Simple transitive 2-representations for two non-fiat 2-categories of projective functors
We show that any simple transitive 2-representation of the 2-category of projective endofunctors for the quiver algebra of $\mathbb{k}(\bullet \rightarrow \bullet )$ and for the quiver algebra of $\mathbb{k}(\bullet \rightarrow \bullet \rightarrow \bullet )$ is equivalent to a cell 2-representati...
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507492086185984 |
|---|---|
| author | Mazorchuk, V. S. Zhang, Xiaoting Мазорчук, В. С. Чжан, Сяотин |
| author_facet | Mazorchuk, V. S. Zhang, Xiaoting Мазорчук, В. С. Чжан, Сяотин |
| author_sort | Mazorchuk, V. S. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:22:46Z |
| description | We show that any simple transitive 2-representation of the 2-category of projective endofunctors for the quiver algebra of
$\mathbb{k}(\bullet \rightarrow \bullet )$ and for the quiver algebra of $\mathbb{k}(\bullet \rightarrow \bullet \rightarrow \bullet )$ is equivalent to a cell 2-representation. |
| first_indexed | 2026-03-24T02:10:10Z |
| format | Article |
| fulltext |
UDC 512.5
V. Mazorchuk* (Uppsala Univ., Sweden),
X. Zhang (East China Normal Univ., Shanghai, China)
SIMPLE TRANSITIVE \bftwo -REPRESENTATIONS
FOR TWO NON-FIAT \bftwo -CATEGORIES OF PROJECTIVE FUNCTORS
ПРОСТI ТРАНЗИТИВНI \bftwo -ЗОБРАЖЕННЯ ДВОХ НЕФIАТНИХ \bftwo -КАТЕГОРIЙ
ПРОЕКТИВНИХ ФУНКТОРIВ
We show that any simple transitive 2-representation of the 2-category of projective endofunctors for the quiver algebra of
\Bbbk (\bullet // \bullet ) and for the quiver algebra of \Bbbk (\bullet // \bullet // \bullet ) is equivalent to a cell 2-representation.
Показано, що будь-яке просте транзитивне 2-зображення 2-категорiї проективних ендофункторiв для алгебри сагай-
дака \Bbbk (\bullet // \bullet ) та алгебри сагайдака \Bbbk (\bullet // \bullet // \bullet ) еквiвалентне клiтинковому 2-зображенню.
1. Introduction and description of the results. Classification problems are interesting and impor-
tant problems in the classical representation theory. For example, classifications of various classes of
simple or indecomposable modules over different classes of algebras played significant role in both
development and applications of modern representation theory.
Higher representation theory is a recent direction of representation theory that takes its origins
from the papers [2, 3, 18, 19]. Of particular interest in higher representation theory is the study of
so-called finitary 2-categories as the latter are natural 2-analogues of finite dimensional algebras.
Initial abstract study of finitary 2-categories and the corresponding 2-representation theory was done
in [12 – 17, 20].
As an outcome of this study, one interesting and important class of 2-representations, called
simple transitive 2-representations, was defined in [16]. These 2-representations are natural 2-
analogues of usual simple modules over algebras. Therefore the problem of classification of simple
transitive 2-representations is natural and interesting. In several cases, it turns out that simple
transitive 2-representations can be classified, see, for example, various results in [5, 16, 17, 22, 23].
We also refer the reader to [6, 7, 11, 21] to related questions and applications. In particular, in [7],
classification of simple transitive 2-representations for the 2-category of Soergel bimodules over the
coinvariant algebra of the symmetric group was crucially used for classification of projective functors
in parabolic category \scrO for sln.
The most basic example of a 2-category is the 2-category CA of projective functors for a finite-
dimensional algebra A over an algebraically closed field \Bbbk , defined in [12] (Subsection 7.3). In
[14, 17], it is shown that categories of the form CA essentially exhausts a natural class of “simple”
finitary 2-categories possessing weak involutions. For such 2-categories, it was shown in [16, 17] that
simple transitive 2-representations are exactly the cell 2-representations, defined in [12]. Existence
of a weak involution on a 2-category restricts the classification result to the case when A is a
self-injective algebra.
* V. Mazorchuk is partially supported by the Swedish Research Council.
c\bigcirc V. MAZORCHUK, X. ZHANG, 2018
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 12 1625
1626 V. MAZORCHUK, X. ZHANG
The aim of the present paper to classify simple transitive 2-representations of CA for the smallest
possible non-self-injective algebra, namely the path algebra A of the quiver 1 - \rightarrow 2, over an
algebraically closed field \Bbbk . It turns out that our approach also extends, with a significantly increased
amount of technical work, to the quiver algebra of \Bbbk ( \bullet // \bullet // \bullet ), where, as usual, the
dotted arrow depicts the corresponding zero relation. Our main result is the following theorem, we
refer the reader to Sections 2 for details on all definitions.
Theorem 1. For A = \Bbbk (\bullet \rightarrow \bullet ) or A = \Bbbk ( \bullet // \bullet // \bullet ), any simple transitive
2-representation of the 2-category CA is equivalent to a cell 2-representation.
Despite of the fact that the formulation of Theorem 1 is rather similar to the corresponding
statement in the case when A is self-injective, considered in [16, 17], our approach to the proof
is fairly different, since the general approach outlined in [16, 17] does not apply. Our approach,
rather, has many similarities with the approach in [23] and is mostly based on a careful analysis of
all possible cases.
In Section 2, we collect all necessary preliminaries for 2-representation theory of the 2-category
CA. In Section 3, we prove some general results about 2-representations of CA under the additional
assumption that the algebra A has a non-zero projective injective module.
In Sections 4, 5 and 6, we collect the proof of Theorem 1 in the case A = \Bbbk (\bullet \rightarrow \bullet ). In more
details, Section 4 contains preliminaries on CA for A = \Bbbk (\bullet \rightarrow \bullet ). Section 5 contains combinatorial
results on certain integer matrices which allow us to specify three essentially different cases which
we have to deal with. In Section 6 we prove Theorem 1 for A = \Bbbk (\bullet \rightarrow \bullet ).
In Sections 7, 8, 9 and 10, we collect the proof of Theorem 1 in the second case of the algebra
A = \Bbbk ( \bullet // \bullet // \bullet ). In more details, in Section 7 one finds preliminaries on CA. Sec-
tions 8 and 9 are devoted to finding out which integer matrix captures the combinatorics of a faithful
simple transitive 2-representation of CA. Finally, Section 10 completes the proof of Theorem 1 for
the algebra \Bbbk ( \bullet // \bullet // \bullet ).
2. \bftwo -Category C\bfitA and its \bftwo -representations. 2.1. Notation and conventions. Throughout
the paper we work over an algebraically closed field \Bbbk and abbreviate \otimes \Bbbk by \otimes . Unless explicitly
stated otherwise, by a module, we mean a left module. We compose maps from right to left. For a
1-morphism \mathrm{F}, we denote by \mathrm{i}\mathrm{d}\mathrm{F} the identity 2-morphism for \mathrm{F}.
2.2. \bftwo -Category C\bfitA . We refer the reader to [9 – 11] for generalities on 2-categories. A 2-
category is a category which is enriched over the monoidal category \bfC \bfa \bft of small categories.
Let A be a connected, basic, finite dimensional \Bbbk -algebra and \scrC a small category equivalent to
A-\mathrm{m}\mathrm{o}\mathrm{d}. Consider the 2-category CA (which depends on \scrC ) defined as follows:
CA has one object \tti which we identify with \scrC ;
1-morphisms in CA are endofunctors of \scrC isomorphic to functors given by tensoring with A-A-
bimodules from the additive closure of both AAA and AA\otimes AA;
2-morphisms in CA are natural transformations of functors.
The 2-category CA is finitary in the sense of [12] (Subsection 2.2).
2.3. \bftwo -Representations C\bfitA . We consider the 2-category CA-afmod of all finitary 2-representations
of CA. In this 2-category,
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SIMPLE TRANSITIVE 2-REPRESENTATIONS FOR TWO NON-FIAT 2-CATEGORIES . . . 1627
an object is a strict additive functorial action of CA, denoted \bfM or similar, on an additive,
idempotent split, Krull – Schmidt, \Bbbk -linear category \bfM (\tti ) with finitely many isomorphism classes of
indecomposable objects and finite dimensional morphism spaces;
1-morphisms are 2-natural transformations;
2-morphisms are modifications.
We refer the reader to [14] for details. Two 2-representations are called equivalent provided that
there is a 2-natural transformation between them whose restriction to each object is an equivalence
of categories.
We also consider the 2-category CA-\mathrm{m}\mathrm{o}\mathrm{d} defined similarly using functorial action on categories
equivalent to module categories of finite dimensional \Bbbk -algebras. There is the diagrammatically
defined abelianization 2-functor
\cdot : CA-\mathrm{a}\mathrm{f}\mathrm{m}\mathrm{o}\mathrm{d} \rightarrow CA-\mathrm{m}\mathrm{o}\mathrm{d}.
Given a functorial action of CA on some \bfM (\tti ) as above, the 2-functor \cdot defines component-wise
a functorial action of CA on the Abelian category \bfM (\tti ) whose objects are diagrams of the form
X \rightarrow Y over \bfM (\tti ) and morphisms are given by the obvious commutative squares in which one
mods out the projective homotopy relations. We refer the reader to [13] (Subsection 4.2) for details.
A finitary 2-representation of CA is called transitive provided that, for any indecomposable
objects X and Y in \bfM (\tti ), there is a 1-morphism \mathrm{F} in CA such that Y is isomorphic to a direct
summand of \bfM (\mathrm{F})X.
A transitive 2-representation \bfM is called simple provided that \bfM (\tti ) does not have non-zero
proper CA-invariant ideals.
For simplicity, we will often use the module notation \mathrm{F}X instead of the representation notation
\bfM (\mathrm{F})X.
We note that each strict monoidal category can be viewed as a 2-category with one object. With
this identification, the above notion of 2-representation corresponds to the notion of strict monoidal
functor. One can view 2-representations (with a fixed target) as homomorphism 2-categories in an
appropriate version of the 3-category 2-\bfC \bfa \bft (here our choice of the level of strictness for transfor-
mations corresponds to strong transformations in the language of [9] (see [14], Subsection 2.3) for
details).
2.4. Cells in C\bfitA . Let 1 = e1 + e2 + . . . + en be a primitive decomposition of 1 \in A. Up
to isomorphism, indecomposable 1-morphisms in CA are given by tensoring with AAA or with
AAei \otimes ejAA, where i, j = 1, 2, . . . , n. We fix a representative \mathrm{F}0 in the isomorphism class of 1-
morphisms which correspond to tensoring with AAA. For i, j = 1, 2, . . . , n, we fix a representative
\mathrm{F}ij in the isomorphism class of 1-morphisms which correspond to tensoring with AAei \otimes ejAA.
The set of isomorphism classes of indecomposable 1-morphisms in CA has the natural structure
of a multisemigroup (see [13], Section 3) and [8]. Combinatorics of this structure is encoded into
so-called left, right and two-sided cells (see [13], Section 3). Two 1-morphisms \mathrm{F} and \mathrm{G} belong to
the same left cell provided that there exist 1-morphisms \mathrm{K}1 and \mathrm{K}2 such that \mathrm{F} is isomorphic to a
direct summand of \mathrm{K}1 \circ \mathrm{G} and \mathrm{G} is isomorphic to a direct summand of \mathrm{K}2 \circ \mathrm{F}. Right and two-sided
cells are defined similarly using composition on the right or from both sides, respectively.
For CA, the two sided-cells are
\scrJ 0 := \{ \mathrm{F}0\} and \scrJ := \{ \mathrm{F}ij : i, j = 1, 2, . . . , n\} .
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 12
1628 V. MAZORCHUK, X. ZHANG
The two-sided cell \{ \mathrm{F}0\} is a left and a right cell as well. Other left cells are
\{ \mathrm{F}ij : i = 1, 2, . . . , n\} , j = 1, 2, . . . , n.
Other right cells are
\{ \mathrm{F}ij : j = 1, 2, . . . , n\} , i = 1, 2, . . . , n.
As usual, we have
\mathrm{F}ij \circ \mathrm{F}st = \mathrm{F}
\oplus \mathrm{d}\mathrm{i}\mathrm{m}(ejAes)
it . (1)
We set
\mathrm{F} :=
n\bigoplus
i,j=1
\mathrm{F}ij
and note that
\mathrm{F} \circ \mathrm{F} \sim = \mathrm{F}\oplus \mathrm{d}\mathrm{i}\mathrm{m}(A). (2)
All 1-morphisms in the additive closure of \mathrm{F} are called projective endofunctors of \scrC . Similarly for
A-\mathrm{m}\mathrm{o}\mathrm{d}.
As usual, we will say that a pair (\mathrm{F}ij ,\mathrm{F}st) of 1-morphisms is a pair of adjoint 1-morphisms
provided that there exist 2-morphisms
\alpha : \mathrm{F}ij \circ \mathrm{F}st \rightarrow \mathrm{F}0 and \beta : \mathrm{F}0 \rightarrow \mathrm{F}st \circ \mathrm{F}ij
such that
(\alpha \circ 0 \mathrm{i}\mathrm{d}\mathrm{F}ij ) \circ 1 (\mathrm{i}\mathrm{d}\mathrm{F}ij \circ 0 \beta ) = \mathrm{i}\mathrm{d}\mathrm{F}ij and (\mathrm{i}\mathrm{d}\mathrm{F}st \circ 0 \alpha ) \circ 1 (\beta \circ 0 \mathrm{i}\mathrm{d}\mathrm{F}st) = \mathrm{i}\mathrm{d}\mathrm{F}st .
The 2-category CA is \scrJ -simple in the sense that any non-zero two-sided 2-ideal of CA contains
the identity 2-morphisms for all 1-morphisms given by projective endofunctors (see [1, 13]).
2.5. Cell \bftwo -representations. The first example of a finitary 2-representation of CA is the
principal 2-representation \bfP := CA(\tti , - ). This has a unique maximal CA-invariant ideal and the
corresponding quotient is the cell 2-representation \bfC \scrL , where \scrL = \{ \mathrm{F}0\} .
For any other left cell \scrL , the additive closure of elements in \scrL gives a 2-subrepresentation of
\bfP . This 2-subrepresentation again has a unique maximal CA-invariant ideal and the corresponding
quotient is the cell 2-representation \bfC \scrL . This latter cell 2-representation is equivalent to the defining
action of CA on the category A-proj of projective objects in A-\mathrm{m}\mathrm{o}\mathrm{d} (see [12] for details).
2.6. Matrices in the Grothendieck group. Let \bfM be a finitary 2-representation of CA and
X1, X2, . . . , Xk be a fixed complete and irredundant list of representatives of isomorphism classes
of indecomposable objects in \bfM (\tti ). For a 1-morphism \mathrm{G} in CA, we denote by [\mathrm{G}] the k \times k
matrix with non-negative integer coefficients where, for i, j = 1, 2, . . . , k, the coefficient in the
intersection of the ith row and the j th column gives the number of indecomposable direct summands
of \bfM (\mathrm{G})Xj which are isomorphic to Xi. Note that [\mathrm{G}\oplus \mathrm{H}] = [\mathrm{G}] + [\mathrm{H}] and [\mathrm{G} \circ \mathrm{H}] = [\mathrm{G}][\mathrm{H}].
2.7. Action on simple transitive \bftwo -representations. The following statement is proved in [16]
(Lemma 12).
Lemma 1. Let \bfM be a simple transitive 2-representation of CA. Then, for any non-zero object
X \in \bfM (\tti ), the object \mathrm{F}X is projective in \bfM (\tti ).
The following statement is proved in [16] (Lemma 13).
Lemma 2. Let B be a finite dimensional \Bbbk -algebra and \mathrm{G} an exact endofunctor of B-mod.
Assume that \mathrm{G} sends each simple object of B-mod to a projective object. Then \mathrm{G} is a projective
functor.
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SIMPLE TRANSITIVE 2-REPRESENTATIONS FOR TWO NON-FIAT 2-CATEGORIES . . . 1629
3. Existence of a projective-injective module guarantees exactness of the action. 3.1. Exact-
ness of the action of some projective functors. Let \bfM be a simple transitive 2-representation of
CA. Consider its abelianization \bfM . For \bfM (\tti ), let L1, L2, . . . , Lk be a complete and irredundant list
of representatives of isomorphism classes of simple objects. For i \in \{ 1, 2, . . . , k\} , denote by Pi the
indecomposable projective cover of Li and by Ii the indecomposable injective envelope of Li.
Lemma 3. Let Q be a finite dimensional \Bbbk -algebra and \mathrm{K} a right exact endofunctor of Q-\mathrm{m}\mathrm{o}\mathrm{d}.
Then the following conditions are equivalent:
(a) The functor \mathrm{K} sends projective objects to projective objects.
(b) The right adjoint \mathrm{K}\prime of \mathrm{K} is exact.
Proof. By adjunction, for a projective generator P \in Q-\mathrm{m}\mathrm{o}\mathrm{d}, we have a natural isomorphism
\mathrm{H}\mathrm{o}\mathrm{m}Q(\mathrm{K}P, - ) \sim = \mathrm{H}\mathrm{o}\mathrm{m}Q(P,\mathrm{K}
\prime
- ). (3)
If \mathrm{K}P is projective, the left-hand side of (3) is exact. Hence, the right-hand side is also exact. As P
is a projective generator, the functor \mathrm{H}\mathrm{o}\mathrm{m}Q(P, - ) detects any non-zero homology. This forces \mathrm{K}\prime to
be exact. Therefore, (a) implies (b).
Conversely, assume that \mathrm{K}\prime is exact. Then the right-hand side of (3) is exact. Hence, the left-hand
side is exact. This means that \mathrm{K}P is projective. Therefore, (b) implies (a). The claim follows.
Lemma 4. Assume that there exist s, t \in \{ 1, 2, . . . , n\} such that the left A-modules Aes and
\mathrm{H}\mathrm{o}\mathrm{m}\Bbbk (etA,\Bbbk ) are isomorphic. Then, for any i \in \{ 1, 2, . . . , n\} , the pair (\mathrm{F}it,\mathrm{F}si) is a pair of adjoint
1-morphisms.
Proof. The functor \mathrm{F}it is given by tensoring with the A-A-bimodule Aei\otimes etA. The right adjoint
of this functor is thus the functor \mathrm{H}\mathrm{o}\mathrm{m}A(Aei\otimes etA, - ). By the computation in [12] (Subsection 7.3),
the exact functor \mathrm{H}\mathrm{o}\mathrm{m}A(Aei \otimes etA, - ) is isomorphic to the functor of tensoring with the A-A-
bimodule
\mathrm{H}\mathrm{o}\mathrm{m}\Bbbk (etA, \Bbbk )\otimes eiA.
The injective A-module It \sim = \mathrm{H}\mathrm{o}\mathrm{m}\Bbbk (etA, \Bbbk ) is isomorphic to the projective A-module Aes, by
assumption. Therefore, \mathrm{H}\mathrm{o}\mathrm{m}\Bbbk (etA,\Bbbk ) \otimes eiA is isomorphic to Aes \otimes eiA. This means that \mathrm{F}si is
isomorphic to the right adjoint of \mathrm{F}it. The claim follows.
Corollary 1. Assume that there exist s, t \in \{ 1, 2, . . . , n\} such that the left A-modules Aes and
\mathrm{H}\mathrm{o}\mathrm{m}\Bbbk (etA,\Bbbk ) are isomorphic. Then, for any i \in \{ 1, 2, . . . , n\} and any 2-representation \bfN of CA,
the pair (\bfN (\mathrm{F}it),\bfN (\mathrm{F}si)) is a pair of adjoint functors.
Proof. This follows directly from Lemma 4 and definitions.
Corollary 2. Assume that there exist s, t \in \{ 1, 2, . . . , n\} such that the left A-modules Aes and
\mathrm{H}\mathrm{o}\mathrm{m}\Bbbk (etA,\Bbbk ) are isomorphic. Then, for any i \in \{ 1, 2, . . . , n\} and any finitary 2-representation \bfN
of CA, the functor \bfN (\mathrm{F}si) is exact.
Proof. This follows from the definitions by combining Lemma 3 and Corollary 1.
3.2. Auxiliary lemma.
Lemma 5. Let Q be a finite dimensional \Bbbk -algebra and \mathrm{K}, \mathrm{H} and \mathrm{G} be three endofunctors of
Q-\mathrm{m}\mathrm{o}\mathrm{d}. Assume that:
(a) \mathrm{H} is a projective functor;
(b) \mathrm{K} is right exact;
(c) \mathrm{K} sends projective objects to projective objects;
(d) \mathrm{K} \circ \mathrm{H} \sim = \mathrm{G}.
Then \mathrm{G} is a projective functor.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 12
1630 V. MAZORCHUK, X. ZHANG
Proof. By assumption (a), the functor \mathrm{H} is given by tensoring with the Q-Q-bimodule X \otimes Y,
for some projective left Q-module X and some projective right Q-module Y. By assumption (b), \mathrm{K}
is given by tensoring with some Q-Q-bimodule V. Using assumption (d), the Q-Q-bimodule that
determines the functor \mathrm{G} is given by
V \otimes Q (X \otimes Y ) \sim = (V \otimes Q X)\otimes Y. (4)
By assumption (c), V \otimes Q X is a projective left Q-module. This implies that (4) is a projective
Q-Q-bimodule and hence \mathrm{G} is a projective functor.
3.3. Exactness of the action.
Proposition 1. Assume that there exist s, t \in \{ 1, 2, . . . , n\} such that the left A-modules Aes
and \mathrm{H}\mathrm{o}\mathrm{m}\Bbbk (etA, \Bbbk ) are isomorphic. Let \bfM be a simple transitive 2-representation of CA. Then the
functor \bfM (\mathrm{F}) is exact.
Proof. Let B be a finite dimensional algebra such that \bfM (\tti ) is equivalent to B-\mathrm{m}\mathrm{o}\mathrm{d}.
For i \in \{ 1, 2, . . . , n\} , consider the 1-morphism \mathrm{F}si. By Corollary 2, the functor \bfM (\mathrm{F}si) is exact.
By Lemma 1, \bfM (\mathrm{F}si) sends any object in \bfM (\tti ) to a projective object. Therefore, by Lemma 2,
\bfM (\mathrm{F}si) is a projective endofunctor of B-\mathrm{m}\mathrm{o}\mathrm{d}.
Now, for any j \in \{ 1, 2, . . . , n\} , we have
\mathrm{F}js \circ \mathrm{F}si
\sim = \mathrm{F}\oplus k
ji ,
where k = \mathrm{d}\mathrm{i}\mathrm{m}(esAes) > 0. Therefore, \bfM (\mathrm{F}\oplus k
ji ) is a projective functor for B-\mathrm{m}\mathrm{o}\mathrm{d} by Lemma 5.
By additivity, \bfM (\mathrm{F}ji) is a projective functor for B-\mathrm{m}\mathrm{o}\mathrm{d} as well. In particular, \bfM (\mathrm{F}ji) is exact.
The claim follows.
4. The algebra \Bbbk (\bullet \rightarrow \bullet ). Let \Bbbk be an algebraically closed field. Denote by A the path
algebra, over \Bbbk , of the quiver 1
\alpha - \rightarrow 2. The algebra A has basis e1, e2 and \alpha and the multiplication
table (x, y) \mapsto \rightarrow x \cdot y is given by
x\setminus y e1 e2 \alpha
e1 e1 0 0
e2 0 e2 \alpha
\alpha \alpha 0 0
Note that e1Ae2 = 0 as A contains no paths from 2 to 1. Note also that the left A-modules Ae1 and
\mathrm{H}\mathrm{o}\mathrm{m}\Bbbk (e2A,\Bbbk ) are isomorphic.
Let \scrC be a small category equivalent to A-\mathrm{m}\mathrm{o}\mathrm{d}. Consider the corresponding finitary 2-category
CA. Up to isomorphism, indecomposable 1-morphisms in CA are \mathrm{F}0 and \mathrm{F}ij , where i, j = 1, 2.
Note that formula (2) for A reads \mathrm{F} \circ \mathrm{F} = \mathrm{F}\oplus 3. Using (1), the table of compositions for the functors
\mathrm{F}ij (up to isomorphism) is as follows:
\circ \mathrm{F}11 \mathrm{F}12 \mathrm{F}21 \mathrm{F}22
\mathrm{F}11 \mathrm{F}11 \mathrm{F}12 0 0
\mathrm{F}12 \mathrm{F}11 \mathrm{F}12 \mathrm{F}11 \mathrm{F}12
\mathrm{F}21 \mathrm{F}21 \mathrm{F}22 0 0
\mathrm{F}22 \mathrm{F}21 \mathrm{F}22 \mathrm{F}21 \mathrm{F}22
(5)
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SIMPLE TRANSITIVE 2-REPRESENTATIONS FOR TWO NON-FIAT 2-CATEGORIES . . . 1631
Set \scrJ 0 := \{ \mathrm{F}0\} and \scrJ := \{ \mathrm{F}ij : i, j = 1, 2\} . Note that the 2-category CA is not weakly fiat in the
sense of [13, 17] as the algebra A is not self-injective.
As CA is \scrJ -simple and A has trivial center, the only proper non-zero quotient of CA contains
just the identity 1-morphism (up to isomorphism) and its scalar endomorphisms (cf. [14]). Therefore
this quotient is fiat with strongly regular \scrJ -classes and hence it has a unique, up to equivalence,
simple transitive 2-representation, namely \bfC \scrL 0 , where \scrL 0 = \scrJ 0 (see [16], Theorem 18). This means
that, in order to prove Theorem 1 for A, it is enough to consider faithful 2-representations of CA.
From the formula
\mathrm{H}\mathrm{o}\mathrm{m}A-A(Aei \otimes ejA,Aes \otimes etA) \sim = eiAes \otimes etAej , (6)
for all i, j, s, t \in \{ 1, 2\} , we get the following table of \mathrm{H}\mathrm{o}\mathrm{m}CA(\tti )(X,Y ) (up to isomorphism), where
X and Y are indecomposable 1-morphisms:
X \setminus Y \mathrm{F}11 \mathrm{F}12 \mathrm{F}21 \mathrm{F}22
\mathrm{F}11 \Bbbk \Bbbk 0 0
\mathrm{F}12 0 \Bbbk 0 0
\mathrm{F}21 \Bbbk \Bbbk \Bbbk \Bbbk
\mathrm{F}22 0 \Bbbk 0 \Bbbk
(7)
5. Integer matrices for \Bbbk (\bullet \rightarrow \bullet ). 5.1. Integer matrices satisfying \bfitM \bftwo = \bfthree \bfitM . In this
section we classify all square matrices M with positive integer coefficients which satisfy M2 = 3M.
Proposition 2. Let M be a k\times k matrix, for some k, with positive integer coefficients, satisfying
M2 = 3M. Then M is one of the following matrices:
M1 := (3), M2 :=
\Biggl(
2 1
2 1
\Biggr)
, M3 :=
\Biggl(
2 2
1 1
\Biggr)
, M4 :=
\Biggl(
1 1
2 2
\Biggr)
,
M5 :=
\Biggl(
1 2
1 2
\Biggr)
, M6 :=
\left(
1 1 1
1 1 1
1 1 1
\right) .
Proof. Clearly, we have M2
i = 3Mi, for each i = 1, 2, 3, 4, 5, 6. So, we need to show that no
other square matrix with positive integer coefficients satisfies M2 = 3M.
Let M be a k \times k matrix, for some k, with positive integer coefficients satisfying M2 = 3M.
Then M is diagonalizable (as x2 - 3x has no multiple roots) and the only possible eigenvalues for M
are 0 and 3. From the Perron – Frobenius theorem it follows that the Perron – Frobenius eigenvalue 3
must have multiplicity one. Therefore, M has rank one and trace three. As all entries in M are
positive integers, we get k \leq 3.
If k = 1, then, clearly, M = M1.
If k = 3, then all diagonal entries in M are 1. As all 2\times 2 minors in M should have determinant
zero and positive integer entries, it follows that all entries in M are 1 and thus M = M6.
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1632 V. MAZORCHUK, X. ZHANG
If k = 2, then the two diagonal entries in M are 1 and 2. As the determinant of M is zero, the
two remaining entries are also 1 and 2. Therefore, M = Mi for some i \in \{ 2, 3, 4, 5\} .
Proposition 2 is proved.
5.2. The matrix [\bfF ] for a faithful simple transitive \bftwo -representation. Let \bfM be a finitary,
simple, transitive and faithful 2-representation of CA. Let M := [\mathrm{F}] be the matrix of \bfM (\mathrm{F}) and, for
i, j = 1, 2, let Mij := [\mathrm{F}ij ] be the matrix of \bfM (\mathrm{F}ij). Note that M = M11 +M12 +M21 +M22.
The symmetric group Sk acts on \mathrm{M}\mathrm{a}\mathrm{t}k\times k(\BbbZ ) by conjugation with permutation matrices. This
action corresponds to permutation of basis elements, whenever the matrix on which we act represents
an endomorphism of some free \BbbZ -module. We will call this action the permutation action.
Proposition 3. In order to respect the multiplication rule (5), up to the permutation action, we
have the following three possibilities:
(a) M = M2 and
M11 =
\Biggl(
1 0
0 0
\Biggr)
, M12 =
\Biggl(
1 1
0 0
\Biggr)
, M21 =
\Biggl(
0 0
1 0
\Biggr)
, M22 =
\Biggl(
0 0
1 1
\Biggr)
;
(b) M = M3 and
M11 =
\Biggl(
0 1
0 1
\Biggr)
, M12 =
\Biggl(
1 0
1 0
\Biggr)
, M21 =
\Biggl(
0 1
0 0
\Biggr)
, M22 =
\Biggl(
1 0
0 0
\Biggr)
;
(c) M = M6 and
M11 =
\left(
1 0 0
1 0 0
0 0 0
\right) , M12 =
\left(
0 1 1
0 1 1
0 0 0
\right) ,
M21 =
\left(
0 0 0
0 0 0
1 0 0
\right) , M22 =
\left(
0 0 0
0 0 0
0 1 1
\right) .
Proof. As \bfM is simple, transitive and faithful, we get that M has positive integer entries. As
\mathrm{F}\circ \mathrm{F} = \mathrm{F}\oplus 3, we have M = Mi for some i \in \{ 1, 2, 3, 4, 5, 6\} , by Proposition 2. As M is the sum of
four non-zero matrices (corresponding to all \mathrm{F}ij ) each of which has non-negative integer entries, we
have M \not = M1. The case M = M4 reduces to the case M = M3 by swapping the basis elements.
The case M = M5 reduces to the case M = M2 by swapping the basis elements. It is easy to check
that the cases (a), (b) and (c) listed in the formulation satisfy (5).
Assume M = M2. Note, from (5), that \mathrm{F}11, \mathrm{F}12 and \mathrm{F}22 are idempotent, while \mathrm{F}21 is nilpotent.
Therefore M11, M12, M22 must have non-zero diagonals, while the diagonal for M21 should be
zero. From M11M22 = 0 it follows that M11 and M22 cannot have common diagonal entries. In
any case, this means that M12 has the non-zero diagonal entry in the left upper corner. Let us first
assume the following:
M11 =
\Biggl(
0 \ast
\ast 1
\Biggr)
, M12 =
\Biggl(
1 \ast
\ast 0
\Biggr)
, M21 =
\Biggl(
0 \ast
\ast 0
\Biggr)
, M22 =
\Biggl(
1 \ast
\ast 0
\Biggr)
.
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SIMPLE TRANSITIVE 2-REPRESENTATIONS FOR TWO NON-FIAT 2-CATEGORIES . . . 1633
From M11M21 = 0, we get
M11 =
\Biggl(
0 0
0 1
\Biggr)
, M12 =
\Biggl(
1 0
\ast 0
\Biggr)
, M21 =
\Biggl(
0 1
0 0
\Biggr)
, M22 =
\Biggl(
1 0
\ast 0
\Biggr)
.
This, however, contradicts M11M12 = M12. Now assume
M11 =
\Biggl(
1 \ast
\ast 0
\Biggr)
, M12 =
\Biggl(
1 \ast
\ast 0
\Biggr)
, M21 =
\Biggl(
0 \ast
\ast 0
\Biggr)
, M22 =
\Biggl(
0 \ast
\ast 1
\Biggr)
.
From M11M21 = M11M22 = 0, we have
M11 =
\Biggl(
1 0
\ast 0
\Biggr)
, M12 =
\Biggl(
1 1
\ast 0
\Biggr)
, M21 =
\Biggl(
0 0
\ast 0
\Biggr)
, M22 =
\Biggl(
0 0
\ast 1
\Biggr)
.
From M12M22 = M12, we obtain
M11 =
\Biggl(
1 0
0 0
\Biggr)
, M12 =
\Biggl(
1 1
0 0
\Biggr)
, M21 =
\Biggl(
0 0
1 0
\Biggr)
, M22 =
\Biggl(
0 0
1 1
\Biggr)
.
Assume M = M3. Note, from (5), that \mathrm{F}11, \mathrm{F}12 and \mathrm{F}22 are idempotent, while \mathrm{F}21 is nilpotent.
Therefore M11, M12, M22 must have non-zero diagonals, while the diagonal for M21 should be
zero. From M11M22 = 0 it follows that M11 and M22 cannot have common diagonal entries. In
any case this means that M12 has the non-zero diagonal entry in the left upper corner. Let us first
assume the following:
M11 =
\Biggl(
1 \ast
\ast 0
\Biggr)
, M12 =
\Biggl(
1 \ast
\ast 0
\Biggr)
, M21 =
\Biggl(
0 \ast
\ast 0
\Biggr)
, M22 =
\Biggl(
0 \ast
\ast 1
\Biggr)
.
From M11M21 = 0, we get
M11 =
\Biggl(
1 0
0 0
\Biggr)
, M12 =
\Biggl(
1 \ast
0 0
\Biggr)
, M21 =
\Biggl(
0 0
1 0
\Biggr)
, M22 =
\Biggl(
0 \ast
0 1
\Biggr)
.
This, however, contradicts M21M12 = M22. Now assume the following:
M11 =
\Biggl(
0 \ast
\ast 1
\Biggr)
, M12 =
\Biggl(
1 \ast
\ast 0
\Biggr)
, M21 =
\Biggl(
0 \ast
\ast 0
\Biggr)
, M22 =
\Biggl(
1 \ast
\ast 0
\Biggr)
.
From M11M21 = M11M22 = 0, we have
M11 =
\Biggl(
0 \ast
0 1
\Biggr)
, M12 =
\Biggl(
1 \ast
1 0
\Biggr)
, M21 =
\Biggl(
0 \ast
0 0
\Biggr)
, M22 =
\Biggl(
1 \ast
0 0
\Biggr)
.
From M11M12 = M12, we obtain
M11 =
\Biggl(
0 1
0 1
\Biggr)
, M12 =
\Biggl(
1 0
1 0
\Biggr)
, M21 =
\Biggl(
0 1
0 0
\Biggr)
, M22 =
\Biggl(
1 0
0 0
\Biggr)
.
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1634 V. MAZORCHUK, X. ZHANG
Assume M = M6. Note, from (5), that \mathrm{F}11, \mathrm{F}12 and \mathrm{F}22 are idempotent, while \mathrm{F}21 is nilpotent.
Therefore M11, M12, M22 must have non-zero diagonals, while the diagonal for M21 should be
zero. Therefore, up to permutation of basis vectors, we may assume that
M11 =
\left(
1 \ast \ast
\ast 0 \ast
\ast \ast 0
\right) , M12 =
\left(
0 \ast \ast
\ast 1 \ast
\ast \ast 0
\right) ,
M21 =
\left(
0 \ast \ast
\ast 0 \ast
\ast \ast 0
\right) , M22 =
\left(
0 \ast \ast
\ast 0 \ast
\ast \ast 1
\right) .
From M11M21 = M11M22 = 0 we thus get that the last column of M11 must be zero and the first
row of both M21 and M22 must be zero. Since the Mij ’s add up to M, the rightmost element in the
first row of M12 must be 1:
M11 =
\left(
1 \ast 0
\ast 0 0
\ast \ast 0
\right) , M12 =
\left(
0 \ast 1
\ast 1 \ast
\ast \ast 0
\right) ,
M21 =
\left(
0 0 0
\ast 0 \ast
\ast \ast 0
\right) , M22 =
\left(
0 0 0
\ast 0 \ast
\ast \ast 1
\right) .
From M11M12 = M12 it follows that the second row of M11 cannot be zero, which yields
M11 =
\left(
1 \ast 0
1 0 0
\ast \ast 0
\right) , M12 =
\left(
0 \ast 1
0 1 \ast
\ast \ast 0
\right) ,
M21 =
\left(
0 0 0
0 0 \ast
\ast \ast 0
\right) , M22 =
\left(
0 0 0
0 0 \ast
\ast \ast 1
\right) .
Now M11M12 = M12 implies that the first and the second rows of M12 should coincide, moreover,
the first element in the third row in M12 should be zero and also the third row in M11 and thus also
in M12 must be zero:
M11 =
\left(
1 0 0
1 0 0
0 0 0
\right) , M12 =
\left(
0 1 1
0 1 1
0 0 0
\right) ,
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SIMPLE TRANSITIVE 2-REPRESENTATIONS FOR TWO NON-FIAT 2-CATEGORIES . . . 1635
M21 =
\left(
0 0 0
0 0 0
\ast \ast 0
\right) , M22 =
\left(
0 0 0
0 0 0
\ast \ast 1
\right) .
Now, from M21M11 = M21, we have
M11 =
\left(
1 0 0
1 0 0
0 0 0
\right) , M12 =
\left(
0 1 1
0 1 1
0 0 0
\right) ,
M21 =
\left(
0 0 0
0 0 0
\ast 0 0
\right) , M22 =
\left(
0 0 0
0 0 0
\ast 1 1
\right) .
Finally, from M12M21 = M11, we obtain
M11 =
\left(
1 0 0
1 0 0
0 0 0
\right) , M12 =
\left(
0 1 1
0 1 1
0 0 0
\right) ,
M21 =
\left(
0 0 0
0 0 0
1 0 0
\right) , M22 =
\left(
0 0 0
0 0 0
0 1 1
\right) .
Proposition 3 is proved.
6. Proof of Theorem 1 for \Bbbk (\bullet \rightarrow \bullet ). Let \bfM be a simple transitive 2-representation of CA.
Let B be a basic finite dimensional algebra such that \bfM (\tti ) is equivalent to B-proj.
As the left A-modules Ae1 and \mathrm{H}\mathrm{o}\mathrm{m}\Bbbk (e2A,\Bbbk ) are isomorphic, from Proposition 1 it follows
that the functor \bfM (\mathrm{F}) is exact. From Lemmas 1 and 2 we thus obtain that \bfM (\mathrm{F}) is a projective
endofunctor of B-\mathrm{m}\mathrm{o}\mathrm{d}.
Case 1. Assume that M = M3 and the Mij ’s are thus given by Proposition 3 (b). Let \bfM be the
abelianization of \bfM . As usual, we write P1 and P2 for indecomposable projectives in \bfM (\tti ) and L1
and L2 for their respective simple tops. Let \epsilon 1 and \epsilon 2 be the corresponding primitive idempotents
in B. For i, j = 1, 2, denote by \mathrm{G}ij an endofunctor of \bfM (\tti ) which corresponds to tensoring with
B\epsilon i \otimes \epsilon jB.
From the form of M21, we see that \mathrm{F}21 acts via \mathrm{G}12. Similarly, \mathrm{F}22 acts via \mathrm{G}11. From the
matrices M21 and M22 it follows that
[P1 : L1] = 1, [P1 : L2] = 0, [P2 : L1] = 0, [P2 : L2] = 1.
This means that B \sim = \Bbbk \oplus \Bbbk . Therefore, all \mathrm{G}ij are isomorphisms between the corresponding \Bbbk -\mathrm{m}\mathrm{o}\mathrm{d}
components. From the matrices M12 and M21 it thus follows directly that there are no nonzero
homomorphisms from \mathrm{F}21 to \mathrm{F}12. This contradicts (7) and hence Case 1 cannot occur.
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1636 V. MAZORCHUK, X. ZHANG
Case 2. Assume that M = M2 and the Mij ’s are thus given by Proposition 3 (a). Let \bfM be the
abelianization of \bfM . As usual, we write P1 and P2 for indecomposable projectives in \bfM (\tti ) and L1
and L2 for their respective simple tops. Let \epsilon 1 and \epsilon 2 be the corresponding primitive idempotents
in B. For i, j = 1, 2, denote by \mathrm{G}ij an endofunctor of \bfM (\tti ) which corresponds to tensoring with
B\epsilon i \otimes \epsilon jB.
From the form of M11, we see that \mathrm{F}11 acts via \mathrm{G}11. Similarly, \mathrm{F}21 acts via \mathrm{G}21.
From the form of M12, we see that \mathrm{F}12 acts either via \mathrm{G}12 or via \mathrm{G}11 or via \mathrm{G}12 \oplus \mathrm{G}11.
However, we already know that the matrix of \mathrm{G}11 is M11. This leaves us with possibilities \mathrm{G}12 or
\mathrm{G}12 \oplus \mathrm{G}11 for \mathrm{F}12.
Assume that \mathrm{F}12 acts via \mathrm{G}12 \oplus \mathrm{G}11. We already know the matrix of \mathrm{G}11, so the matrix of
\mathrm{G}12 is \Biggl(
0 1
0 0
\Biggr)
.
This and the matrix M11 imply that
\mathrm{G}11 P1
\sim = P1, \mathrm{G}11 P2 = 0, \mathrm{G}12 P1 = 0, \mathrm{G}12 P2
\sim = P2.
Therefore,
[P1 : L1] = 1, [P1 : L2] = 0, [P2 : L1] = 0, [P2 : L2] = 1
and we have B \sim = \Bbbk \oplus \Bbbk . This leads to the same contradiction as in Case 1 above. Therefore, \mathrm{F}12
acts via \mathrm{G}12. Similarly one shows that \mathrm{F}22 acts via \mathrm{G}22.
From the matrices for all \mathrm{G}ij ’s it follows that the Cartan matrices of A and B coincide which
implies that A and B are isomorphic (that is special for our case, but the algebra A is very
small, so this claim is clear). Furthermore, all \mathrm{F}ij ’s act via the corresponding \mathrm{G}ij . It now fol-
lows by the usual arguments (see [16], Proposition 9), that \bfM is equivalent to a cell 2-representation
of CA.
Case 3. Assume that M = M6 and the Mij ’s are thus given by Proposition 3 (c). Let \bfM be
the abelianization of \bfM . As usual, we write P1, P2 and P3 for indecomposable projectives in \bfM (\tti )
and L1, L2 and L3 for their respective simple tops. Let \epsilon 1, \epsilon 2 and \epsilon 3 be the corresponding primitive
idempotents in B. For i, j = 1, 2, 3, denote by \mathrm{G}ij an endofunctor of \bfM (\tti ) which corresponds to
tensoring with B\epsilon i \otimes \epsilon jB.
From the form of M21, we see that \mathrm{F}21 acts via \mathrm{G}31. From the form of M11, we see that \mathrm{F}11
acts via \mathrm{G}11 \oplus \mathrm{G}21. This implies
[P1 : L1] = 1, [P2 : L1] = [P3 : L1] = 0.
From the form of M22, we see that \mathrm{F}22 acts either via \mathrm{G}32 or via \mathrm{G}33 or via \mathrm{G}32 \oplus \mathrm{G}33. In the
latter case, we have that the matrices of \mathrm{G}32 and \mathrm{G}33 are, respectively:\left(
0 0 0
0 0 0
0 1 0
\right) and
\left(
0 0 0
0 0 0
0 0 1
\right) .
It follows that
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SIMPLE TRANSITIVE 2-REPRESENTATIONS FOR TWO NON-FIAT 2-CATEGORIES . . . 1637
[P2 : L2] = 1, [P1 : L2] = [P3 : L2] = 0
and
[P3 : L3] = 1, [P1 : L3] = [P2 : L3] = 0.
This implies that B \sim = \Bbbk \oplus \Bbbk \oplus \Bbbk and leads to a similar contradiction as in Case 1.
Subcase 3.1. Assume that \mathrm{F}22 acts via \mathrm{G}32. This implies
[P2 : L2] = [P3 : L2] = 1, [P1 : L2] = 0.
From [P1 : L2] = 0 we have \epsilon 2B\epsilon 1 = 0. This means that
\mathrm{H}\mathrm{o}\mathrm{m}B-B(B\epsilon 3 \otimes \epsilon 1B,B\epsilon 3 \otimes \epsilon 2B) = 0,
that is, \mathrm{H}\mathrm{o}\mathrm{m}(\mathrm{G}31,\mathrm{G}32) = 0. This contradicts \mathrm{H}\mathrm{o}\mathrm{m}C (\mathrm{F}21,\mathrm{F}22) \not = 0, see (7).
Subcase 3.2. Assume that \mathrm{F}22 acts via \mathrm{G}33. This implies
[P3 : L3] = [P2 : L3] = 1, [P1 : L3] = 0.
From [P1 : L3] = 0 we obtain \epsilon 3B\epsilon 1 = 0. This means that
\mathrm{H}\mathrm{o}\mathrm{m}B-B(B\epsilon 3 \otimes \epsilon 1B,B\epsilon 3 \otimes \epsilon 3B) = 0,
that is, \mathrm{H}\mathrm{o}\mathrm{m}(\mathrm{G}31,\mathrm{G}33) = 0. This contradicts \mathrm{H}\mathrm{o}\mathrm{m}C (\mathrm{F}21,\mathrm{F}22) \not = 0, see (7). The proof is now
complete.
7. The algebra \Bbbk (\bullet \bfitalpha - \rightarrow \bullet \bfitbeta - \rightarrow \bullet )/(\bfitbeta \bfitalpha ). Let \Bbbk be an algebraically closed field. Denote by
A the path algebra, over \Bbbk , of the quiver
\Bbbk
\Bigl(
1
\alpha - \rightarrow 2
\alpha - \rightarrow 3
\Bigr)
modulo the relations \beta \alpha = 0.
The algebra A has basis e1, e2, e3, \alpha and \beta and the multiplication table (x, y) \mapsto \rightarrow x \cdot y is given by
x\setminus y e1 e2 e3 \alpha \beta
e1 e1 0 0 0 0
e2 0 e2 0 \alpha 0
e3 0 0 e3 0 \beta
\alpha \alpha 0 0 0 0
\beta 0 \beta 0 0 0
Note that e1Ae2 = 0, e1Ae3 = 0, e2Ae3 = 0 and e3Ae1 = 0. Note also that the left A-modules Ae1
and \mathrm{H}\mathrm{o}\mathrm{m}\Bbbk (e2A,\Bbbk ) are isomorphic and the left A-modules Ae2 and \mathrm{H}\mathrm{o}\mathrm{m}\Bbbk (e3A,\Bbbk ) are isomorphic.
Let \scrC be a small category equivalent to A-\mathrm{m}\mathrm{o}\mathrm{d}. Consider the corresponding finitary 2-category
CA. Up to isomorphism, indecomposable 1-morphisms in CA are \mathrm{F}0 and \mathrm{F}ij , where i, j = 1, 2, 3.
Note that formula (2) for A reads \mathrm{F} \circ \mathrm{F} \sim = \mathrm{F}\oplus 5. Using (1), the table of compositions for the functors
\mathrm{F}ij (up to isomorphism) is as follows:
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1638 V. MAZORCHUK, X. ZHANG
\circ \mathrm{F}11 \mathrm{F}12 \mathrm{F}13 \mathrm{F}21 \mathrm{F}22 \mathrm{F}23 \mathrm{F}31 \mathrm{F}32 \mathrm{F}33
\mathrm{F}11 \mathrm{F}11 \mathrm{F}12 \mathrm{F}13 0 0 0 0 0 0
\mathrm{F}12 \mathrm{F}11 \mathrm{F}12 \mathrm{F}13 \mathrm{F}11 \mathrm{F}12 \mathrm{F}13 0 0 0
\mathrm{F}13 0 0 0 \mathrm{F}11 \mathrm{F}12 \mathrm{F}13 \mathrm{F}11 \mathrm{F}12 \mathrm{F}13
\mathrm{F}21 \mathrm{F}21 \mathrm{F}22 \mathrm{F}23 0 0 0 0 0 0
\mathrm{F}22 \mathrm{F}21 \mathrm{F}22 \mathrm{F}23 \mathrm{F}21 \mathrm{F}22 \mathrm{F}23 0 0 0
\mathrm{F}23 0 0 0 \mathrm{F}21 \mathrm{F}22 \mathrm{F}23 \mathrm{F}21 \mathrm{F}22 \mathrm{F}23
\mathrm{F}31 \mathrm{F}31 \mathrm{F}32 \mathrm{F}33 0 0 0 0 0 0
\mathrm{F}32 \mathrm{F}31 \mathrm{F}32 \mathrm{F}33 \mathrm{F}31 \mathrm{F}32 \mathrm{F}33 0 0 0
\mathrm{F}33 0 0 0 \mathrm{F}31 \mathrm{F}32 \mathrm{F}33 \mathrm{F}31 \mathrm{F}32 \mathrm{F}33
(8)
Set \scrJ 0 := \{ \mathrm{F}0\} and \scrJ := \{ \mathrm{F}ij : i, j = 1, 2, 3\} . Note that the 2-category CA is not weakly fiat in
the sense of [13, 17] as the algebra A is not self-injective.
As CA is \scrJ -simple and A has trivial center, the only proper non-zero quotient of CA contains
just the identity 1-morphism (up to isomorphism) and its scalar endomorphisms (cf. [14]). Therefore
this quotient is fiat with strongly regular \scrJ -classes and hence it has a unique, up to equivalence,
simple transitive 2-representation, namely \bfC \scrL 0 , where \scrL 0 = \scrJ 0 (see [16], Theorem 18). This means
that, in order to prove Theorem 1 for A, it is enough to consider faithful 2-representations of CA.
From (6), we get the following table of \mathrm{H}\mathrm{o}\mathrm{m}CA(\tti )(X,Y ) (up to isomorphism), where X and Y
are indecomposable 1-morphisms:
X \setminus Y \mathrm{F}11 \mathrm{F}12 \mathrm{F}13 \mathrm{F}21 \mathrm{F}22 \mathrm{F}23 \mathrm{F}31 \mathrm{F}32 \mathrm{F}33
\mathrm{F}11 \Bbbk \Bbbk 0 0 0 0 0 0 0
\mathrm{F}12 0 \Bbbk \Bbbk 0 0 0 0 0 0
\mathrm{F}13 0 0 \Bbbk 0 0 0 0 0 0
\mathrm{F}21 \Bbbk \Bbbk 0 \Bbbk \Bbbk 0 0 0 0
\mathrm{F}22 0 \Bbbk \Bbbk 0 \Bbbk \Bbbk 0 0 0
\mathrm{F}23 0 0 \Bbbk 0 0 \Bbbk 0 0 0
\mathrm{F}31 0 0 0 \Bbbk \Bbbk 0 \Bbbk \Bbbk 0
\mathrm{F}32 0 0 0 0 \Bbbk \Bbbk 0 \Bbbk \Bbbk
\mathrm{F}33 0 0 0 0 0 \Bbbk 0 0 \Bbbk
(9)
8. Integer matrices for \Bbbk (\bullet \bfitalpha - \rightarrow \bullet \bfitbeta - \rightarrow \bullet )/(\bfitbeta \bfitalpha ). 8.1. Integer matrices satisfying \bfitM \bftwo =
= \bffive \bfitM . In this section we classify all square matrices M with positive integer coefficients which
satisfy M2 = 5M.
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SIMPLE TRANSITIVE 2-REPRESENTATIONS FOR TWO NON-FIAT 2-CATEGORIES . . . 1639
Proposition 4. Let M be a k\times k matrix, for some k, with positive integer coefficients, satisfying
M2 = 5M. Then, up to permutation action, M is one of the following matrices:
N1 := (5), N2 :=
\Biggl(
4 1
4 1
\Biggr)
, N3 :=
\Biggl(
4 4
1 1
\Biggr)
, N4 :=
\Biggl(
4 2
2 1
\Biggr)
,
N5 :=
\Biggl(
3 6
1 2
\Biggr)
, N6 :=
\Biggl(
3 3
2 2
\Biggr)
, N7 :=
\Biggl(
3 2
3 2
\Biggr)
, N8 :=
\Biggl(
3 1
6 2
\Biggr)
,
N9 :=
\left(
3 1 1
3 1 1
3 1 1
\right) , N10 :=
\left(
3 3 3
1 1 1
1 1 1
\right) , N11 :=
\left(
2 2 2
2 2 2
1 1 1
\right) ,
N12 :=
\left(
2 4 2
1 2 1
1 2 1
\right) , N13 :=
\left(
2 2 1
2 2 1
2 2 1
\right) , N14 :=
\left(
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
\right)
,
N15 :=
\left(
2 1 1 1
2 1 1 1
2 1 1 1
2 1 1 1
\right) , N16 :=
\left(
2 2 2 2
1 1 1 1
1 1 1 1
1 1 1 1
\right) .
We note an important difference with Proposition 2: to make our list shorter, Proposition 4 gives
classification only up to permutation action.
Proof. Clearly, we have N2
i = 5Ni, for each i = 1, 2, . . . , 16. So, we need to show that any
other square matrix with positive integer coefficients satisfying M2 = 5M can be reduced to one of
the above using permutation action.
Let M be a k \times k matrix, for some k, with positive integer coefficients satisfying M2 = 5M.
Then M is diagonalizable (as x2 - 5x has no multiple roots) and the only possible eigenvalues for M
are 0 and 5. From the Perron – Frobenius theorem it follows that the Perron – Frobenius eigenvalue
5 must have multiplicity one. Therefore, M has rank one and trace five. As all entries in M are
positive integers, we get k \leq 5. Using the permutation action, we may assume that the entries on the
main diagonal of M weakly decrease from the top left corner to the bottom right corner.
If k = 1, then, clearly, M = N1.
If k = 2, then the diagonal of M is either (4, 1) or (3, 2). In the first case, as the determinant of
M is zero, the two remaining entries are either 2 and 2 or 4 and 1. This gives M = N2, M = N3
or M = N4. In the second case, as the determinant of M is zero, the two remaining entries are
either 2 and 3 or 1 and 6. This gives M = N5, M = N6, M = N7 or M = N8.
If k = 3, then the diagonal of M is either (3, 1, 1) or (2, 2, 1). In the first case, as M has rank
one, any 2\times 2 minor in M has determinant zero. This means that all entries which are neither in the
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1640 V. MAZORCHUK, X. ZHANG
first row nor in the first column are equal to 1. If the first row contains more than one entry different
from 1, then all entires in this row are 3 and we get M = N10. If the first column contains more
than one entry different from 1, then all entires in this column are 3 and we get M = N9.
In the second case write
M =
\left(
2 m12 m13
m21 2 m23
m31 m32 1
\right) .
Then m32m23 = 2, m31m13 = 2 and m21m12 = 4. Hence, both (m32,m23) and (m31,m13) are in
\{ (1, 2), (2, 1)\} . We can choose them independently and the fact that M has rank one then uniquely
determines the pair (m21,m12). This gives us M = N11, M = N12 and M = N13 and also the
possibility
M = N \prime
12 :=
\left(
2 1 1
4 2 2
2 1 1
\right)
which reduces to M = N12 by permutation action.
If k = 4, then the diagonal of M is (2, 1, 1, 1). As M has rank one, any 2\times 2 minor in M has
determinant zero. This means that all entries which are neither in the first row nor in the first column
are equal to 1. If the first row contains more than one entry different from 1, then all entires in this
row are 2 and we get M = N16. If the first column contains more than one entry different from 1,
then all entires in this column are 2 and we obtain M = N15.
If k = 5, then all diagonal entries in M are 1. As all 2\times 2 minors in M should have determinant
zero and positive integer entries, it follows that all entries in M are 1 and thus M = N14.
Proposition 4 is proved.
8.2. Filtering “easy cases” out. Let \bfM be a finitary, simple, transitive and faithful 2-
representation of CA. Let M := [\mathrm{F}] be the matrix of \bfM (\mathrm{F}) and, for i, j = 1, 2, 3, let Mij := [\mathrm{F}ij ]
be the matrix of \bfM (\mathrm{F}ij). We have Mij \not = 0, for all i, j = 1, 2, 3. By Proposition 4, up to permuta-
tion action, we have M = Ni, for some i \in \{ 1, 2, . . . , 16\} as in Proposition 4. Note that trace of M
is five.
As usual, we call “position (i, j)” the intersection of the ith row and the j th column of a matrix.
From now on, we assume that \bfM (\tti ) is equivalent to B-\mathrm{m}\mathrm{o}\mathrm{d}, for some basic algebra B. Let L1,
L2,. . . , Lk be a complete and irredundant list of representatives of isomorphism classes of simple
objects in \bfM (\tti ). For i \in \{ 1, 2, . . . , k\} , denote by Pi the indecomposable projective cover of Li and
by Ii the indecomposable injective envelope of Li. The matrices Mij are given with respect to this
fixed ordering of isomorphism classes of simple objects.
Lemma 6. (i) All diagonal elements in M13, M21, M31 and M32 are zero.
(ii) Each of the matrices M11, M12, M22, M23 and M33, has exactly one entry equal to 1 on
the diagonal and all other diagonal entries are zero.
Proof. From (8), we see that \mathrm{F}ij is idempotent if and only if
(i, j) \in \{ (1, 1), (1, 2), (2, 2), (2, 3), (3, 3)\}
and \mathrm{F}2
ij = 0 otherwise. As the trace of a non-zero idempotent with non-negative coefficients is
non-zero, each idempotent Mij has trace at least one. As trace of M is five, it follows that all
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SIMPLE TRANSITIVE 2-REPRESENTATIONS FOR TWO NON-FIAT 2-CATEGORIES . . . 1641
idempotents Mij have trace one. This proves claim (ii). Claim (i) follows from claim (ii) as M is
the sum of the Mij ’s.
Corollary 3. The matrix M cannot be equal to Ni, where i \in \{ 1, 2, . . . , 10\} .
Proof. Each of the matrices Ni, where i \in \{ 1, 2, . . . , 10\} , contains a diagonal element which is
greater than or equal to 3. If M = Ni would be possible, at least three idempotents Mij would have
this diagonal element non-zero. But then any product of any two such matrices would be non-zero.
However, from (8) we have that, for any three different idempotents \mathrm{F}ij , one of the products of two
of these elements is zero. The obtained contradiction completes the proof.
8.3. Auxiliary adjunction. We will need the following easy observations:
Lemma 7. Let D be a finite dimensional algebra and (\mathrm{G},\mathrm{H}) an adjoint pair of right exact
endofunctors of D-mod. Let L and L\prime be simple D-modules and P and P \prime their corresponding
indecomposable projective covers. Assume that L\prime appears in the top of \mathrm{G}P. Then \mathrm{H}P \prime \not = 0.
Proof. By adjunction, we have
0 \not = \mathrm{H}\mathrm{o}\mathrm{m}B(\mathrm{G}P,L\prime ) \sim = \mathrm{H}\mathrm{o}\mathrm{m}B(P,\mathrm{H}L\prime ),
which implies \mathrm{H}L\prime \not = 0. As \mathrm{H} is right exact, this forces \mathrm{H}P \prime \not = 0.
Lemma 8. We have the following pairs of adjoint 1-morphisms in CA :
(\mathrm{F}33,\mathrm{F}23), (\mathrm{F}23,\mathrm{F}22), (\mathrm{F}22,\mathrm{F}12), (\mathrm{F}12,\mathrm{F}11).
Proof. As both, the left A-modules Ae1 and \mathrm{H}\mathrm{o}\mathrm{m}\Bbbk (e2A,\Bbbk ) are isomorphic and the left A-
modules Ae2 and \mathrm{H}\mathrm{o}\mathrm{m}\Bbbk (e3A, \Bbbk ) are isomorphic, the claim follows from Lemma 4.
8.4. Idempotent integral matrices of rank one. Recall from [4] (Theorem 2) that, up to
permutation action, idempotent matrices of rank one with non-negative integer entries have the form\left(
\bfzero v vwt
\bfzero 1 wt
\bfzero \bfzero \bfzero
\right)
where \bfzero denotes the zero matrix (of an appropriate size), and v and w are arbitrary vectors with
non-negative integer entries. In particular, if the diagonal entry 1 is in the ith row, then the whole
matrix can be written as the product of its ith column with its ith row.
8.5. Filtering matrices \bfitN \bfone \bffour , \bfitN \bfone \bffive and \bfitN \bfone \bfsix out.
Proposition 5. The matrix M cannot be equal to N14, N15 or N16.
Proof. Assume that the diagonals of the matrices M11 and M12 are different. This means that
M11 has 1 in row i, that M12 has 1 in row j, and that i \not = j.
Then \mathrm{F}12 Pj has Pj as a direct summand. Therefore, by combining Lemmas 7 and 8, we have
that M11 must have a non-zero element in column j. From Subsection 8.4 it follows that M11 has a
non-zero entry in position (i, j).
As M11 has a non-zero entry in position (i, j) and the matrix M12 has 1 in position (j, j), it
follows that M11M12 has a non-zero entry in position (i, j). From (8), we have M11M12 = M12,
which means that M12 has a non-zero entry in position (i, j). This already means that the case
M = N14 is not possible.
Assume M = N15. Then we must have j = 1. Exactly the same argument as above applied to
M22 and M23 shows that M23 has 1 in position (1, 1). This contradicts M23M12 = 0 as follows
from (8). Therefore, M = N15 is not possible.
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1642 V. MAZORCHUK, X. ZHANG
Assume M = N16. Then we must have i = 1. Exactly the same argument as above applied to
M22 and M23 shows that M22 has 1 in position (1, 1). This contradicts M11M22 = 0 as follows
from (8). Therefore, M = N16 is not possible. This completes the proof.
The remaining cases for M will be studied on a case-by-case basis.
9. Filtering matrices \bfitN \bfone \bfone and \bfitN \bfone \bftwo out. 9.1. Statement. The main aim of this section is to
prove the following proposition.
Proposition 6. The matrix M cannot be equal to N11 or N12.
We start with the following observation.
Lemma 9. The only unordered pairs of idempotent 1-morphisms of the form \mathrm{F}ij such that the
product of any two elements in the pair is non-zero are
\{ \mathrm{F}11,\mathrm{F}12\} , \{ \mathrm{F}12,\mathrm{F}22\} , \{ \mathrm{F}22,\mathrm{F}23\} , \{ \mathrm{F}23,\mathrm{F}33\} .
Proof. This follows directly from (8).
9.2. Proof for \bfitM = \bfitN \bfone \bfone . We will arrange matrices Mij , where i, j = 1, 2, 3, as follows:
M11 M12 M13
M21 M22 M23
M31 M32 M33.
(10)
Assume M = N11. The diagonal elements in N11 are (2, 2, 1). Therefore, two pairs of idempotent
matrices of the form Mij would have common diagonal elements. Any product of matrices in any
such pair would be non-zero. Therefore, using Lemma 9, we have three cases to consider.
Case 1. Suppose first that the pairs of idempotent matrices which share diagonal elements are
\{ \mathrm{F}11,\mathrm{F}12\} and \{ \mathrm{F}22,\mathrm{F}23\} . Up to permutation action, we may assume that\left(
1 \ast \ast
\ast 0 \ast
\ast \ast 0
\right) ,
\left(
1 \ast \ast
\ast 0 \ast
\ast \ast 0
\right) ,
\left(
0 \ast \ast
\ast 0 \ast
\ast \ast 0
\right) ,
\left(
0 \ast \ast
\ast 0 \ast
\ast \ast 0
\right) ,
\left(
0 \ast \ast
\ast 1 \ast
\ast \ast 0
\right) ,
\left(
0 \ast \ast
\ast 1 \ast
\ast \ast 0
\right) ,
\left(
0 \ast \ast
\ast 0 \ast
\ast \ast 0
\right) ,
\left(
0 \ast \ast
\ast 0 \ast
\ast \ast 0
\right) ,
\left(
0 \ast \ast
\ast 0 \ast
\ast \ast 1
\right) .
Using all possible zero products which appear in (8), we obtain that the Mij ’s look as follows:\left(
1 0 0
0 0 0
0 0 0
\right) ,
\left(
1 \ast 0
0 0 0
0 0 0
\right) ,
\left(
0 \ast \ast
0 0 0
0 0 0
\right) ,
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SIMPLE TRANSITIVE 2-REPRESENTATIONS FOR TWO NON-FIAT 2-CATEGORIES . . . 1643\left(
0 0 0
\ast 0 0
\ast 0 0
\right) ,
\left(
0 0 0
\ast 1 0
\ast \ast 0
\right) ,
\left(
0 0 0
0 1 \ast
0 \ast 0
\right) ,
\left(
0 0 0
0 0 0
\ast 0 0
\right) ,
\left(
0 0 0
0 0 0
\ast \ast 0
\right) ,
\left(
0 0 0
0 0 0
0 \ast 1
\right) .
As all matrices must be non-zero and add up to M, we obtain\left(
1 0 0
0 0 0
0 0 0
\right) ,
\left(
1 \ast 0
0 0 0
0 0 0
\right) ,
\left(
0 \ast 2
0 0 0
0 0 0
\right) ,
\left(
0 0 0
\ast 0 0
0 0 0
\right) ,
\left(
0 0 0
\ast 1 0
0 0 0
\right) ,
\left(
0 0 0
0 1 2
0 0 0
\right) ,
\left(
0 0 0
0 0 0
1 0 0
\right) ,
\left(
0 0 0
0 0 0
0 1 0
\right) ,
\left(
0 0 0
0 0 0
0 0 1
\right) .
This contradicts M13M31 = M11 which is a consequence of (8). Therefore, this case is not possible.
Case 2. Suppose now that the pairs of idempotent matrices which share diagonal elements are
\{ \mathrm{F}12,\mathrm{F}22\} and \{ \mathrm{F}23,\mathrm{F}33\} . Up to permutation action, we may assume that the Mij ’s look as follows:\left(
0 \ast \ast
\ast 0 \ast
\ast \ast 1
\right) ,
\left(
1 \ast \ast
\ast 0 \ast
\ast \ast 0
\right) ,
\left(
0 \ast \ast
\ast 0 \ast
\ast \ast 0
\right) ,
\left(
0 \ast \ast
\ast 0 \ast
\ast \ast 0
\right) ,
\left(
1 \ast \ast
\ast 0 \ast
\ast \ast 0
\right) ,
\left(
0 \ast \ast
\ast 1 \ast
\ast \ast 0
\right) ,
\left(
0 \ast \ast
\ast 0 \ast
\ast \ast 0
\right) ,
\left(
0 \ast \ast
\ast 0 \ast
\ast \ast 0
\right) ,
\left(
0 \ast \ast
\ast 1 \ast
\ast \ast 0
\right) .
Using all possible zero products which appear in (8), we obtain that the Mij ’s look as follows:\left(
0 0 \ast
0 0 0
0 0 1
\right) ,
\left(
1 0 \ast
0 0 0
\ast 0 0
\right) ,
\left(
0 \ast 0
0 0 0
0 \ast 0
\right) ,
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1644 V. MAZORCHUK, X. ZHANG\left(
0 0 \ast
0 0 \ast
0 0 0
\right) ,
\left(
1 0 \ast
\ast 0 \ast
0 0 0
\right) ,
\left(
0 \ast 0
0 1 0
0 0 0
\right) ,
\left(
0 0 0
0 0 \ast
0 0 0
\right) ,
\left(
0 0 0
\ast 0 \ast
0 0 0
\right) ,
\left(
0 0 0
0 1 0
0 0 0
\right) .
Here we have that P2 is a direct summand of \mathrm{F}23 P2. From Lemmas 7 and 8, it follows that
\mathrm{F}22 P2 \not = 0, which is a contradiction. Therefore, this case cannot occur either.
Case 3. Suppose first that the pairs of idempotent matrices which share diagonal elements are
\{ \mathrm{F}11,\mathrm{F}12\} and \{ \mathrm{F}23,\mathrm{F}33\} . Up to permutation action, we may assume that the Mij ’s look as follows:\left(
1 \ast \ast
\ast 0 \ast
\ast \ast 0
\right) ,
\left(
1 \ast \ast
\ast 0 \ast
\ast \ast 0
\right) ,
\left(
0 \ast \ast
\ast 0 \ast
\ast \ast 0
\right) ,
\left(
0 \ast \ast
\ast 0 \ast
\ast \ast 0
\right) ,
\left(
0 \ast \ast
\ast 0 \ast
\ast \ast 1
\right) ,
\left(
0 \ast \ast
\ast 1 \ast
\ast \ast 0
\right) ,
\left(
0 \ast \ast
\ast 0 \ast
\ast \ast 0
\right) ,
\left(
0 \ast \ast
\ast 0 \ast
\ast \ast 0
\right) ,
\left(
0 \ast \ast
\ast 1 \ast
\ast \ast 0
\right) .
Using all possible zero products which appear in (8), we obtain that the Mij ’s look as follows:\left(
1 0 0
0 0 0
\ast 0 0
\right) ,
\left(
1 0 \ast
0 0 0
\ast 0 0
\right) ,
\left(
0 \ast \ast
0 0 0
0 \ast 0
\right) ,
\left(
0 0 0
\ast 0 0
\ast 0 0
\right) ,
\left(
0 0 0
\ast 0 \ast
\ast 0 1
\right) ,
\left(
0 0 0
0 1 \ast
0 \ast 0
\right) ,
\left(
0 0 0
\ast 0 0
0 0 0
\right) ,
\left(
0 0 0
\ast 0 \ast
0 0 0
\right) ,
\left(
0 0 0
0 1 \ast
0 0 0
\right) .
Here we have that P2 is a direct summand of \mathrm{F}23 P2. From Lemmas 7 and 8, it follows that
\mathrm{F}22 P2 \not = 0, which is a contradiction. Therefore, this case cannot occur either.
This completes the proof of Lemma 9 for M = N11.
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SIMPLE TRANSITIVE 2-REPRESENTATIONS FOR TWO NON-FIAT 2-CATEGORIES . . . 1645
9.3. Proof for \bfitM = \bfitN \bfone \bftwo . Let
N \prime
12 :=
\left(
2 1 1
4 2 2
2 1 1
\right) .
The matrix N \prime
12 reduces to M = N12 by permutation action, however, it is convenient to use the
freedom of permutation action in another way, see below. Because of Lemma 9, we have three cases
to consider.
Case 1. Suppose first that the pairs of idempotent matrices which share diagonal elements are
\{ \mathrm{F}11,\mathrm{F}12\} and \{ \mathrm{F}22,\mathrm{F}23\} . Then, using permutation action and all possible zero products which
appear in (8), we obtain that the Mij ’s look as follows:\left(
1 0 0
0 0 0
0 0 0
\right) ,
\left(
1 \ast 0
0 0 0
0 0 0
\right) ,
\left(
0 \ast \ast
0 0 0
0 0 0
\right) ,
\left(
0 0 0
\ast 0 0
\ast 0 0
\right) ,
\left(
0 0 0
\ast 1 0
\ast \ast 0
\right) ,
\left(
0 0 0
0 1 \ast
0 \ast 0
\right) ,
\left(
0 0 0
0 0 0
\ast 0 0
\right) ,
\left(
0 0 0
0 0 0
\ast \ast 0
\right) ,
\left(
0 0 0
0 0 0
0 \ast 1
\right) .
If M = N12, then the (1, 3)-entry of M13 equals 2 while the (3, 1)-entry of M31 equals 1. As the
(1, 1)-entry of M11 is 1, we get a contradiction to M13M31 = M11, which follows from (8). This
implies that M = N \prime
12 and, using also M12M21 = M11, we have\left(
1 0 0
0 0 0
0 0 0
\right) ,
\left(
1 1 0
0 0 0
0 0 0
\right) ,
\left(
0 0 1
0 0 0
0 0 0
\right) ,
\left(
0 0 0
1 0 0
\ast 0 0
\right) ,
\left(
0 0 0
\ast 1 0
\ast \ast 0
\right) ,
\left(
0 0 0
0 1 \ast
0 \ast 0
\right) ,
\left(
0 0 0
0 0 0
1 0 0
\right) ,
\left(
0 0 0
0 0 0
\ast \ast 0
\right) ,
\left(
0 0 0
0 0 0
0 \ast 1
\right) .
As the Mij ’s must add up to M, we get
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1646 V. MAZORCHUK, X. ZHANG\left(
1 0 0
0 0 0
0 0 0
\right) ,
\left(
1 1 0
0 0 0
0 0 0
\right) ,
\left(
0 0 1
0 0 0
0 0 0
\right) ,
\left(
0 0 0
1 0 0
\ast 0 0
\right) ,
\left(
0 0 0
3 1 0
\ast \ast 0
\right) ,
\left(
0 0 0
0 1 2
0 \ast 0
\right) ,
\left(
0 0 0
0 0 0
1 0 0
\right) ,
\left(
0 0 0
0 0 0
\ast \ast 0
\right) ,
\left(
0 0 0
0 0 0
0 \ast 1
\right) .
This contradicts M23M31 = M21, which follows from (8). Therefore, this case cannot occur.
Case 2. Suppose now that the pairs of idempotent matrices which share diagonal elements
are \{ \mathrm{F}12,\mathrm{F}22\} and \{ \mathrm{F}23,\mathrm{F}33\} . This gives the same contradiction as in Case 2 in Subsection 9.2.
Therefore, this case cannot occur either.
Case 3. Suppose first that the pairs of idempotent matrices which share diagonal elements
are \{ \mathrm{F}11,\mathrm{F}12\} and \{ \mathrm{F}23,\mathrm{F}33\} . This gives the same contradiction as in Case 3 in Subsection 9.2.
Therefore, this case cannot occur either. This completes the proof of Lemma 9.
10. Proof of Theorem 1 for \Bbbk (\bullet \bfitalpha - \rightarrow \bullet \bfitbeta - \rightarrow \bullet )/(\bfitbeta \bfitalpha ). 10.1. Finding the matrices. Com-
bining Proposition 4 with Corollary 3, Propositions 5 and 6, we have M = N13. We will arrange our
matrices similarly to (10).
We will need the following easy and general observation:
Lemma 10. Let M be any of the Nm’s and i, j \in \{ 1, 2, 3\} . If, for some s, the column s in the
matrix Mij is non-zero, then the column s is non-zero in Mtj , for any t \in \{ 1, 2, 3\} .
Proof. The fact that the column s in Mij is non-zero is equivalent to saying that \mathrm{F}ij Ps \not = 0
(and similarly for \mathrm{F}tj ). We have \mathrm{F}it \circ \mathrm{F}tj
\sim = \mathrm{F}ij from (8). Therefore, \mathrm{F}tj Ps = 0 implies \mathrm{F}ij Ps = 0
and the claim follows.
Proposition 7. The only possibility for the Mij ’s is\left(
1 0 0
0 0 0
0 0 0
\right) ,
\left(
1 1 0
0 0 0
0 0 0
\right) ,
\left(
0 1 1
0 0 0
0 0 0
\right) ,
\left(
0 0 0
1 0 0
0 0 0
\right) ,
\left(
0 0 0
1 1 0
0 0 0
\right) ,
\left(
0 0 0
0 1 1
0 0 0
\right) ,
\left(
0 0 0
0 0 0
1 0 0
\right) ,
\left(
0 0 0
0 0 0
1 1 0
\right) ,
\left(
0 0 0
0 0 0
0 1 1
\right) .
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SIMPLE TRANSITIVE 2-REPRESENTATIONS FOR TWO NON-FIAT 2-CATEGORIES . . . 1647
Proof. Due to Lemma 9, we have to consider three cases.
Case 1. Suppose that the pairs of idempotent matrices which sharediagonal elements are
\{ \mathrm{F}12,\mathrm{F}22\} and \{ \mathrm{F}23,\mathrm{F}33\} . This gives the same contradiction as in Case 2 in Subsection 9.2. There-
fore, this case cannot occur.
Case 2. Suppose that the pairs of idempotent matrices which share diagonal elements are
\{ \mathrm{F}11,\mathrm{F}12\} and \{ \mathrm{F}23,\mathrm{F}33\} . This gives the same contradiction as in Case 3 in Subsection 9.2. There-
fore, this case cannot occur either.
Case 3. Suppose that the pairs of idempotent matrices which share diagonal elements are
\{ \mathrm{F}11,\mathrm{F}12\} and \{ \mathrm{F}22,\mathrm{F}23\} . Then, using permutation action and all possible zero products which
appear in (8), we obtain that the Mij ’s look as follows:\left(
1 0 0
0 0 0
0 0 0
\right) ,
\left(
1 \ast 0
0 0 0
0 0 0
\right) ,
\left(
0 \ast \ast
0 0 0
0 0 0
\right) ,
\left(
0 0 0
\ast 0 0
\ast 0 0
\right) ,
\left(
0 0 0
\ast 1 0
\ast \ast 0
\right) ,
\left(
0 0 0
0 1 \ast
0 \ast 0
\right) ,
\left(
0 0 0
0 0 0
\ast 0 0
\right) ,
\left(
0 0 0
0 0 0
\ast \ast 0
\right) ,
\left(
0 0 0
0 0 0
0 \ast 1
\right) .
Using that all matrices must be non-zero and add up to M and also M13M31 = M11, M21M12 =
= M22 and M21M13 = M23, given by (8), we have\left(
1 0 0
0 0 0
0 0 0
\right) ,
\left(
1 1 0
0 0 0
0 0 0
\right) ,
\left(
0 1 1
0 0 0
0 0 0
\right) ,
\left(
0 0 0
1 0 0
\ast 0 0
\right) ,
\left(
0 0 0
1 1 0
\ast \ast 0
\right) ,
\left(
0 0 0
0 1 1
0 \ast 0
\right) ,
\left(
0 0 0
0 0 0
1 0 0
\right) ,
\left(
0 0 0
0 0 0
\ast \ast 0
\right) ,
\left(
0 0 0
0 0 0
0 \ast 1
\right) .
Comparing the first and the second columns in M32 with those of M22 and also the third column in
M33 with that of M23 and using Lemma 10 we get exactly the arrangement in the formulation of our
proposition.
Proposition 7 is proved.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 12
1648 V. MAZORCHUK, X. ZHANG
10.2. Connecting to the cell \bftwo -representation. Now we know that the Mij ’s have the form as
specifies in Proposition 7. For i, j = 1, 2, 3, we denote by \mathrm{G}ij the corresponding indecomposable
projective endofunctor of \bfM (\tti ).
From the form of Mi1, where i = 1, 2, 3, we see that \mathrm{F}i1 acts via \mathrm{G}i1 (up to isomorphism).
Moreover, we also have [Pi : L1] = \delta i,1.
From the form of M12, we see that \mathrm{F}12 acts via either \mathrm{G}12 or \mathrm{G}11 or \mathrm{G}12 \oplus \mathrm{G}11. However,
we already know that \mathrm{G}11 has matrix M11. This leaves us with the only possibilities of \mathrm{G}12 or
\mathrm{G}12 \oplus \mathrm{G}11.
Assume that \mathrm{F}12 acts via \mathrm{G}12 \oplus \mathrm{G}11. Then the matrix of \mathrm{G}12 is\left(
0 1 0
0 0 0
0 0 0
\right) .
This implies that [Pi : L2] = \delta i,2, for i = 1, 2, 3.
According to (9), there is a non-zero 2-morphism \alpha : \mathrm{F}21 \rightarrow \mathrm{F}12. As \bfM is faithful, \bfM (\alpha ) is
non-zero. Evaluation of the latter at
P3 is zero as P3 is annihilated by both \mathrm{F}21 and \mathrm{F}12;
P2 is zero as P2 is annihilated by \mathrm{F}21;
P1 is zero as \mathrm{F}12 P1
\sim = P1, \mathrm{F}21 P1
\sim = P2 and we also have
\mathrm{H}\mathrm{o}\mathrm{m}\bfM (\tti )(P2, P1) = [P1 : L2] = 0,
by the previous paragraph.
Therefore, \bfM (\alpha ) must be zero, a contradiction. Consequently, \mathrm{F}12 acts via \mathrm{G}12, which also
implies [P1 : L2] = 1. From this, it follows that \mathrm{F}i2 acts via \mathrm{G}i2, for i = 1, 2, 3.
A similar argument shows that \mathrm{F}i3 acts via \mathrm{G}i3, for i = 1, 2, 3, and that
[P2 : L3] = [P3 : L3] = 1, [P3 : L1] = [P3 : L2] = 0.
This means that B \sim = A and that each \mathrm{F}ij acts via the corresponding \mathrm{G}ij . It now follows by the
usual arguments (see [16], Proposition 9) that \bfM is equivalent to a cell 2-representation of CA. The
claim of Theorem 1 for the algebra \Bbbk ( \bullet // \bullet // \bullet ) follows.
Acknowledgment. The authors thank Vanessa Miemietz for stimulating discussions.
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SIMPLE TRANSITIVE 2-REPRESENTATIONS FOR TWO NON-FIAT 2-CATEGORIES . . . 1649
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Received 28.03.16
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 12
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| id | umjimathkievua-article-1665 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:10:10Z |
| publishDate | 2018 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/a7/f0aea82551f803f71787ac1941a287a7.pdf |
| spelling | umjimathkievua-article-16652019-12-05T09:22:46Z Simple transitive 2-representations for two non-fiat 2-categories of projective functors Простi транзитивнi 2 -зображення двох нефiатних 2 -категорiй проективних функторiв Mazorchuk, V. S. Zhang, Xiaoting Мазорчук, В. С. Чжан, Сяотин We show that any simple transitive 2-representation of the 2-category of projective endofunctors for the quiver algebra of $\mathbb{k}(\bullet \rightarrow \bullet )$ and for the quiver algebra of $\mathbb{k}(\bullet \rightarrow \bullet \rightarrow \bullet )$ is equivalent to a cell 2-representation. Показано, що будь-яке просте транзитивне 2-зображення 2-категорiї проективних ендофункторiв для алгебри сагайдака $\mathbb{k}(\bullet \rightarrow \bullet )$ та алгебри Сагайдака $\mathbb{k}(\bullet \rightarrow \bullet \rightarrow \bullet )$ еквiвалентне клiтинковому 2-зображенню. Institute of Mathematics, NAS of Ukraine 2018-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1665 Ukrains’kyi Matematychnyi Zhurnal; Vol. 70 No. 12 (2018); 1625-1649 Український математичний журнал; Том 70 № 12 (2018); 1625-1649 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1665/647 Copyright (c) 2018 Mazorchuk V. S.; Zhang Xiaoting |
| spellingShingle | Mazorchuk, V. S. Zhang, Xiaoting Мазорчук, В. С. Чжан, Сяотин Simple transitive 2-representations for two non-fiat 2-categories of projective functors |
| title | Simple transitive 2-representations for two non-fiat 2-categories of projective functors |
| title_alt | Простi транзитивнi 2 -зображення двох нефiатних 2 -категорiй проективних функторiв |
| title_full | Simple transitive 2-representations for two non-fiat 2-categories of projective functors |
| title_fullStr | Simple transitive 2-representations for two non-fiat 2-categories of projective functors |
| title_full_unstemmed | Simple transitive 2-representations for two non-fiat 2-categories of projective functors |
| title_short | Simple transitive 2-representations for two non-fiat 2-categories of projective functors |
| title_sort | simple transitive 2-representations for two non-fiat 2-categories of projective functors |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1665 |
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