Simple transitive 2representations of some 2categories of projective functors
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Abstract
We show that every simple transitive 2representation of the 2category of projective functors for a certain quotient of the quadratic dual of the preprojective algebra associated with a tree is equivalent to a cell 2representation.
Keywords
Representation theory 2category Simple transitive 2representation Cell 2representationMathematics Subject Classification
18D05 16G10 16D901 Introduction
Mazorchuk and Miemietz (2011) started a systematic study of 2representations for certain 2categories which should be thought of as analogues of finite dimensional algebras. They introduced the notion of cell 2representations as a possible 2analogue of the notion of simple modules. This was revised in Mazorchuk and Miemitz (2016b, c) where the notion of a simple transitive 2representation was introduced. A weak version of the Jordan–Hölder theory was developed in Mazorchuk and Miemitz (2016b) for simple transitive 2representations which was a convincing argument that simple transitive 2representations are proper 2analogues of simple modules. In many important cases, for example for the 2category of Soergel bimodules in type A, it turns out that every simple transitive 2representations is equivalent to a cell 2representation.
Another class of natural 2categories, for which every simple transitive 2representations is equivalent to a cell 2representation, is the class of 2categories of projective bimodules for a finite dimensional selfinjective associative algebra, see Mazorchuk and Miemitz (2016b, c). After Mazorchuk and Miemitz (2016b, c) there were several attempts to extend this results to other associative algebras. Two particular algebras were considered in Mazorchuk and Zhang (2017a) and one more in Mazorchuk et al. (2017a). These two papers have rather different approaches: the approach of Mazorchuk and Zhang (2017a) is based on existence of a nonzero projective–injective module while Mazorchuk et al. (2017a) treats the smallest algebra which does not have any projective–injective modules. Recently, Mazorchuk and Zhang (2017b) extended the approach of Mazorchuk and Zhang (2017a) and completely covered the case of directed algebras which have a nonzero projective–injective module. We refer the reader to Mazorchuk (2017) for a general overview of the problem and related results.
In this note we show that the method developed in Mazorchuk and Zhang (2017b) can also be extended to some interesting algebras which are not directed (but which have a nonzero projective–injective module). The algebras we consider are certain quotients of quadratic duals of preprojective algebras associated with trees (cf. Ringel 1998). These kinds of algebras appear naturally in Lie theory (see Stroppel 2005; Martirosyan 2014), in diagram algebras (see Huerfano and Khovanov 2001) and in the theory of Koszul algebras (see Dubsky 2017). Our main result is that, for our algebras (which are defined in Sect. 2.1), every simple transitive 2representation of the corresponding 2category of projective bimodules is equivalent to a cell 2representation.
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.
After the first version of the paper appeared on the arxiv, a more general result was proved in Mazorchuk et al. (2017b) by completely different methods. The results of Mazorchuk et al. (2017b) work for arbitrary finite dimensional algebras without any additional assumptions on existence of projective–injective modules. The proofs of Mazorchuk et al. (2017b) are based on two new ideas: a new version of the category of complexes, called the category of pyramids, and on an embedding of the original 2category into a bigger 2category which has some partial adjunction morphisms.
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 labelled 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.
 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 we set \(a_2a_1 = 0\).
 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 we set \(a_1b_1 = b_2a_2\).
 For \(v \in V\) and \(s \in S\) such that there are arrows \(a, b \in \hat{E}\) with we set \(ab = 0\).

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 multiplicityfree 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 nonisomorphic top and socle.

If \(i\in S\ne V\), 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 \(\{i,j\}\in E\).
The motivation for the above definition stems from the following examples.
Example 2.1
A second example is:
Example 2.2
2.2 The 2category \(\mathscr {C}_A\)

it has finitely many objects;

each \(\mathscr {C}(\texttt {i},\texttt {j})\) is equivalent to the category of projective modules over some finite dimensional \(\Bbbk \)algebra;

all compositions are additive and \(\Bbbk \)linear, when applicable;

all identity 1morphisms are indecomposable.

the regular AAbimodule \(_AA_A\) (this corresponds to the identity 1morphism \(\mathbbm {1}_{{\mathrm{i}}}\));

the indecomposable AAbimodule \(Ae_i \otimes _{\Bbbk } e_jA\), for some \(i,j\in \{1,2,\dots ,n\}\), we denote such a 1morphism by \(F_{ij}\).
A finitary 2representation of \(\mathscr {C}_A\) is a (strict) 2functor from \(\mathscr {C}_A\) to the 2category of small finitary additive \(\Bbbk \)linear categories. In other words, \(\mathbf {M}\) is given by an additive and \(\Bbbk \)linear functorial action on a category \(\mathbf {M}(\texttt {i})\) which is equivalent to the category Bproj 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 2representation of \(\mathscr {C}_A\) form a 2category \(\mathscr {C}_A\)afmod where 1morphisms are strong 2natural transformations and 2morphisms are modifications, see Mazorchuk and Miemietz (2016a, Section 2.3) for details.
We call a finitary 2representation \(\mathbf {M}\) of \(\mathscr {C}_A\) transitive if, for every nonzero object \(X \in \mathbf {M}(\texttt {i})\), the additive closure of \(\{FX\}\), where F runs through all 1morphisms in \(\mathscr {C}_A\), equals \(\mathbf {M}(\texttt {i})\). We call a transitive 2representation \(\mathbf {M}\) simple if \(\mathbf {M}(\texttt {i})\) has no proper \(\mathscr {C}_A\)invariant ideals. For more details on this, we refer the reader to Mazorchuk and Miemitz (2016b, c).

The cell 2representations \(\mathbf {C}_{\mathbbm {1}_{\texttt {i}}}\) 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. Mazorchuk and Miemitz 2014, Section 6).

Each cell 2representations \(\mathbf {C}_{\mathcal {L}_j}\), where \(j=1,2,\dots ,n\), is equivalent to the defining 2representation (the defining action on \(\mathcal {C}\)).
2.3 Positive idempotent matrices
One of the ingredients in our proofs is the following classification of nonnegative idempotent matrices, see Flor (1969).
Theorem 2.3
Remark 2.4
This theorem can be applied to quasiidempotent (but not nilpotent) matrices as well. If \(I^2 = \lambda I\) and \(\lambda \ne 0\), 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 2representation of \(\mathscr {C}_A\), then \(\mathbf {M}\) is equivalent to a cell 2representation.
Note that, if \(S = \varnothing \) or \(S=V\), then the algebra A is selfinjective and hence \(\mathscr {C}_A\) is a weakly fiat 2category. In this case the statement follows from Mazorchuk and Miemietz (2016b, Theorem 15) and Mazorchuk and Miemietz (2016c, Theorem 33). Therefore, in what follows, we assume that \(S \ne \varnothing ,V\).
3.1 Some notation
For a simple transitive 2representation \(\mathbf {M}\) of \(\mathscr {C}_A\), we denote by B a basic \(\Bbbk \)algebra such that \(\mathbf {M}(\texttt {i})\) is equivalent to Bproj. 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 \(1 \le i,j \le r\), by \(G_{ij}\) the endofunctor of Bmod given by tensoring with the indecomposable projective BBbimodule \(B\epsilon _i \otimes \epsilon _jB\). 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 Bmodule \(B\epsilon _i\).
We may, without loss of generality, assume that \(\mathbf {M}\) is faithful since \(\mathscr {C}_A\) is simple which was shown in Mazorchuk et al. (2017a, 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 Mazorchuk andMiemitz (2016b, Theorem 18) and therefore \(\mathbf {M}\) is equivalent to the cell 2representation \(\mathbf {C}_{\mathbbm {1}_{\texttt {i}}}\) 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 nonzero. As we have seen above, A has a nonzero projective–injective module and thus, combining Mazorchuk and Zhang (2017a, Section 3) and Kildetoft et al. (2016, Theorem 2), we deduce that each \(\mathbf {M}(F_{ij})\) is a projective endofunctor of Bmod and, as such, is isomorphic to a nonempty direct sum of \(G_{st}\), for some \(1 \le s,t \le r\), possibly with multiplicities.
3.2 The sets \(X_i\) and \(Y_i\)

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

\( Y_{ij} := \{t\;\;\) \(G_{st}\) is isomorphic to a direct summand of \(\mathbf {M}(F_{ij})\), for some \(1 \le s \le r\}\).
In Mazorchuk and Zhang (2017b, 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 Mazorchuk and Zhang (2017b, Lemmas 21, 22), 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 1morphism 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
Let now \(i\not \in S\). We may restrict the action of \(\mathscr {C}_A\) to the 2full finitary 2subcategory \(\mathscr {D}\) of \(\mathscr {C}_A\) whose indecomposable 1morphisms are the ones which are isomorphic to either \(\mathbbm {1}_\texttt {i}\) or \(F_{ii}\). This 2category, clearly, has only strongly regular twosided cells. As \(i\not \in S\), the projective module \(P_i\) is also injective and hence \(F_{ii}\) is a selfadjoint functor (see Mazorchuk and Miemietz 2011, Subsection 7.3). Therefore \(\mathscr {D}\) satisfies all assumptions of Mazorchuk andMiemitz (2016b, Theorem 18) and hence every simple transitive 2representation of \(\mathscr {D}\) is equivalent to a cell 2representation.
The 2category \(\mathscr {D}\) has two left cells (both are also twosided cells) and each left cell contains a unique indecomposable 1morphism. The matrix of \(F_{ii}\) in these 2representations is either (0) or (2). This implies that, for \(i\not \in S\), all diagonal elements in \([F_{ii}]\) are either equal to 0 or to 2. As \([F_{ii}]\) has trace 2, it follows again that \([F_{ii}]\) contains a unique nonzero diagonal element and thus \(X_i = 1\). \(\square \)
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 \(i \ne j\), we have \(X_i \cap X_j = \varnothing \).
Proof
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 \(\dim {e_iAe_j} = \dim {\epsilon _iB\epsilon _j}\).
Proof
3.4 Proof of Theorem 3.1
With the results of Sect. 3.3 at hand, the proof of Theorem 3.1 can now be done using similar arguments as used inMazorchuk and Zhang (2017b, Section 5) or Mackaay and Mazorchuk (2017, Subsection 4.9). Consider the principal 2representation \(\mathbf {P}_{\texttt {i}} := \mathscr {C}_A(\texttt {i}, \_ )\) 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 2representation of \(\mathscr {C}_A\). By Mazorchuk and Miemitz (2014, 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 2representation \(\mathbf {C}_{\mathcal {L}_1}\).
Now, mapping \(\mathbbm {1}_{\texttt {i}}\) to the simple object corresponding to \(Q_1\) in the abelianization of \(\mathbf {M}\), induces a 2natural transformation \(\Phi : \mathbf {N} \rightarrow \mathbf {M}\). Due to the results of the previous subsection, we know that \(\Phi \) maps indecomposable 1morphisms 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 \(\mathbf {N}\slash \mathbf {I}\cong \mathbf {C}_{\mathcal {L}_1}\) and \(\mathbf {M}\). The claim of the theorem follows.
Notes
Acknowledgements
The author wants to thank his supervisor Volodymyr Mazorchuk for many helpful discussions.
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