable $A$-$A$-bimodules which contribute to $1$-morphisms in $\sc\mbox{C}\hskip 1.0pt_{A}$ consists of the regular bimodule ${}_{A}A_{A}$, which corresponds to the identity $1$-morphism $\mathbbm{1}_{\mathtt{i}}$, and indecomposable projective bimodules $A\varepsilon_{i}\otimes\varepsilon_{j}A$, where $i,j=1,2,\dots,k$, which correspond to indecomposable $1$-morphisms respectively denoted by $\mathrm{F}_{ij}$. Finally, $2$-morphisms in $\sc\mbox{C}\hskip 1.0pt_{A}$ are given by homomorphisms of $A$-$A$-bimodules.

A finitary $2$-representation of $\sc\mbox{C}\hskip 1.0pt_{A}$ is a functorial action, denoted $\mathbf{M}$, on a category $\mathbf{M}(\mathtt{i})$ equivalent to $B$-proj of projective modules over some finite dimensional $\Bbbk$-algebra $B$. All such $2$-representations form a $2$-category, denoted $\sc\mbox{C}\hskip 1.0pt_{A}$-afmod, where $1$-morphisms are $2$-natural transformations and $2$-morphisms are modifications, see [MM3] for details.

A finitary $2$-representation $\mathbf{M}$ is called transitive if $\mathbf{M}(\mathtt{i})$ has no proper $\sc\mbox{C}\hskip 1.0pt_{A}$-invariant, idempotent split and isomorphism closed additive subcategories. A transitive $2$-representation $\mathbf{M}$ is called simple if $\mathbf{M}(\mathtt{i})$ has no proper $\sc\mbox{C}\hskip 1.0pt_{A}$-invariant ideals, see [MM5, MM6] for details.

Classical examples of simple transitive $2$-representations are so-called cell $2$-representations as defined in [MM1, MM2]. The $2$-category $\sc\mbox{C}\hskip 1.0pt_{A}$ has, up to equivalence, two cell $2$-representations:

• •

The cell $2$-representation $\mathbf{C}_{\{\mathbbm{1}_{\mathtt{i}}\}}$ which is given as the quotient of the left regular action of $\sc\mbox{C}\hskip 1.0pt_{A}$ on $\sc\mbox{C}\hskip 1.0pt_{A}(\mathtt{i},\mathtt{i})$ by the unique maximal $\sc\mbox{C}\hskip 1.0pt_{A}$-invariant left ideal.

• •

The cell $2$-representation $\mathbf{C}_{\{\mathrm{F}_{i1}\}}$ which is given (up to equivalence) by the defining action of $\sc\mbox{C}\hskip 1.0pt_{A}$ on $A$-proj.

5.2. The main result of the section

The main result of this section is the following:

Theorem 19**.**

Assume that $A$ has a non-zero projective injective module and is directed in the sense that $\varepsilon_{i}A\varepsilon_{j}=0$ whenever $i<j$ and $\varepsilon_{i}A\varepsilon_{i}=\Bbbk\varepsilon_{i}$, for all $i$. Then every simple transitive $2$-representation of $\sc\mbox{C}\hskip 1.0pt_{A}$ is equivalent to a cell $2$-representation.

The algebra $A_{n}$ from Subsection 2.1 obviously satisfies both assumption of Theorem 19. Therefore Theorem 19 provides a classification of simple transitive $2$-representation of $\sc\mbox{C}\hskip 1.0pt_{A_{n}}$. In fact, any quotient of the path algebra of the quiver (1) satisfies both assumption of Theorem 19. There are of course many other algebras which satisfy these assumptions, for example incidence algebras of finite posets having the minimum and the maximum element (for example, the Boolean of a finite set) and many others. In the cases $A=A_{2}$ and $A=A_{3}$, Theorem 19 is proved in [MZ].

The rest of this section is devoted to the proof of Theorem 19.

5.3. Notational preparation

We let $\mathbf{M}$ be a simple transitive $2$-representation of $\sc\mbox{C}\hskip 1.0pt_{A}$ and denote by $B$ a basic $\Bbbk$-algebra such that $\mathbf{M}(\mathtt{i})$ is equivalent to $B$-proj. Let $\epsilon_{1}+\epsilon_{2}+\dots+\epsilon_{r}=1$ be a primitive decomposition of the identity in $B$. For $i,j=1,2,\dots,r$, we denote by $\mathrm{G}_{ij}$ the endofunctor of $B$-mod given by tensoring with the indecomposable projective $B$-$B$-bimodule $B\epsilon_{i}\otimes\epsilon_{j}B$. We note that $r\neq k$, in general.

Without loss of generality we may assume that $\mathbf{M}$ is faithful. Indeed, the $2$-category $\sc\mbox{C}\hskip 1.0pt_{A}$ is simple, see [MMZ, Subsection 3.2]. Therefore, if $\mathbf{M}$ is not faithful, then $\mathbf{M}(\mathrm{F}_{ij})=0$, for all $i,j$. The quotient of $\sc\mbox{C}\hskip 1.0pt_{A}$ by the $2$-ideal generated by all $\mathrm{F}_{ij}$ satisfies the assumptions of [MM5, Theorem 18] and hence $\mathbf{M}$ is equivalent to a cell $2$-representation by [MM5, Theorem 18].

So, from now on, $\mathbf{M}$ is faithful, in particular, each $\mathbf{M}(\mathrm{F}_{ij})$ is non-zero. As $A$ has a non-zero projective-injective module, by [MZ, Section 3], each $\mathbf{M}(\mathrm{F}_{ij})$ is a projective endofunctor of $B$-mod, that is, is isomorphic to a direct sum of some $\mathrm{G}_{st}$, $s,t\in\{1,2,\dots,r\}$, possibly even with multiplicities. For $i,j=1,2,\dots,k$, we denote by

• •

$X_{ij}$ the set of all $s\in\{1,2,\dots,r\}$ such that $\mathrm{G}_{st}$ is isomorphic to a direct summand of $\mathbf{M}(\mathrm{F}_{ij})$, for some $t\in\{1,2,\dots,r\}$;

• •

$Y_{ij}$ the set of all $t\in\{1,2,\dots,r\}$ such that $\mathrm{G}_{st}$ is isomorphic to a direct summand of $\mathbf{M}(\mathrm{F}_{ij})$, for some $s\in\{1,2,\dots,r\}$.

As each $\mathbf{M}(\mathrm{F}_{ij})$ is non-zero, each $X_{ij}$ and each $Y_{ij}$ is not empty.

Recall that $A$ is assumed to have a non-zero projective-injective module. Then there exist $i_{0},j_{0}\in\{1,2,\dots,k\}$ such that $A\varepsilon_{i_{0}}\cong\mathrm{Hom}_{\Bbbk}(\varepsilon_{j_{0}}A,\Bbbk)$. In this case, for every $q\in\{1,2,\dots,k\}$, the pair

[TABLE]

is an adjoint pair of $1$-morphisms, see [MZ, Lemma 5].

For $s=1,2,\dots,r$, we denote by $P_{s}$ the indecomposable projective $B$-module $B\epsilon_{s}$ and by $L_{s}$ the simple top of $P_{s}$. Whenever it does not lead to any confusion, we will use action notation and simply write e.g. $\mathrm{F}_{ij}\,M$, for $M\in\mathbf{M}(\mathtt{i})$ and $M\in B$-mod, instead of $\mathbf{M}(\mathrm{F}_{ij})(M)$.

5.4. Analysis of the sets $X_{ij}$ and $Y_{ij}$

Lemma 20**.**

For $i=1,2,\dots,k$, we have $X_{ij_{1}}=X_{ij_{2}}$, for all $j_{1},j_{2}\in\{1,2,\dots,k\}$. Similarly, for $j=1,2,\dots,k$, we have $Y_{i_{1}j}=Y_{i_{2}j}$, for all $i_{1},i_{2}\in\{1,2,\dots,k\}$.

Proof.

We prove the first claim, the proof of the second one is similar. We have

[TABLE]

Note that $\dim(\varepsilon_{j_{1}}A\varepsilon_{j_{1}})>0$. Therefore $\mathrm{F}_{ij_{2}}$ is isomorphic to a direct summand of $\mathrm{F}_{ij_{1}}\circ\mathrm{F}_{j_{1}j_{2}}$. This implies $X_{ij_{2}}\subset X_{ij_{1}}$ (as multiplication on the right cannot create new indexing idempotents on the left). By symmetry, we also have $X_{ij_{1}}\subset X_{ij_{2}}$. The claim follows. ∎

After Lemma 20, for $i=1,2,\dots,k$, we may denote by $X_{i}$ the common value of all $X_{ij}$, where $j\in\{1,2,\dots,k\}$. Similarly, for $j=1,2,\dots,k$, we may denote by $Y_{j}$ the common value of all $Y_{ij}$, where $i\in\{1,2,\dots,k\}$.

Lemma 21**.**

We have $X_{1}\cup X_{2}\cup\dots\cup X_{k}=\{1,2,\dots,r\}$.

Proof.

The set $X_{1}\cup X_{2}\cup\dots\cup X_{k}$ indexes those projectives that can be obtained using the action of $\sc\mbox{C}\hskip 1.0pt_{A}$. Therefore the claim follows immediately from transitivity of $\mathbf{M}$. ∎

Lemma 22**.**

For every $q=1,2,\dots,k$, we have $X_{q}=Y_{q}$.

Proof.

Consider the pair $(\mathrm{F}_{qj_{0}},\mathrm{F}_{i_{0}q})$ of adjoint $1$-morphisms given by (10). By adjunction, for $s\in\{1,2,\dots,r\}$, we have

[TABLE]

in particular, the left hand side is non-zero if and only if the right hand side is non-zero. At the same time, the left hand side is non-zero if and only if $P_{s}$ appears in $\mathrm{F}_{qj_{0}}\,B$, that is, $s\in X_{q}$; while the right hand side is non-zero if and only if $\mathrm{F}_{i_{0}q}\,L_{s}\neq 0$, that is, $s\in Y_{q}$. The claim follows. ∎

As an immediate consequence of Lemmata 21 and 22, we have:

Corollary 23**.**

We have $Y_{1}\cup Y_{2}\cup\dots\cup Y_{k}=\{1,2,\dots,r\}$.

Note that, until this point, we never used that $A$ is directed. It will, however, be crucial for the following two lemmata.

Lemma 24**.**

For $i_{1}\neq i_{2}\in\{1,2,\dots,k\}$, we have $X_{i_{1}}\cap X_{i_{2}}=\varnothing$.

Proof.

Let $s\in X_{i_{1}}\cap X_{i_{2}}$. Then $s\in Y_{i_{1}}\cap Y_{i_{2}}$ by Lemma 22. Without loss of generality we may assume $i_{1}<i_{2}$. Then we have that $\mathbf{M}(\mathrm{F}_{i_{2}i_{1}})\circ\mathbf{M}(\mathrm{F}_{i_{2}i_{1}})$ contains a direct summand isomorphic to $\mathrm{G}_{xs}\circ\mathrm{G}_{sy}$, for some $x,y\in\{1,2,\dots,r\}$. Note that $\mathrm{G}_{xs}\circ\mathrm{G}_{sy}\neq 0$ as $\epsilon_{s}B\epsilon_{s}\neq 0$. Consequently, we obtain

[TABLE]

At the same time, we have $\mathrm{F}_{i_{2}i_{1}}\circ\mathrm{F}_{i_{2}i_{1}}=0$ since $\varepsilon_{i_{1}}A\varepsilon_{i_{2}}=0$ by our assumption that $A$ is directed, a contradiction. The claim follows. ∎

Lemma 25**.**

For $i=1,2,\dots,k$, we have $|X_{i}|=1$.

Proof.

Let $X_{i}=\{s_{1},s_{2},\dots,s_{m}\}$. For a fixed $s_{q}$, the additive closure of all $\mathrm{F}_{xi}L_{s_{q}}$, where $x\in\{1,2,\dots,n\}$, is a non-zero $\sc\mbox{C}\hskip 1.0pt_{A}$-invariant subcategory of $B$-proj and hence must coincide with $B$-proj by transitivity of $\mathbf{M}$. This implies that all $\mathrm{G}_{s_{y}s_{q}}$, where $y\in\{1,2,\dots,m\}$, do appear as direct summands of $\mathbf{M}(\mathrm{F}_{ii})$.

As $A$ is directed, we have $\varepsilon_{i}A\varepsilon_{i}=\Bbbk\varepsilon_{i}$ and hence $\mathrm{F}_{ii}\circ\mathrm{F}_{ii}\cong\mathrm{F}_{ii}$. Therefore

[TABLE]

as well. Let $h$ denote the multiplicity of $\mathrm{G}_{s_{1}s_{1}}$ in $\mathbf{M}(\mathrm{F}_{ii})$. From the previous paragraph we know that $h>0$. Clearly, $\mathrm{G}_{s_{1}s_{1}}$ appears as a direct summand of $\mathrm{G}_{s_{1}s_{1}}\circ\mathrm{G}_{s_{1}s_{1}}$. Therefore (11) implies $h^{2}\leq h$, that is $h=1$.

If $m>1$, then $\mathrm{G}_{s_{1}s_{1}}$ appears as a direct summand of $\mathrm{G}_{s_{1}s_{2}}\circ\mathrm{G}_{s_{2}s_{1}}$. Therefore in this case (11) fails and we are done. ∎

As a consequence of Lemmata 21 and 25, we have $k=r$. Without loss of generality we may now choose $\epsilon_{i}$’s such that each $\mathrm{F}_{ij}$ acts via $\mathrm{G}_{ij}$.

Corollary 26**.**

For $i,j=1,2,\dots,k$, we have $\dim(\varepsilon_{i}A\varepsilon_{j})=\dim(\epsilon_{i}B\epsilon_{j})$.

Proof.

As each $\mathrm{F}_{st}$ acts via $\mathrm{G}_{st}$, the claim follows by comparing

[TABLE]

∎

5.5. Completing the proof of Theorem 19

After the preparation in Subsection 5.4, the proof of Theorem 19 is similar to the arguments in [MZ, Section 6] or [MaMa, Subsection 4.9]. Consider the principal $2$-representation $\mathbf{P}_{\mathtt{i}}:=\sc\mbox{C}\hskip 1.0pt_{A}(\mathtt{i},{}_{-})$ of $\sc\mbox{C}\hskip 1.0pt_{A}$, that is the left regular action of $\sc\mbox{C}\hskip 1.0pt_{A}$ on $\sc\mbox{C}\hskip 1.0pt_{A}(\mathtt{i},\mathtt{i})$. The additive closure of all $\mathrm{F}_{i1}$, where $i=1,2,\dots,k$, in $\sc\mbox{C}\hskip 1.0pt_{A}(\mathtt{i},\mathtt{i})$ is $\sc\mbox{C}\hskip 1.0pt_{A}$-invariant and gives a $2$-representation which we denote by $\mathbf{N}$. The latter has a unique $\sc\mbox{C}\hskip 1.0pt_{A}$-invariant left ideal $\mathbf{I}$ and the corresponding quotient is exactly the cell $2$-representation, see [MM2, Subsection 6.5].

Mapping $\mathbbm{1}_{\mathtt{i}}$ to $L_{1}\in B$-mod, gives rise to a $2$-natural transformation $\Phi$ from $\mathbf{N}$ to $\mathbf{M}$ which, because of the results in Subsection 5.4, sends indecomposable objects to indecomposable objects. Since, by Corollary 26, the Cartan matrices of $A$ and $B$ coincide, it follows that $\Phi$ must annihilate $\mathbf{I}$ and hence induce an equivalence between the cell $2$-representation $\mathbf{N}/\mathbf{I}$ and $\mathbf{M}$. This completes the proof.