Simple transitive 2-representations of bimodules over radical square zero Nakayama algebras via localization

We study the classiﬁcation problem of simple transitive 2-representations of the 2-category of ﬁnite-dimensional bi-modules over a radical square zero Nakayama algebra. This results in a complete classiﬁcation of simple transitive 2-representations whose apex is a ﬁnitary two-sided cell. We deﬁne a notion of localization of 2-representations. We construct previously unknown simple transitive 2-representations as localizations of cell 2-representations. Using the universal property of our construction we prove that any simple transitive 2-representation with ﬁnitary apex is equivalent to a localization of a cell 2-representation. © 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).


Introduction
The main aim of this paper is to prove a significant generalization of [12,Conjecture 2].The conjectured result concerns the classification of a class of simple transitive 2-representations of the 2-category of all bimodules over the algebra D = k[x]/ x 2 .We prove not only the aforementioned conjecture, but also an analogous statement for the case where the algebra D is replaced by an arbitrary radical square Nakayama algebra.
Ever since the notion of categorification was first formulated in [4,5], the role of categorical actions in various areas of mathematics has been increasingly important.Among the instances of categorical actions are celebrated results such as the categorification of the Jones polynomial in [15] and the proof of Broué's conjecture for symmetric groups in [3].The increasing abundance of categorical actions led to the emergence of systematic approaches to higher representation theory, such as those initiated in [3], [28], and [22].
The program initiated by [22] developed the setting of so-called finitary 2-categories and their finitary 2-representations.It provides an axiomatic framework for 2representation theory which crucially admits an abstract, well-behaved categorification of a simple module, known as a simple transitive 2-representation.The resulting 2representation theory is general enough to include many interesting previously known 2-representations, especially those coming from Lie theory, such as the actions of projective functors on the BGG category O.At the same time, due to the multiple finiteness assumptions, it is also restrictive enough to allow for classification of simple transitive 2-representations for various finitary 2-categories.
Given a finite-dimensional algebra A, a monoidal subcategory A of (A -mod-A, ⊗ A ) which admits an additive generator gives rise to a finitary 2-category A, by delooping and strictifying.The 2-category of 2-representations of A is biequivalent to the 2-category of 2-representations of A. An important example of a finitary 2-category is C A , which is obtained from (add {A, A ⊗ k A} , ⊗ A ) via the procedure described above.One of the earliest complete classification results in finitary 2-representation theory, [23,Theorem 15], states that, for self-injective A, any simple transitive 2-representation of C A is equivalent to a so-called cell 2-representation.The generalization, [25,Theorem 12], allows us to remove the assumption about A being self-injective, but it is a much more difficult result.This can be explained by the fact that C A is fiat (in the terminology of [28,6], it admits left and right duals) if and only if A is self-injective.
Most of the finitary 2-categories that brought the subject its initial interest are fiat.However, from the point of view of 2-representation theory, the question of removing the assumption about A being self-injective in the example above is natural.Further, it is easy to construct other non-fiat 2-categories of bimodules over finite-dimensional algebras.The simple transitive 2-representations of an example of such a 2-category were studied in [34], and later classified in [31].In the fiat case, the main tool for classification of simple transitive 2-representations is the study of coalgebra 1-morphisms, as described in [18].In the non-fiat case, [31] constructs 2-representations as 2-categorical weighted colimits of previously known 2-representations.
In view of the preceding paragraphs, an immediate question to ask is whether it is possible to classify simple transitive 2-representations of A -mod-A itself, for some well-chosen A. This 2-category is finitary if and only if A ⊗ k A op is of finite type.All such algebras belong to a single countable family, see [27,Theorem 1] for a detailed account.In this paper we consider a countable family {Λ n } of algebras such that Λ n ⊗ k Λ op n is of tame type, consisting of radical square zero Nakayama algebras.
Our object of study is the family of 2-categories D n associated to Λ n -mod-Λ n .In particular, these 2-categories are not finitary, neither do they fit the generalized finitary setting of [20,21].We use and generalize the results of [11,12].From [11, Theorem 1], we know that all except for one of the idempotent J -cells of D n are finitary.
Our main result, Theorem 8, gives a complete classification of simple transitive 2representations of D n with finitary apex, for all n ≥ 1.
To prove this, we use the approach of [31] and construct the 2-representations indicated in [12, Conjecture 2] as 2-categorical colimits.Interpreted in the 2-category Cat, the colimit we consider gives the classical notion of localization of categories, as described in [7].Indeed, the 2-representations we construct can be described as localizations of cell 2-representations, acting on localizations of the target categories of said cell 2-representations.In particular, we prove that, generally, the localization of a simple transitive 2-representation is simple transitive (Proposition 5).
In our case, the use of colimits can be argued to be more essential to the classification than in the case of [31].Indeed, [12,Theorem 21] shows that the 2-representations we construct cannot be obtained using coalgebra 1-morphisms.This illustrates a fundamental difference between the general (locally) finitary 2-representation theory and the 2-representation theory of fiat 2-categories.
We remark that, following [19], we pass from the 2-categorical setting to the bicategorical setting.As explained in [19,Section 2], the resulting classification problem for simple transitive 2-representations is the same as that in the strict 2-categorical setting.
This paper is organized as follows: in Section 2 we introduce the bicategorical setting in which we work and recall necessary definitions and facts regarding birepresentations.In Section 3 we introduce localization of birepresentations and prove some elementary results about it.In Section 4 we introduce and give a detailed description of the bicategory D n of finite-dimensional bimodules over a radical square zero Nakayama algebra and state the main theorem.Section 5 determines the possible action matrices associated to a simple transitive 2-representation of D n .Finally, in Section 6 we use localization to construct a family of new birepresentations, and prove that any simple transitive birepresentation with finitary apex is equivalent to a member of this family.Thereby we complete the proof of our main result, Theorem 8.

Bicategorical setup
Throughout we assume k to be a field.In Section 3 this assumption could be relaxed by letting k be an integral domain, whereas in Sections 4, 5, 6 we will further assume that k is algebraically closed and of characteristic zero.
A bicategory C is k-linear if, for all i, j ∈ Ob C , the category C (i, j) is k-linear and if composition of 1-morphisms is given by k-bilinear functors.Observe that we do not implicitly assume k-linear categories to be additive or idempotent split -these are properties we treat separately, and, indeed, many of the k-linear categories we consider (e.g.free k-linear categories) will not have these properties.Equivalently, a k-linear bicategory is a category enriched in the monoidal 2-category Cat k , consisting of klinear categories, k-linear functors and natural transformations.For detailed accounts of monoidal bicategories and of enriched bicategories, see [9], [8].
A category C is finitary if it is small and equivalent to the category of finitely generated projective modules over a finite-dimensional associative k-algebra.A k-linear bicategory C is finitary if, for all i, j ∈ Ob C , the category C (i, j) is finitary.
Given k-linear bicategories C, D , a k-linear pseudofunctor M : C → D is a pseudofunctor of the underlying bicategories such that the functor M i,j is k-linear, for all i, j ∈ Ob C .
By A k we denote the (1, 2)-full 2-subcategory of Cat k whose objects are finitary categories.By R k we denote the 2-full 2-subcategory of Cat k whose objects are abelian klinear categories and whose 1-morphisms are right exact functors between such categories.By Cat D k we denote the (1, 2)-full 2-subcategory of Cat k whose objects are k-linear, additive and idempotent split categories.Note that such a category is finitary if and only if it is hom-finite and has finitely many isomorphism classes of indecomposable objects.
For a k-linear bicategory C, we refer to k-linear pseudofunctors C → Cat k as birepresentations of C .In particular, a finitary birepresentation is a k-linear pseudofunctor C → A k .Similarly, an abelian birepresentation of C is a k-linear pseudofunctor C → R k .We will mainly be focusing on finitary birepresentations.
Given k-linear bicategories C, D , we denote by [C, D ] the k-linear bicategory of klinear pseudofunctors from C to D , strong transformations between such pseudofunctors, and modifications of such transformations.Similarly to k-linear categories and functors, the suitable "linearizations" of standard bicategorical notions, as presented for instance in [17], coincide with the enriched notions given in [8].
We remark that, in view of the strictification results of [29, Section 4.2], [19, Section 2.3], classification problems for 2-representations (in the sense of [22]) of a k-linear 2-category C are equivalent to the same classification problems concerning birepresentations of C , or any bicategory biequivalent to C.
A finitary birepresentation M of C is called simple transitive if it does not admit a non-trivial C -stable ideal, i.e. a family of ideals I = (I(i) ⊆ M(i)) i∈Ob C such that MF(I(i)) ⊆ I(j), for all F ∈ C (i, j).
If C has a unique object i, the rank of M is the number of isomorphism classes of indecomposable objects of M(i).If C has more objects, one may consider a function rank M : Ob C → N.
With the exception of Section 3, we will let C be a finitary bicategory and M a finitary birepresentation of C, unless otherwise stated.

Cells
The left preorder ≤ L on the set of isomorphism classes of indecomposable 1-morphisms of C is defined by setting F ≤ L G if there is a 1-morphism H such that G is a direct summand of H • F .We denote the resulting equivalence relation by ∼ L , and refer to its equivalence classes as left cells.Similarly one defines the right and two-sided preorders ≤ R , ≤ J , together with right and two-sided equivalence relations and right and two-sided cells.
Let M be a simple transitive birepresentation of C. By [2, Lemma 1], the collection of two-sided cells of C which are not annihilated by M admits a unique maximal element J with respect to the two-sided order.We refer to J as the apex of M.
A two-sided cell J is called idempotent given that there exist F, G, H ∈ J such that F is a direct summand of G • H.The apex is necessarily idempotent, see [2, Lemma 1].
Let L be a left cell of C.There is then a unique object i of C which is the domain of all 1-morphisms in L. To L we associate a simple transitive subquotient C L of the principal birepresentation P i = C (i, −); see [23,Subsection 3.3] for details.We call C L is called the cell birepresentation corresponding to L.

Action matrices
Let C, D be a pair of finitary categories and let F : C → D be a k-linear functor.Let X 1 , . . ., X n be a complete, irredundant list of isomorphism classes of indecomposable objects in C and let Y 1 , . . ., Y m be such a list for D. With respect to these, the action matrix [F ] of F is the m × n matrix with non-negative integer entries, defined by In particular, given a finitary birepresentation M of C and a 1-morphism F of C, the functor MF satisfies the above assumptions.Hence we obtain an action matrix [MF ].If there is no risk of ambiguity, we may sometimes write [F ] for [MF ].

Abelianization
Given a finitary birepresentation M of C , we have its (projective) abelianization M as defined in [18,Section 3].It is a pseudofunctor C → R k , so that M is an abelian birepresentation of C .Up to equivalence, M is recovered by restricting to the subcategories of projective objects in the underlying (abelian) categories of M.

Additive and Karoubi envelopes
For more detailed accounts of the envelope constructions and of the bicategorical adjunction given above, see [30] and [31, Section 3].

Bicategorical weighted colimits and localization
We now give a brief recollection of the general treatment of bicategorical weighted colimits of birepresentations given in [31].
Given a small k-linear bicategory I, a k-linear bicategory B, a k-linear pseudofunctor F : I → B and a k-linear pseudofunctor W : strongly natural in b.Combining various results of [8], [13], [14] and [30], the main conclusions of [31,Section 3] are the following: • the pointwise computation can be facilitated by using the Karoubi and additive envelope 2-functors, which preserve bicategorical colimits.
One of the earliest studied bicategorical colimits is localization of categories, as described in [7].In the setup described above, we may formulate it as follows: • let I be the 2-category i j S T , with two objects, two parallel non-identity 1morphisms, and a unique non-identity 2-morphism between these.
• Let W : I op → Cat be the 2-functor depicted by s t j to the terminal category 1, i to the walking isomorphism category, -S to the functor choosing the domain of the walking isomorphism f , -T to the codomain of f , -the unique non-identity 2-morphism to the natural transformation given by f .• Let C be a category and let 2 be the walking arrow category, We now describe its universal property, using the description of bicategorical colimits above (or e.g. the description of coinverters in [16,Section 6.6]).Given a category D, let Cat(C, D) S be the full subcategory of Cat(C, D), an object of which is a functor F such that F • ω C • I is a natural isomorphism.Equivalently, F(s) is an isomorphism in D, for any s ∈ S. The universal property is given by an equivalence of 2-functors: Using the bicategorical Yoneda lemma, we may also obtain the localization functor Q Given a k-linear, additive, idempotent split category C and a small category I, the functor category Cat(I, C) also is k-linear, additive and idempotent split.In fact, we have the canonical isomorphism Cat(I, C) Cat k (kI, C), where kI is the free k-linear category on I.We thus obtain a k-linear 2-functor Consider again the particular case I = 2.

MF (S(i)) ⊆ S(j).
We say that S is multiplicative if S(i) is a subcategory of M(i), for all i ∈ Ob C .
As an immediate consequence of the definition, there is a canonical correspondence between locally full subbirepresentations of M → and C-stable collections in M. Indeed, a locally full subbirepresentation K of a birepresentation N of C is uniquely determined by a tuple (K(i)) i∈Ob C of collections of objects of N(i) such that NF(K(i)) ⊆ K(j).
Let S I − → M → be a locally full subbirepresentation of M → , and let S be its corresponding C -stable collection.
Consider again the 2-category I which we used in the above definition of localization of categories.Let k I be the free k-linear 2-category on I (its objects and 1-morphisms coincide with those of I, and its spaces of 2-morphisms are linearizations of the sets of 2-morphisms in I).A k-linear pseudofunctor from k I to a k-linear bicategory D can be canonically identified with an ordinary pseudofunctor from I to (the underlying bicategory of) D. Hence, if D is a k-linear 2-category, then a k-linear 2-functor k I → D is given by a diagram in D of shape I.
In Since bicategorical colimits in [C, Cat D k ] are constructed pointwise in Cat D k , we conclude that, for every i ∈ Ob C , we have M[S −1 ](i) M(i)[S(i) −1 ].
Reading off the universal property from (1), we find the following: Equivalently, Θ is a strong transformation such that Θ i (s) is invertible, for any i ∈ Ob C and any s ∈ S(i).
By Yoneda lemma for bicategories, the components Υ i : M(i) → M(i)[S(i) −1 ] of the localization transformation Υ : M → M[S −1 ] are given by the indicated localization functors.
Example 4. Consider the quiver A 2 : 1 a − → 2. Let kA 2 be the free k-linear category on A 2 .The k-linear localization of kA 2 by the morphism a is the free k-linear category k A 2 on the category A 2 , which admits the following presentation: Indeed, it is easy to verify that if kA 2 F − → D is a functor such that F(a) is invertible, then we may uniquely extend F to F : k A 2 → D by setting F(a −1 ) = F(a) −1 .It is also clear that k A 2 is equivalent to kA 1 , where A 1 is the quiver with a unique vertex and no arrows. Hence, Similarly, we may extend the above argument to the case of a quiver of the form For I ⊆ {1, . . ., m}, localizing the category of projectives over the path algebra of this quiver by {s i | i ∈ I} gives the category of projectives over the path algebra of the quiver obtained by contracting the arrows {s i | i ∈ I} and replacing the connected component i s i − → i by a single vertex s i .

Localization and finitary birepresentations
We now give two properties of localization of birepresentations which are particularly relevant for the study of finitary birepresentations.
Consider the ideal Ker π of M. Since M is simple transitive, this ideal is zero or all of M.
If Ker π = M, then π = 0 and hence, since π determines π up to invertible modification, we see that also π = 0, showing that I is all of M[S −1 ].
If Ker π = 0, then π is locally faithful.But π being locally faithful implies that also Q is locally faithful.However, since M is finitary, the category M(i) is balanced (mono and epi implies iso), for all i ∈ Ob C .A faithful functor from a balanced category reflects isomorphisms, since faithful functors reflect monomorphisms and epimorphisms.We thus see that Q is given by conservative functors, which implies that S consists of isomorphisms, and so M M[S −1 ], proving that I is zero.Beyond the finitary case, the same argument holds whenever M(i) is balanced, for all i ∈ Ob C .
Recall that a k-linear category is finitary if it is hom-finite, additive, idempotent split and with finitely many isomorphism classes of indecomposable objects.We now show that once we have established hom-finiteness, the last condition follows automatically: Proposition 6.Let M be a finitary birepresentation of C, and let S be a Proof.Under the assumption above, M[S −1 ](i) is hom-finite, additive and idempotent split, for every i ∈ Ob C .As a consequence, each such category is Krull-Schmidt.For each i, fix a complete list of representatives X i 1 , . . ., X i m(i) of the isomorphism classes of indecomposable objects in M(i).Since M Q − → M[S −1 ] is locally essentially surjective, we see that

. , m(i) .
Since this category is Krull-Schmidt, we can decompose each of the objects Q i (X i j ) into a direct sum of finitely many indecomposable objects.We may thus write and so it follows that . ., n(j) and j = 1, . . ., m(i) .
From this we see that indeed there are only finitely many isomorphism classes of indecomposable objects in M[S −1 ](i) and the result follows.

The bicategory of finite-dimensional Λ n -Λ n -bimodules and the main result
From now on we assume k to be an algebraically closed field of characteristic 0. Further, we assume all modules to be finite-dimensional.Throughout we use the notions of Λ-Λ-bimodules and left Λ ⊗ k Λ op -modules interchangeably.

The algebra Λ n
Let Λ 1 be the path algebra of the quiver 1 α modulo the relation α 2 = 0. Then Λ 1 is isomorphic to the algebra of dual numbers For n ≥ 2, let Q n be the following quiver: Let Λ n be the path algebra kQ n modulo the ideal generated by the relations that composition of any two arrows is 0. We denote the orthogonal, primitive idempotents associated to the vertices of Q n by ε 1 , ε 2 , . . ., ε n .
Given a positive integer n, let D n be the bicategory which has a unique object i such that D n (i, i) = Λ n -mod-Λ n , where the composition of 1-morphisms is given by tensoring over Λ n .
For each n ≥ 1, the algebra Λ n ⊗ k Λ op n is special biserial in the sense of [1].The isomorphism classes of indecomposable finite-dimensional modules over special biserial algebras were classified in [1], [33].We use the notation from [11] (up to a small change of indexing, see Remark 7).The case n = 1 requires slightly different notation, so we do not describe it in detail here but refer the reader to [12].All statements still hold for n = 1.
For n ≥ 2, the algebra Λ n ⊗ k Λ op n is isomorphic to the path algebra of the discrete torus where we identify the first row with the last row, and the first column with the last column, modulo the following relations: • composition of any two horizontal arrows is 0; • composition of any two vertical arrows is 0; • all squares commute.
Vertical arrows are of the form α i ⊗ε j , whereas horizontal arrows are of the form ε i ⊗α op j .Since we presented Λ n ⊗ k Λ op n using a discrete torus, it is natural that in some arguments we write α k , ε k for k > n, in which case we set α k := α k , for k ∈ {1, . . ., n} such that k ≡ k mod n, and similarly for ε.
We describe Λ n -Λ n -bimodules diagrammatically as representation of the above quiver with relations.For readability we only present the part of the quiver at which the value of a representation is non-zero.All arrows in the diagrams indicate action via identity operators.
The band bimodules form a three-parameter family indexed by triples (j, m, λ), where j ∈ {1, . . ., n}, m is a positive integer and λ ∈ k \ {0} is a non-zero scalar.Similarly to [12], we do not need to consider band bimodules in our arguments, hence omit a detailed exposition (which can be found for instance in [11]).
Projective-injectives: for each i, j ∈ {1, . . ., n}, there is an indecomposable bimodule P i|j = I i+1|j−1 , with the following diagrammatic presentation: String bimodules: for each i, j ∈ {1, . . ., n} and all nonnegative integers k, there are four string bimodules Below are examples of small dimensions to illustrate, for more details see [11].
i|j : The index i|j is called the initial vertex.The index k counts the number of valleys, i.e. sinks of indegree 2.
Remark 7. Compared to [11], the indexing of the string bimodules of shapes N and M is shifted.The bimodules respectively.This change in notation gives easier formulas for tensor products in Subsection 5.2.
Following [24], we call an indecomposable Λ n -Λ n -bimodule k-split if it is of the form U ⊗ k V , for indecomposable left and right A-modules U and V .The k-split Λ n -Λ nbimodules are • the projective-injective bimodules P i|j , • the simple bimodules The two-sided cells in the set of isomorphism classes of indecomposable Λ n -Λ nbimodules are the following: • the cell J split consisting of all k-split bimodules, • the cell J M 0 consisting of all bimodules M (0) i|j where i, j ∈ {1, . . ., n}, • for each positive integer k, the cell J k consisting of all string bimodules with exactly k valleys, • the cell J band consisting of all band bimodules.Moreover, the two-sided cells are linearly ordered as follows: All two-sided cells except J M 0 are idempotent.Moreover, all cells are finite, apart from J band , which has the same cardinality as the field k.

The main result
From now on, we fix a positive integer n.The main goal of this paper is to prove the following result.(2) Generalizing [12, Theorem 1(iv)], we note that the cell birepresentation corresponding to one of the left cells in J M 0 is a simple transitive birepresentation of D n of rank n with apex J 1 .In particular, recall that by [2, Lemma 1], the apex of a transitive 2-representation must be idempotent.The cell J M 0 is not idempotent, and thus does not appear in the classification of Theorem 8. (3) The minimal J -cell of the bicategory of Λ n -Λ n -bimodules, J band , is even further from the finitary setting than the remaining J -cells.Indeed, the collection of isomorphism classes of indecomposable band bimodules is parametrized by pairs (n, λ), for n ∈ Z ≥0 and λ ∈ k.The classification problem for simple transitive birepresentations with apex J band goes beyond the scope of this article, and since in the case of finitary apex our approach allows us to focus on the apex only, the band bimodules will not play a great role in our considerations.
The remainder of this paper is devoted to the proof of Theorem 8.It is structured as follows: Claim (i) is proved in Subsection 4.4.In Section 5 we take a closer look at the two-sided cells J k and prove Claim (ii).Finally, Claim (iii) is proved in Section 6.

The two-sided cell J k
We fix a positive integer k.In this section we shall establish some facts about the two-sided cell J k in order to better understand the simple transitive birepresentations of D n having it as apex.For readability we sometimes omit the upper index (k) on the elements of J k , writing N i|j for N (k) i|j and so on.

One-sided cells in J k
In [11], it was shown that every left cell in J k contains either bimodules of type W and S, or bimodules of type M and N .Similarly, every right cell contains either bimodules of type W and N , or bimodules of type S and M .More precisely, we have the egg-box diagram below.The columns of the diagram are the left cells of J k , and the rows are the right cells of J k .
In other words, in each left cell all elements have the same second coordinate of the initial vertex (i.e. the lower index i|j).Similarly, in each right cell all elements have the same first coordinate of the initial vertex.

Multiplication table
With calculations similar to those in [10], one can show that, modulo direct summands from two-sided cells strictly J-greater than J k , the multiplication table of J k is given by for all i, j, l ∈ {1, . . ., n}, together with for all U, V ∈ {M, N, W, S} whenever j = r.In particular, modulo two-sided cells strictly J-greater than J k , the 1-morphisms Moreover, setting and F i|j ⊗ F r|s = 0 for j = r.

Adjoint pairs
In contrast to the fiat/fiab setting, not every 1-morphism of D n admits a left or right adjoint.We now describe the adjoint pairs in J k .Proposition 10.For any non-negative integer k and any i, j = 1, . . ., n, the pair (S Proof.By [27, Lemma 13], it is enough to show that S (k) i|j is projective as a left Λ n -module and that Hom Λ n -mod (S Consider now the diagrammatic representation of the bimodules S (k) i|j and N (k) j|i in standard bases and with standard Λ n -action.
Recall that we have and so on, but also and so on.Now, define linear maps S (k) i|j → Λ n as follows: with all basis vectors not indicated above mapped to 0. It is easy to check that

Action matrices
Let M be a simple transitive 2-representation of D n with apex J k .In this section we completely classify all possible action matrices for MU , where U is in the cell J k .This is done by block decomposition of the action matrix of MF (with F as in the previous subsection), and reduction of diagonal blocks to the case n = 1, which was considered in [12].As a by-product of this computation, we prove Theorem 8(ii).Thereafter we consider off-diagonal blocks.
To simplify notation, given a 1-morphism G, we write [G] instead of [M(G)] for the action matrix of M(G).

Block decomposition and diagonal blocks
Using the notation and results from Subsection 5.2, together with elementary linear algebra, we make the following observations.By the above argument, we can write [F ] as a block matrix where the block F i|j is the nonzero part of [F i|j ].For example, [F 1|2 ] is the block matrix 0 F 1|2 0 0 0 0 .
Then each of the blocks F i|j is a matrix with only positive integer entries.We have where δ denotes the Kronecker delta.In particular each F i|i must satisfy the relation i|j be the part of the action matrix [U There is now a decomposition of the matrix F i|j as Using the above calculations and Proposition 10 in the case i = j, we see that M

. Simple transitive birepresentations of D n
In this section we prove Theorem 8(iii).Let M be a simple transitive birepresentation of D n .Let B be a basic algebra such that M(i) B-proj.In the arguments that follow, we identify these two categories, in particular, we will write Be ∈ M(i), for an idempotent e ∈ B. Further, the indecomposable 1-morphisms of the form U i|j , for U ∈ {M, N, W, S}, are always assumed to lie in the apex of M.
From the earlier matrix calculations, we know that, for any i, j ∈ {1, . . ., n} and U ∈ {M, N, W, S}, the action matrix [U i|j ] of the indecomposable 1-morphism U i|j has a unique non-zero row.Hence there is an indecomposable object Be i U of M(i) such that the essential image of MU i|j lies in add {Be i U }. From our matrix analysis it also follows that, for all i, we have add {Be i N } = add {Be i W } and add {Be i M } = add {Be i S }, so Be i N Be i W and Be i M Be i S .Similarly, we have Be i S Be i N if and only if the block matrix A i|i is 1 × 1.In this case, we choose a representative Be i S,N of the resulting common isomorphism class.We will often indicate this case by writing e i S = e i N , and it's negation by e i S = e i N .In the former case, we denote the common idempotent by e i S,N .
Finally, we have also shown that, for any j, all the rows in the matrix Proof.The first two cases are an immediate consequence of Proposition 14.
If there is any morphism α : Be i U → Be j V outside the identity morphisms and those of the form Be i S → Be i N , then again from our earlier description of e k S B we see that e k S B ⊗ B α = 0, for k = 1, . . ., n.
In view of the characterization given in Proposition 13, this shows that the D n -stable ideal of M(i) generated by α does not coincide with all of M(i), and hence is a proper ideal, contradicting M being simple transitive.
Corollary 16.The quiver of B, together with the labelling we use above, is given as follows: • it consists of n connected components, labelled by i ∈ {1, . . ., n}, each of type Proof.The result is an immediate consequence of Propositions 14 and 15.
The labelled quiver described in Corollary 16 should be compared with the quiver underlying the basic algebra C satisfying C-proj C(i), where C is a cell birepresentation with apex J k .The choice of j below does not matter as the different cell birepresentations are equivalent, hence we suppress it from our notation, writing C rather than C j .The quiver underlying C is of the following form: Its labelling illustrates the fact that the indecomposable objects of C(i) are indecomposable 1-morphisms of L, and the arrows correspond to the bimodule epimorphisms M i|j → N i|j .Consider the strong transformation Θ L j S : P i → M, uniquely determined by the assignment 1 i → L j S .For a 1-morphism U of D n , we have Θ L j S (U ) MU (L j S ), and so Proposition 13 implies that the codomain of Θ L j S can be restricted to M, since M is canonically embedded as the category of projective objects of M.
Similarly to [34,Theorem 6.2] and [23, Proposition 9], we conclude that Θ L j S induces a locally faithful strong transformation Σ : C → M which satisfies Σ(U i|j ) MU i|j (L j S ).Lemma 17.For any i ∈ {1, . . ., n}, the morphism Mα i is an isomorphism if and only if e i S = e i N .
Proof.If e i S = e i N , then we have MM i|j MN i|j , and so Mα is a morphism between isomorphic objects: Mα ∈ Hom M(i) MM i|j (L j S ), MN i|j (L j S ) .The morphism Mα i is non-zero since Σ is faithful.Thus, Mα i is an isomorphism, since the endomorphism algebra of MN i|j (L j S ) Be i N is simple.
If e i S = e i N , then MM i|j (L j S ) MN i|j (L j S ), which completes the proof.Proof.From MN i|j (L j S ) Be i N and MM i|j (L j S ) Be i S , we see that Σ is essentially surjective.Further, we have previously observed that Σ is faithful.From Lemma 17 and the universal property of C S −1 , we see that there is a strong transformation Σ[S −1 ] such that commutes up to invertible modification.Since Σ is essentially surjective, so is Σ[S −1 ].We now claim that, given indecomposable objects X, Y ∈ C[S −1 ], the hom-spaces Hom C[S −1 ] (X, Y ) and Hom C[S −1 ] Σ[S −1 ](X), Σ[S −1 ](Y ) are equidimensional.To see this, we recall that we have explicitly described the quiver for the category C[S −1 ](i) in Example 4 and that it is the same as the quiver for M(i).Following the description from Example 4, we also find that, on the level of quivers, Υ is given by "contracting" the A 2 -components of the quiver of C(i) labelled by indices i such that e i S = e i N .Using Proposition 15 and Lemma 17, the same conclusion can be made about Σ, which proves the claim about equidimensionality.
Next we show that Σ[S −1 ] is faithful.To do so, we show that it is such on the level of indecomposable objects.Since all the Hom-spaces in C[S −1 ] are at most 1-dimensional, respectively.Further, the natural transformation ω : dom ⇒ cod gives rise to the natural transformation ω C : dom C ⇒ cod C .• Let S be a collection of morphisms of C; equivalently, S gives a collection of objects of C → .Let S be the full subcategory of C → satisfying Ob S = S, and let I : S → C → be its inclusion functor.• We define a 2-functor F : I → Cat by the diagram S C dom C •I cod C •I ω C •I .• The bicategorical colimit W F, known as the coinverter of F, is the localization C[S −1 ].

Definition 2 .
particular, we have the diagram Cat(2, −) 1 Cat D k Dom Cod w which we may precompose with M to obtain the diagram M → M, Let M be a birepresentation of C, let S be a multiplicative C-stable collection in M and let S be its corresponding locally full subbirepresentation of M → .We define the localization M → M[S −1 ] of M by S as the coinverter of Diagram (2) in [C , Cat D k ].

Proposition 5 .
Let M be a finitary birepresentation of C and let S be a C-stable collection in M. If M is simple transitive, then so is M[S −1 ].Proof.Let I be an ideal in M[S −1 ], and consider the canonical strong transformation M[S −1 ] π − → M[S −1 ]/I.There is a strong transformation π : M → M[S −1 ]/I which sends S to isomorphisms and makes the following diagram commute up to invertible modification:

Theorem 8 . 9 .( 1 )
Fix a positive integer k.Then the following holds.(i) Any simple transitive birepresentation of D n with apex J split is equivalent to a cell birepresentation.(ii) Any simple transitive birepresentation of D n with apex J k has rank between n and 2n.(iii) For each j = 0, . . ., n, there exist exactly n j pairwise non-equivalent simple transitive birepresentations of D n with apex J k which have rank n + j.Every such birepresentation can be constructed by localizing a cell birepresentation with apex J k by a suitable D n -stable collection.Remark In the case n = 1, Theorem 8(i) −(ii) are parts of [12, Theorem 1], and Theorem 8(iii) is [12, Conjecture 2].

Corollary 11 . 1 .
Let M be a simple transitive 2-representation of D n with apex J k , for some k ≥ Then for all i, j ∈ {1, . . ., n}, the functor M(N (k) i|j ) is a projective functor.Proof.By Proposition 10, each M(N (k) i|j ) is left exact.Therefore the result follows by [12, Theorem 4].

•
Since M is simple transitive, each entry of [F ] is a positive integer.Moreover, it satisfies [F ] 2 = 4n[F ].By the Perron-Frobenius theorem, its trace is therefore 4n, cf.[32] (in particular Proposition 4.1).• As a simple transitive 2-representation does not annihilate any element of its apex, each [U(k) i|j ] is nonzero.• For i = 1, . . ., n and U ∈ {M, N, S, W }, each entry of [U (k)i|i ] is a nonnegative integer, and the matrix itself is idempotent.Thus these matrices have diagonal elements 0 and/or 1, and trace equal to rank.•For i = j, each [U (k)i|j ] has nonnegative integer entries and squares to 0, and therefore has zero diagonal.As the number of 1-morphisms of the form U (k) i|i is exactly 4n, we conclude that the diagonal of each [U (k) i|i ] contains exactly one entry equal to 1, and all the remaining diagonal entries are zeros.If [U (k) i|i ] and [V (k) j|j ] have their unique 1 on the diagonal in the same position, then [U(k) i|i ][V (k) j|j ] = [U (k) i|i ⊗ Λ n V (k) j|j ] = 0.This implies i = j.We can now choose an ordering of the indecomposable objects in M(i) such that the first elements of diag[F ] are the nonzero elements of diag[F 1|1 ], the next elements of diag[F ] are the nonzero elements of diag[F 2|2 ], and so on.Let n 1 be the number of nonzero elements in diag[F 1|1 ].Assume that [F i|j ] has a nonzero element in one of the first n 1 rows.Then [F 1|1 ][F i|j ] = 0, implying that i = 1.Similarly, if [F i|j ] has a nonzero element in on of the first n 1 columns, then [F i|j ][F 1|1 ] = 0, implying j = 1.The same arguments can be applied to the rows and columns where [F 2|2 ], . . ., [F n|n ] have their nonzero diagonal entries.
i|i satisfy all the relations of the corresponding action matrices[M k ], [N k ], [S k ] and [W k ] from[12, Section 4], where n = 1.Consequently, we have, for each i, either
subcategory of Cat k whose objects are additive k-linear categories.Similarly, let Cat K k denote the (1, 2)-full 2-subcategory of Cat k whose objects are idempotent split k-linear categories.The respective inclusion 2functors Cat ⊕ k → Cat k and Cat K k → Cat k admit bicategorically left adjoint 2-functors (−) ⊕ , (−) K , known as the additive and Karoubi envelopes, respectively.The Karoubi envelope restricts to a bicategorical left adjoint to the inclusion Cat The functors dom, cod induce k-linear 2transformations Dom, Cod : Cat(2, −) → 1 Cat D k and the natural transformation ω gives a modification w : Dom → Cod.Let M : C → Cat D k be a birepresentation of C and let M → := Cat(2, −) • M. Definition 1.A tuple S = (S(i)) i∈Ob C, where S(i) is a collection of morphisms of M(i), is said to be a C-stable collection S in M if, for any i, j ∈ Ob C and any F ∈ C (i, j), we have Proposition 12. Let U ∈ M i|j , N i|j .Then MU is an indecomposable projective functor.Proposition 15.For U, V ∈ {S, N } and i, j ∈ {1, . . ., n}, we have: [26,1(N i|j ⊕ S i|j ) are non-zero, so any indecomposable object in M(i) is isomorphic to an object of the form Be i N or Be i S , for some i.Thus, the collectione i S =e i N {Be i S , Be i N } ∪ e i S =e i N Be i S,Nis a complete and irredundant collection of isomorphism classes of indecomposable objects of M(i).Immediately, we obtainrank M = 2n − |{i | e i S = e i N }| = n + |{i | e i S = e i N }| ,Proof.From Corollary 11 we know that MN i|j is a projective functor.From the multiplication table in Subsection 5.2, together with[26, Lemma 8], we conclude that also MM i|j is a projective functor.If the action matrix of MU has a unique non-zero entry, it must equal 1 and it immediately follows that MU is indecomposable.
component indexed by i is of type A 1 if and only if e i S = e i N .Its unique vertex is labelled by i S,N ; • the connected component indexed by i is of type A 2 if and only if e i S = e i N .It is labelled as i N → i S .