HICAS OF LENGTH ≤ 4

VANESSA MIEMIETZ AND WILL TURNER

Abstract. A hica is a highest weight, homogeneous, indecomposable, Calabi-Yau category of dimension 0. A hica has length l if its objects have Loewy length l and smaller. We classify hicas of length ≤ 4, up to equivalence, and study their properties. Over a fixed field F, we prove that hicas of length 4 are in one-one correspondence with bipartite graphs. We prove that an algebra A_{Γ} controlling the hica associated to a bipartite graph Γ is Koszul, if and only if Γ is not a simply laced Dynkin graph, if and only if the quadratic dual of A_{Γ} is Calabi-Yau of dimension 3.

1. Introduction

Once a mathematical definition has been made, the theory surrounding that definition usually begins with a study of small examples. A striking violation of this principle occurred at the birth of category theory, where early theory was concerned with establishing results valid for large and floppy mathematical structures like the category of sets, or the category of groups, or the category of topological spaces. But time has passed, categories have begun to be taken seriously, and they are now objects of detailed study. Since categories are often large and floppy, the 2-category of all categories is very large and very floppy. To prove theorems about categories, it is necessary to make strong restrictions on their structure. To prove classification theorems for categories, it is necessary to make very strong restrictions on their structure. There is by now an extensive collection of categorical classification theorems. A category with one object and invertible morphisms is a group, and there are many examples of classification theorems in group theory. Rings are endowed with various categories, like their module categories. Classification theorems for commutative rings can be thought of as classification theorems in algebraic geometry. There are a number of classification theorems for rings of finite homological dimension, to which the term noncommutative geometry is applied. For example, hereditary algebras over an algebraically closed field can be parametrised by quivers. Calabi-Yau algebras of dimensions 2 and 3 can be loosely parametrised by quivers with a superpotential [2], [5], [8]. Categorical classification theorems also appear in higher representation theory: irreducible integrable representations of 2-Kac-Moody Lie algebras can be parametrised by integral dominant weights [17]. Our paper runs in this vein. A hica is a \underline{hi}ghest weight, \underline{h}omogeneous, \underline{i}ndecomposable, \underline{Ca}labi-Yau \underline{ca}tegory of dimension 0. Here, we say a highest weight category is homogeneous if its standard objects all have the same Loewy length, and its costandard objects all have the same Loewy length. We say a hica has length l if its projective objects have Loewy length l and smaller. We classify hicas of length ≤ 4 up to equivalence.

Hicas show up naturally in group representation theory and in the theory of tilings [19, 3, 13, 14]. A multitude of examples of hicas were constructed by Mazorchuk and Miemietz [12]. Every hica can be realised as the module category of some symmetric quasi-hereditary algebra. If the hica is not semisimple, the corresponding algebra is necessarily infinite dimensional, noncommutative, of infinite homological dimension. Let us fix a field F, and consider hicas over F, up to equivalence. The only hica of length 1 is the category of vector spaces over F. There are no hicas of length 2. There is a unique hica of length 3, which is the module category of the Brauer tree algebra on a bi-infinite line. Our first main result is

Theorem 1. There is a natural one-one correspondence

{bipartite graphs} ↔ {hicas of length 4}.

The one-one correspondence of Theorem 1 is obtained from a sequence of three oneone correspondences: a one-one correspondence between bipartite graphs and topsyturvy quivers; a one-one correspondence between topsy-turvy quivers and basic self-injective directed algebras of Loewy length 3; and a one-one correspondence between basic self-injective directed algebras of Loewy length 3 and hicas of length 4. Let us describe here the construction of a hica C_{Γ} of length 4 from the following bipartite graph Γ:

[figure: •; •; •; •; •]

First, we construct a quiver Q_{Γ}, by consecutively gluing together opposite orientations of this bipartite graph, one next to the other:

\xymatrix{ & • \ar[r] \ar[dr] & • \ar[r] \ar[dr] & • \ar[r] \ar[dr] & • \ar[r] \ar[dr] & • \ar[r] \ar[dr] & • \ar[r] \ar[dr] & • \\ ... & • \ar[ur] \ar[r] & • \ar[r] \ar[ur] \ar[dr] \ar[ur] \ar[dr] & • \ar[ur] \ar[r] & • \ar[r] \ar[ur] \ar[dr] \ar[dr] & • \ar[ur] \ar[r] & • \ar[r] \ar[ur] \ar[dr] \ar[dr] & • & ... \\  & • \ar[ur] \ar[ur] &  & • \ar[ur] \ar[ur] \ar[ur] &  & • \ar[ur] \ar[ur] \ar[ur] &  & •}

This quiver has an automorphism ϕ which shifts a vertex to a vertex which is two steps horizontally to its right. We take the path algebra of this quiver. We now construct a self-injective directed algebra B_{Γ} of Loewy length 3, factoring the path algebra by relations which insist that all squares commute, and that paths u → v → w of length 2 are zero, unless w = ϕ(u). We define A_{Γ} to be the trivial extension B_{Γ} ⊕ B_{Γ}^{∗} of B_{Γ} by its dual. The module category C_{Γ} of A_{Γ} is a hica of length 4. Its quiver is

\xymatrix{ & • \ar[r] \ar[dr] & • \ar[r] \ar[dr] & • \ar[ll] \ar[r] \ar[dr] & • \ar[ll] \ar[r] \ar[dr] & • \ar[ll] \ar[r] \ar[dr] & • \ar[ll] \ar[r] \ar[dr] & • \ar[ll] \\ ... & • \ar[ur] \ar[r] & • \ar[r] \ar[ur] \ar[dr] \ar[dr] & • \ar[ll] \ar[ur] \ar[r] & • \ar[ll] \ar[r] \ar[ur] \ar[dr] \ar[dr] & • \ar[ll] \ar[ur] \ar[r] & • \ar[ll] \ar[r] \ar[ur] \ar[dr] \ar[dr] & • \ar[ll] & ... \\  & • \ar[ur] \ar[ur] \ar[ur] &  & • \ar[ll] \ar[ur] \ar[ur] \ar[ur] &  & • \ar[ll] \ar[ur] \ar[ur] \ar[ur] &  & • \ar[ll]}

The relations for A_{Γ} are those for B_{Γ}, along with relations which insist that the product of two leftwards pointing arrows is zero whilst squares involving a pair of parallel leftwards pointing arrows commute. The algebra A_{Γ} has some pleasing properties. It admits a derived self-equivalence ψ_{γ} for every vertex γ of Γ. It also admits a number of ℤ_{+}^{3}-gradings, one for each orientation of the graph Γ. It is Koszul and its quadratic dual A_{Γ}^{!} is Calabi-Yau of dimension 3. More generally, we have the following theorem.

Theorem 2. Suppose Γ is a bipartite graph, and C_{Γ} = A_{Γ} -mod the associated hica of length 4. The following are equivalent:

1. Γ is not a simply laced Dynkin graph.

2. A_{Γ} is Koszul.

3. The quadratic dual of A_{Γ} is Calabi-Yau of dimension 3.

The way this paper evolved was surprising to us. We began with the problem of classifying small hicas, categories whose structural features (Calabi-Yau 0, highest weight) were motivated by exposure to group representation theory. We ended having made contact with mathematics of different kin: bipartite graphs, Calabi-Yau 3s, and Dynkin classifications. The hica restrictions indeed capture some features of Lie theoretic representation theory, but they can also be thought of as noncommutative geometric restrictions: highest weight categories were invented to capture stratification properties appearing in algebraic geometry, whilst 0-Calabi-Yau categories are categories possessing a homological duality with trivial Serre functor.

2. Preliminaries

Our main objects of study, hicas, are a species of abelian categories. As we study them, we will use freely the languages of abelian categories, algebras, and triangulated categories. Here we give a short phrasebook for these languages. Let F be a field. The collection of F -algebras is a 2-category, whose arrows are exact bimodules and 2-arrows are bimodule homomorphisms. We have a 2-functor

𝔄𝔩𝔤𝔢𝔟𝔯𝔞 → 𝔄𝔟𝔢𝔩𝔦𝔞𝔫

from the 2-category 𝔄𝔩𝔤𝔢𝔟𝔯𝔞 of F -algebras to the 2-category 𝔄𝔟𝔢𝔩𝔦𝔞𝔫 of abelian categories. This 2-functor takes an algebra A to its module category, a bimodule _{A}M_{B} to the functor M ⊗_{B}−, and a bimodule homomorphism to a natural transformation. We have a 2-functor

𝔄𝔟𝔢𝔩𝔦𝔞𝔫 → 𝔗𝔯𝔦𝔞𝔫𝔤𝔲𝔩𝔞𝔱𝔢𝔡

taking values in the 2-category of triangulated categories, which takes an abelian category A to its derived category D(A).

If X is an object of an abelian category of finite composition length, we define the Loewy length of X (or length of X, or l(X)) to be the smallest number l for which there exists a filtration of X with l nonzero sections, all of which are semisimple. If A is an abelian category, we define the length of A to be the supremum over all lengths of objects in A. If A is an algebra, we define the length of A to be the length of the abelian category A -mod of A-modules. Given a finite dimensional F -vector space V, we denote by V^{∗} the dual Hom_{F} (V, F) of V. We say an F -linear triangulated category T is Calabi-Yau of dimension d if Hom_{T} (P, X) is finite dimensional for objects X ∈ T , and compact objects P ∈ T , and

Hom_{T} (P, X) ≅ Hom_{T} (X, P [d])^{∗}

naturally in objects X ∈ T , and compact objects P ∈ T . For background, we recommend a survey article of B. Keller concerning Calabi-Yau triangulated categories [8]. To avoid confusion here, let us emphasise that the definition of a Calabi-Yau triangulated category Keller uses is slightly different from this one since he makes no compactness assumption on P. We say an F -linear abelian category A is Calabi-Yau of dimension d if its derived category D(A) is Calabi-Yau of dimension d. We say an F -algebra A is Calabi-Yau of dimension d if its module category A -mod is Calabi-Yau of dimension d.

Suppose A is a basic (not necessarily unital) F -algebra satisfying the following assumptions:

(i) A has a set {e_{λ} | λ ∈ Λ} of orthogonal primitive idempotents, which is countable;

(ii) for any x, y ∈ Λ the F -vector space e_{x}Ae_{y} is finite dimensional; (iii) for any x ∈ Λ there exist only finitely many y ∈ Λ such that e_{x}Ae_{y} ≠ 0; (iv) for any x ∈ Λ there exist only finitely many y ∈ Λ such that e_{y}Ae_{x} ≠ 0.

Under these assumptions all indecomposable projective A-modules Ae_{x} and all injective A-modules Hom_{F} (e_{x}A, F) are finite-dimensional. A-modules M = _{A}M will be left A-modules unless they carry a right subscript as in M_{A} in which case they will be right A-modules. We denote by A -mod the collection of all finitedimensional left A-modules and by mod- A the collection of all finite-dimensional right A-modules. We denote by A -perf the subcategory of the derived category of A -mod consisting of perfect complexes, that is the smallest thick subcategory of the derived category of A -mod containing all projective objects of A -mod, or equivalently the subcategory of compact objects in the derived category of A. We define A^{∗} to be the A-A-bimodule ⨁_{x∈Λ} Hom_{F} (Ae_{x}, F). We say A is a symmetric algebra if A ≅ A^{∗} as A-A-bimodules. Then A is symmetric if and only if A -mod is Calabi-Yau of dimension 0 (cf. [16], Theorem 3.1). Suppose A is an algebra satisfying the above conditions, and Λ is ordered. For λ ∈ Λ, let J_{≥λ} = ∑_{μ≥λ} Ae_{μ}A and J_{>λ} = ∑_{μ>λ} Ae_{μ}A. Let J_{λ} = J_{≥λ}/J_{>λ}. We say A is quasi-hereditary if the product map J_{λ}e_{λ} ⊗_{F} e_{λ}J_{λ} → J_{λ} is an isomorphism for every λ ∈ Λ [4].

Now suppose A is an abelian category over F, with enough projective objects, a countable set Λ indexing the isomorphism classes of simple objects of A, such that all objects of A have a finite composition series with sections in Λ. We denote by P (λ) the minimal projective cover of λ in A.

We call A a highest weight category [4] if there is an ordering < on Λ, and a collection of objects Δ(λ), for λ ∈ Λ, such that

(i) there is an epimorphism Δ(λ) ↠ λ whose kernel X(λ) has composition factors μ < λ;

(ii) P (λ) has a filtration with a single section isomorphic to Δ(λ) and every

other section isomorphic to Δ(μ), for μ > λ. If A is quasi-hereditary, then A -mod is a highest weight category, with standard objects _{A}Δ(λ) = J_{λ}e_{λ}, and mod- A is a highest weight category with standard modules Δ_{A}(λ) = e_{λ}J_{λ}. Thus A has a filtration by ideals, whose sections are isomorphic to

AΔ(λ) ⊗_{F} Δ_{A}(λ).

Conversely, if C is a highest weight category with ordered set Λ, then the algebra A = ⊕_{λ,μ} Hom(P_{λ}, P_{μ}) is quasi-hereditary. The left and right costandard modules _{A}∇(λ), ∇(λ)_{A} of A are defined to be the duals of the right and left standard modules ∇(λ)_{A}, _{A}∇(λ) of A. We write Δ = ⊕_{λ}Δ(λ) and ∇ = ⊕_{λ}∇(λ).

Lemma 3. Let A be a selfinjective quasi-hereditary algebra. If A is not semisimple, then A is infinite dimensional.

Proof. Nonsemisimple selfinjective algebras have infinite homological dimension, since Heller translation is invertible. Finite dimensional quasi-hereditary algebras have finite homological dimension. □

We say a highest weight category C is homogeneous if its standard objects all have the same Loewy length, and its costandard objects all have the same Loewy length. Equivalently, C = A -mod, where A is a quasi-hereditary algebra whose left standard modules all have the same Loewy length, and whose right standard modules all have the same Loewy length.

Definition 4. A hica is is a \underline{hi}ghest weight, \underline{h}omogeneous, \underline{i}ndecomposable \underline{Ca}labi-Yau \underline{ca}tegory of dimension 0.

The collection ℌ𝔦𝔠𝔞 of hicas forms a 2-category (arrows are exact functors, 2-arrows are natural transformations). We denote by ℌ𝔦𝔠𝔞_{l} the 2-category of hicas of length l.

Lemma 5. The 2-functor

{symmetric, homogeneous, quasihereditary algebras } ↠ ℌ𝔦𝔠𝔞

which takes an algebra to its module category is essentially bijective on objects.

Proof. If A is a symmetric, Δ-homogeneous quasihereditary algebra then A -mod is a hica by ([4], [16], Theorem 3.1). If C is a hica, then A = ⊕_{λ∈Λ} Hom(P (λ), P (μ)) is an algebra such that A -mod = C. □

A highest weight category C has a collection of indecomposable tilting modules T (λ) indexed by Λ, characterised as indecomposable objects with a Δ-filtration and a ∇-filtration. The Ringel dual C′ of C is the module category A′ -mod of the algebra

A′ = ⊕_{λ,μ} Hom_{C}(T (λ), T (μ)).

The Ringel dual C′ of C is a highest weight category. If C = A -mod, we call A′ the Ringel dual of A. If C ≅ C′ then we say C and A are Ringel self-dual.

Lemma 6. Suppose C = A -mod is a hica. Then

l(A) = l(_{A}Δ) + l(Δ_{A}) − 1.

Proof. The length of A is the least number l such that the product of any l elements of A is zero. This can be otherwise defined as the radical length of the A ⊗ A^{op}- module A. Since A is quasi-hereditary, _{A}A_{A} has a bimodule filtration with sections AΔ(λ)⊗_{F} Δ_{A}(λ). These sections have radical length l(_{A}Δ)+l(Δ_{A})−1, as A⊗A^{op}- modules. The tops of all of these sections lie in the top of_{A}A_{A}. Therefore the Loewy length of A is equal to l(_{A}Δ) + l(Δ_{A}) − 1, as required. □

We also wish to consider graded algebras, which may satisfy weaker assumptions than those given above. If G is a group, and A an algebra, then a G-grading of A is a decomposition A = ⊕_{g∈G}A^{g}, such that A^{g}.A^{h} ⊂ A^{gh}. A graded A-module is an A module with a decomposition M = ⊕_{g∈G}M^{g}, such that A^{g}.M^{h} ⊂ M^{gh}; a homomorphism ϕ : M → N of graded modules is an A-module homomorphism sending M^{g} to N^{g}, for g ∈ G. We say A is ℤ_{+}-graded if it is ℤ-graded, with A^{i} = 0 for i < 0. Suppose A a ℤ-graded algebra, whose degree 0 part A^{0} satisfies the conditions (i)-(iv) above. Then we denote by A -mod the abelian subcategory of the category of all A-modules generated by A^{0} -mod, and by A -gr the abelian subcategory of the category of all graded A-modules generated by the category of finite dimensional A^{0} -mod⟨i⟩, for i ∈ ℤ. We denote by A -grperf the thick subcategory of the the derived category of graded A-modules generated by objects of the form A⊗_{A^{0}} X⟨i⟩, where X ∈ A^{0} -mod and i ∈ ℤ.

3. Elementary constructions

Let us give some elementary constructions of symmetric algebras. Suppose B is an algebra. Let A = T (B) denote the trivial extension of B by B^{∗}. Then A is symmetric, and A -mod is Calabi-Yau of dimension 0. Suppose B is an algebra and M is a B-B-bimodule such that e_{λ}Me_{μ} is finitedimensional for every λ, μ ∈ Λ and such that for every λ only finitely many of e_{λ}Me_{μ} and e_{μ}Me_{λ} are non-zero. Define M^{∗} := ⨁_{λ∈Λ} Hom_{F} (Me_{λ}, F) and assume we have a fixed bimodule isomorphism M ≅ M^{∗}. Then we have a sequence of bimodule homomorphisms

B → Hom_{B}(M, M) ≅ Hom_{B}(M, M^{∗}) = Hom_{B}(M, ⨁_{λ∈Λ} Hom(Me_{λ}, F))

≅ ⨁_{λ∈Λ} Hom_{B}(M, Hom(Me_{λ}, F))

≅ ⨁_{λ∈Λ} Hom_{F} (M ⊗_{B} Me_{λ}, F)

= ⨁_{λ∈Λ} Hom_{F} ((M ⊗_{B} M)e_{λ}, F)

= (M ⊗_{B} M)^{∗},

noting that M ⊗_{B} M satisfies that e_{λ}M ⊗_{B} Me_{μ} = ⨁_{ν∈Λ} e_{λ}Me_{ν} ⊗_{B} e_{ν}Me_{μ} is finitedimensional (finitely many finite-dimensional direct summands) for all λ, μ, and for every λ only finitely many of e_{λ}M ⊗_{B} Me_{μ} and e_{μ}M ⊗_{B} Me_{λ} are nonzero. The obtained bimodule homomorphisms compose to give a bimodule homomorphism B → (M ⊗_{B} M)^{∗}. Let μ : M ⊗_{B} M → B^{∗} denote the dual map. Associated to the data (B, M), we have a ℤ-graded algebra U = U(B, M) concentrated in degrees 0,1, and 2 whose degree 0,1,2 part is B,M, B^{∗} respectively. The product map U^{0} ⊗ U^{i} → U^{i} is given by the left action of B on the bimodule U^{i}, for i = 0, 1, 2. The product map U^{i} ⊗ U^{0} → U^{i} is given by the right action of B on the bimodule U^{i}. We define the product U^{1} ⊗_{U^{0}} U^{1} → U^{2} to be given by μ.

Lemma 7. U(B, M) -mod is Calabi-Yau of dimension zero.

Proof. We have a bimodule isomorphism U ≅ U^{∗} which exchanges U^{0} and U^{2}, and sends U^{1} to U^{1∗} via the fixed isomorphism M ≅ M^{∗}. □

4. Topsy-turvy quivers

Given a vertex w in a quiver Q, let P(w) denote the collection of vertices v of Q for which there is an arrow pointing from v to w (the past of w), counted with multiplicity. Let F(u) denote the collection of vertices v of Q for which there is an arrow pointing from u to v (the future of u), counted with multiplicity.

Definition 8. A connected quiver is topsy-turvy if it contains at least one arrow, and there is an automorphism ϕ of the vertices of Q such that F(u) = P(u^{ϕ}) for every vertex u of Q.

For any topsy-turvy quiver, the automorphism ϕ extends to a quiver automorphism, since arrows from x to y can be placed in bijection with arrows from y to x^{ϕ}, which can be placed in bijection with arrows from x^{ϕ} to y^{ϕ}.

Lemma 9. If Q is a topsy-turvy quiver, then PF(w) = FP(w) for all vertices w of Q.

Proof. Any x in FP(w) lies in the future of some u in the past of w, and therefore lies in the past of u^{ϕ}; since Q is topsy-turvy, u^{ϕ} also lies in the future of w and x lies in PF(w). By symmetry, if x lies in PF(w) then x also lies in FP(w). □

A directed topsy-turvy quiver Q can be ℤ-graded in the following way: take an arbitrary vertex u of Q and place it in degree 0. We say another vertex v in Q is in degree k if there exist i_{1}, …, i_{r} and j_{1}, …, j_{r} such that v ∈ P^{i_{1}}F^{j_{1}} ⋯ P^{i_{r}}F^{j_{r}}(u) and ∑_{1≤s≤r} j_{s} − ∑_{1≤s≤r} i_{s} = k. This is well-defined since PF(w) = FP(w). It follows that all arrows in Q point from degree i to degree i + 1 and that ϕ has degree 2. A bipartite graph is a connected graph Γ whose set V of vertices decomposes into two nonempty subsets V = V_{l} ∪ Γ_{r} such that no edges of Γ connect V_{l} to V_{l}, or V_{r} to V_{r}. Note that we do not call the graph with one vertex and no arrows bipartite. Given a graph Γ with a bipartite decomposition of vertices V = V_{l} ∪ V_{r}, we have an associated directed topsy-turvy quiver Q_{Γ}, obtained by orienting ℤ copies of Γ, identifying, for i even, the r-vertices of i^{th} copy of Γ with the r-vertices of the i+1^{th} copy of Γ, the l-vertices of i^{th} copy of Γ with the l-vertices of the i − 1^{th} copy of Γ, and insisting that arrows in the i^{th} copy of Γ point from the i − 1^{th} copy to the i + 1^{th} copy, for i ∈ ℤ. Note that if we label our bipartite decomposition with the opposite orientation, we obtain an isomorphic topsy-turvy quiver.

Lemma 10. We have a one-one correspondence Γ ↔ Q_{Γ} between bipartite graphs and directed topsy-turvy quivers.

Proof. Given a directed topsy-turvy quiver, we have a ℤ-grading of the set of vertices V = ⨿_{i∈ℤ}V_{i}, see above. Let A_{i} denote the set of arrows from V_{i} to V_{i+1}. The set of arrows of our quiver is graded A = ⨿_{i∈ℤ}A_{i}. The automorphism ϕ defines isomorphisms between V_{i} and V_{j} and between A_{i} and A_{j} when i and j are both even, or when i and j are both odd. We can thus identify the V_{i} for i even with a single vertex set V_{even}, the V_{i} for i odd with a single vertex set V_{odd}, the A_{i} for i even with a single arrow set A_{eo} from V_{even} to V_{odd}, the A_{i} for i odd with a single arrow set A_{oe} from V_{odd} to V_{even}. The topsy-turviness of the quiver means precisely that A_{eo} is the opposite of A_{oe}. We thus obtain a graph with vertices V_{even} ∪ V_{odd}, and with edges between V_{even} and V_{odd}, such that directing edges from V_{even} to V_{odd} gives us A_{eo} and directing edges from V_{odd} to V_{even} gives us A_{oe}. This is a bipartite graph, by definition. Reversing the above argument, from any bipartite quiver, we obtain a directed topsy-turvy quiver. □

Example 11 The bipartite graph • • with two vertices and a single edge results in a topsy-turvy quiver which can be depicted as an oriented line:

\xymatrix{... & • \ar[r] & • \ar[r] & • \ar[r] & • \ar[r] & • \ar[r] & • \ar[r] & • & ...}

The bipartite graph

...  •  •  •  •  •  •  •  ...

results in a topsy-turvy quiver which can be depicted as a directed square lattice in ℝ^{2}:

\xymatrix{... & • \ar[dr] &  & • \ar[dr] &  & • & ... \\  &  & • \ar[ur] \ar[dr] &  & • \ar[ur] \ar[dr] \\ ... & • \ar[ur] &  & • \ar[ur] &  & • & ...}

The bipartite graph whose vertices are elements of the square lattice lattice in ℝ^{2} results in a topsy-turvy quiver whose arrows can be thought of as the diagonals of a face-centred cubic lattice in ℝ^{3}.

5. Self-injective directed algebras of length ≤ 3

Throughout the following, let B be an indecomposable self-injective directed algebra. Here self-injective means that B ≅ ⨁_{x∈Λ} Hom_{F} (Ae_{x}, F) as left B-modules or, equivalently, that all projective B-modules are also injective, and vice versa. Directed is understood to mean that the Ext^{1}-quiver of B is a directed quiver. Note that such an algebra is necessarily infinite-dimensional, since directed implies quasi-hereditary which, in the finite-dimensional case, implies finite global dimension, contradicting self-injectivity.

Lemma 12. If B is radical-graded, all projective B-modules have the same Loewy length.

Proof. For finite-dimensional algebras, this was shown in [11, Theorem 3.3]. We remark that the same proof holds for algebras in our setup, as the comparisons of Loewy length only need to be done using neighbouring projectives in the Extquiver. □

Let us now assume that B be an indecomposable self-injective algebra of Loewy length ≤ 3.

Lemma 13. B is radical-graded.

Proof. Set A_{0} := ⨁_{x∈Λ}F e_{x} ≅ A/ Rad A realized by the semisimple algebra generated

by the idempotents, this is obviously a subalgebra. It acts naturally on the bimodule A_{1} ≅ Rad A/ Rad^{2} A given by the arrows in the Ext-quiver and on A_{2} := Rad^{2} A. Obviously the multiplication maps A_{1} ⊗ A_{1} to A_{2}, so A is radical-graded. □

Corollary 14. All projectives of B have the same Loewy length.

Lemma 15. The quiver of B is a topsy-turvy quiver.

Proof. A projective indecomposable B-module P (λ) can be identified with an injective indecomposable B-module I(λ^{ϕ}). Here ϕ is a quiver isomorphism, corresponding to the Nakayama automorphism of B. Since B is selfinjective Loewy length 3, elements of F(λ) correspond to composition factors in the heart of P (λ) =_{B} Be_{λ}. Switching from left action to right action, we find elements of P(λ^{ϕ}) correspond to composition factors in the heart of e_{λ}B_{B}. Taking duals, we find elements of P(λ^{ϕ}) correspond to composition factors in the heart of I(λ^{ϕ}). Since P (λ) = I(λ^{ϕ}), we conclude F(λ) = P(λ^{ϕ}). Thus B has a topsy-turvy quiver, as required. □

To any topsy-turvy quiver Q, we can associate a self-injective algebra R(Q) of Loewy length 3 by factoring out relations from the path algebra as follows: make products of arrows of Q which do not lie in some F(u) ∪ P(u^{ϕ}) equal to zero; make squares in F(u) ∪ P(u^{ϕ}) commute. Let us now assume B is directed.

Lemma 16. (a) If B has Loewy length 2, it is isomorphic to the F Q/I, where Q is the infinite quiver

\xymatrix{... & • \ar[r] & • \ar[r] & • \ar[r] & • \ar[r] & • \ar[r] & • \ar[r] & • & ...}

and I is the quadratic ideal generated by all paths of length two.

(b) If B has Loewy length 3, it is given by R(Q), where Q is a directed, topsyturvy quiver.

Proof. (a) Obvious.

(b) Since projectives are injectives, both have irreducible head and socle. Since B is directed, projectives have structure

λ
μ_{1} ⊕ ... ⊕ μ_{n}
ν,

where ν < μ_{i} < λ all i. We only have to worry about the nonzero relations. These take the form ac = ξbd, for ξ ∈ F^{×}, where a, b are arrows in F (u) and c, d are arrows in P (u^{ϕ}) for some u. We want to remove the scalars ξ from this description.

Let us write B = F Q/I. Then Q is topsy-turvy with ϕ described by the Nakayama automorphism of B. Since Q is directed as well, we can give the collection of vertices of our quiver a ℤ-grading, so that arrows have degree 1, and ϕ has degree 2. We now alter scalars inductively. Arrows from vertices of degree 0 to vertices of degree 1 we leave alone. An arrow a from degree 1 to degree 2 lies in P (t(a)), and in no other P (w). Therefore, multiplying arrows between vertices of degree 1 and degree 2 by nonzero scalars if necessary, we can force squares in quiver degree 0, 1, 2 to commute. Similarly, multiplying arrows in degree 2, 3 by scalars, we can force squares in quiver degree 1, 2, 3 to commute. And so on. Working backwards, make squares in degree −1, 0, 1 commute and so on. □

Suppose Γ is a bipartite graph. The double quiver of Γ is the quiver which has vertices as Γ and a pair of opposing arrows running along each edge of Γ.

Definition 17. Let B_{Γ} denote the self-injective directed algebra R(Q_{Γ}). Let A_{Γ} denote the trivial extension T (B_{Γ}) of B_{Γ}. Let C_{Γ} denote the category A_{Γ} -mod.

We define Z_{Γ} to be the zigzag algebra associated to Γ [7]. It is the path algebra of the double quiver associated to Γ modulo relations insisting that all quadratic paths based at a single vertex are equal, whilst all other quadratic relations are zero. Since the relations are homogeneous, Z_{Γ} is a ℤ_{+}-graded algebra with homogeneous elements graded by path length.

Lemma 18. The category Z_{Γ} -mod is Calabi-Yau of dimension 0. We have an equivalence

Z_{Γ} -gr ≃ B_{Γ} -mod^{⊕2}

between the category Z_{Γ} -gr of graded modules of Z_{Γ}, taken with respect to the ℤ_{+}- grading by path length, and the direct sum of two copies of B_{Γ} -mod. Under this equivalence, twisting by the automorphism ϕ of Q_{Γ} corresponds to a degree shift by 2 in Z_{Γ} -gr.

Proof. The irreducible objects of Z_{Γ} -gr are S⟨i⟩, where S is an irreducible Z_{Γ}- module concentrated in degree 0. There are homomorphisms in Z_{Γ} -gr between S⟨i⟩ and T ⟨j⟩ precisely when S = T and i = j. There is an extension in Z_{Γ} -gr of S⟨i⟩ by T ⟨j⟩ precisely when there is an extension between S by T in Z_{Γ} -mod and j = i + 1. In particular when there exists such an extension, S corresponds to a vertex in V_{l} and T corresponds to a vertex in V_{r}. We thus have two blocks in Z_{Γ} -gr: one block is generated by S⟨i⟩ where S lies in V_{l} and i is even or S lies in V_{r} and i is odd; the other block is generated by S⟨i⟩ where S lies in V_{r} and i is even or S lies in V_{l} and i is odd. It is not difficult to see that each block is isomorphic to B_{Γ} -mod so that the automorphism ϕ corresponds to a degree shift ⟨2⟩. □

For a quiver Q, we define P_{Q} to be the path algebra of Q, modulo the ideal of all paths of length ≥ 2.

Lemma 19. For every orientation Γ^{→} of the bipartite graph Γ, we have an isomorphism

Z_{Γ} ≅ T (P Γ^{→})

between Z_{Γ} and the trivial extension algebra T (P Γ^{→}) of P Γ^{→} by its dual.

Proof. Projectives for P Γ^{→} take two shapes: they are either of Loewy length two, hence have a simple top with a certain number of extensions, or they are simple. Similarly injectives are simple in the first case or of length two with a simple socle and a certain number of simples in the top in the second case. Projectives for T (P Γ^{→}) are extensions of projectives for P Γ^{→} by injectives for the same algebra, hence either of a module of Loewy length two with a certain number of simples in the socle by a simple or of a simple by a module of Loewy length two with a simple socle and some composition factors in the top. In both cases top and socle of the resulting extension have to be simple which forces, in the first case, all of the simples in the socle of the P Γ^{→}-projective to extend the simple P Γ^{→}-injective, and in the second case, the simple P Γ^{→}-projective to extend all the simples in the top of the P Γ^{→}-injective. This is the

same as saying that for every arrow in Γ^{→} the quiver for T (P Γ^{→}) has an arrow in the opposite direction as well, and that all quadratic paths based at a single vertex are the same (we can easily get rid of scalars by rescaling the arrows) while all other quadratic relations are zero. This exactly describes the algebra Z_{Γ}. □

In this way, every orientation Γ^{→} of the graph Γ defines a ℤ_{+}^{{f,a}}-grading on Z_{Γ}, whose f component corresponds to the ℤ_{+}-grading of P Γ^{→} by path length, and whose a component corresponds to the ℤ_{+}-grading of T (P Γ^{→}) which puts P Γ^{→} in degree 0 and its dual in degree 1. Correspondingly, the orientation Γ^{→} of Γ gives rise to a ℤ_{+}^{{f,a}}-grading of the associated selfinjective directed algebra B_{Γ} as follows: define a bigrading of the corresponding topsy-turvy quiver by grading arrows with an f if they run with the orientation Γ^{→} of Γ, and grading them a if they run against the orientation. This grading extends to a ℤ_{+}^{{f,a}}-grading of B_{Γ}.

6. Hicas of length ≤ 4

The following is a classical statement which hold for any quasi-hereditary algebra:

Lemma 20. (a) _{A}Δ ≅ (∇_{A})^{∗} (b) _{A}∇ ≅ (Δ_{A})^{∗}

Lemma 21. Suppose C = A -mod is a highest weight category which is Calabi-Yau of dimension 0, and Ringel self-dual. Then A is quasi-hereditary with respect to two orders, denoted ▲ and ▼, and we have

(a) _{A}Δ^{▲} ≅ _{A}∇^{▼}

(b) _{A}Δ^{▼} ≅ _{A}∇^{▲}

Proof. Let us suppose the quasi-hereditary structure on A is given by the partial order ▲, and the one induced by Ringel duality is ▼. Since A is Ringel self-dual, we have an isomorphism A ≅ A′. Say that under this homomorphism the right projective e_{x}A corresponding to x ∈ Λ goes to the right projective e′_{y}A′ for some y ∈ Λ. Then by Hom_{A}(Ae_{x}, A) ≅ e_{x}A ≅ e′_{y}A′ = Hom_{A}(T (y), A) for T (y) the tilting module for y and the fact that any projective for A is also injective and therefore tilting, it follows that T (y) = P (x). So all tilting modules are projective A-modules. So, there is a 1-1-correspondence between tilting modules and projective modules for A, say it is, in the above scenario given by y = ♯x. In particular this gives a oneto-one correspondence between standard modules and their socles x = soc Δ^{▲}(♯x).

This makes the definition Δ^{▼}(x) := ∇^{▲}(♯x) well-defined. Filtrations of projectives by Δ^{▼}s as well as the respective ordering conditions follow immediately from the dual statments for injectives (=projectives) and ∇^{▲}s. □

Lemma 22. Hicas of length 1 are semisimple. There are no hicas of length 2. There is a unique hica of length 3, which is the module category of the Brauer tree algebra associated to a bi-infinite line.

Let us now assume C is a hica of length 4. Thus C = A -mod for a symmetric quasi-hereditary Δ-homogeneous algebra A of Loewy length 4.

Lemma 23. The endomorphism ring of a projective indecomposable object in C is isomorphic to F [d]/d^{2}.

Proof. Top and socle of each projective indecomposable are isomorphic and this simple cannot appear in any of the middle radical layer as this would imply a self-extension of this simple - contradicting quasi-heredity. □

Lemma 24. Either _{A}Δ has length 3 and Δ_{A} has length 2, or else _{A}Δ has length 2 and Δ_{A} has length 3.

Proof. Since A is a hica, we have

l(_{A}Δ) + l(Δ_{A}) = 5.

It is impossible that l(_{A}Δ) = 1 since this would imply that A_{A} = Δ_{A}, which contradicts Lemma 23. Likewise it is impossible that l(Δ_{A}) = 1. It follows that {l(_{A}Δ), l(Δ_{A})} = {2, 3}, as required. □

We use < to mean “less than, in the order ▲”.

Lemma 25. C is Ringel self-dual.

Proof. To say that C is Ringel self-dual is to say that _{A}A is a full tilting module for A. This is equivalent to saying that A_{A} is a full tilting module for A (consider finite dimensional quotients/subalgebras, and pass to a limit). In other words, A is left Ringel self-dual if and only if A is right Ringel self-dual. To establish the Ringel self-duality of C, we may therefore assume that _{A}Δ has length 3, by Lemma 24. Suppose C is not Ringel self-dual. Then we have a nonprojective indecomposable tilting module T (λ), which has a filtration with sections

Δ(λ), Δ(λ_{2}), ..., Δ(λ_{n}).

Note that Δ(λ) is the bottom section, and up to scalar we have a unique homomorphism from P (λ) to T whose image is Δ(λ) (reference Ringel). Since T is nonprojective, it has length < 4. Since the sections all have length 3, the tilting module has length 3, and the tops of the sections all lie in the top of T. The module T also has a ∇-filtration since it is tilting. Any simple in the top of T must lie in the top of some ∇ of length 2. In particular, λ itself must lie in the top of some ∇(μ) of length 2. The surjective homomorphism P (λ) ↠ ∇(μ) must lift to a homomorphism P (λ) → T. Up to scalar, there is a unique such homomorphism whose image is Δ(λ), implying that μ is a factor of Δ(λ). Thus, λ is a factor of ∇(μ) and μ is a factor of Δ(λ). Thus λ > μ > λ which is a contradiction. □

Lemma 26. Standard modules for A have irreducible head and socle.

Proof. A standard module in one ordering is isomorphic to a costandard module in another ordering, by Lemmas 21 and 25. □

If there is a nonsplit extension of λ by μ then either λ > μ or λ < μ. We define Rel to be the set of relations λ > μ or λ < μ of this kind. We define ↑ to be the partial order on Λ generated by Rel. The order ▲ is a refinement of ↑. We define ↓ to be the ordering on Λ which is Ringel dual to ↑.

Lemma 27. C is a highest weight category with respect to the partial order ↑ on Λ.

Proof. C has length 4, which implies that either left or right standard modules have length two and, by Lemma 26, are therefore uniserial. The quasi-hereditary structure induced by ▲ is already determined by these non-split extensions and therefore the order ↑ already induces the same quasi-hereditary structure as its refinement ▲. □

From now on, whenever we refer to standard or costandard modules, or to orderings, without specifying the order, we mean the order ↑.

Lemma 28. Standard A-modules are directed.

Proof. We want that all standard modules are directed, which means for any subquotient of a standard module which is a non-split extension of simple modules λ by μ, λ is greater than μ. This is trivial for a standard module of Loewy length 2. Let Δ(x) be a standard module of Loewy length 3. It must have an irreducible socle y by Lemma 26. Thus Δ(x) appears in a Δ-filtration of P (y). Δ(y) appears as the top factor of a Δ-filtration of P (y). Indeed, since P (y) has length 4 with irreducible top and socle, a Δ-filtration of P (y) has precisely two factors, namely Δ(x) and Δ(y). The module Δ(y) must have irreducible socle z, where y > z, by Lemma 26. Since P (y) has length 4 and Δ(y) has length 3, we conclude there is an extension of z by y. Since ∇(y) is dual to a Δ which has length 2, ∇(y) itself has length 2, and it must in fact be this extension of z by y. For any other nonsplit extension of an irreducible modules w by y, we must have w > y by Lemma 27. These are precisely the extensions of w by y contained in Δ(x). The extensions of x by w contained in Δ(x) imply x > w by definition of a standard module. Thus any extension of λ by μ in Δ(x) implies λ > μ as required. □

Corollary 29. The orders ↑ and ↓ on Λ are opposite.

Proof. Just as standard modules are directed in the ↑ ordering, costandard modules are directed in the ↓ ordering. But standard modules in the ↑ ordering are equal to costandard modules in the ↓ ordering by Lemmas 21 and 25. Therefore ↑ and ↓ orderings are opposite, as required. □

Remark 30 If a finite-dimensional algebra is quasi-hereditary with respect to two opposing orders then it must be directed, in which case the standard modules are projectives in one ordering, and simples in the opposite ordering. This can easily be proved by induction on the size of the indexing set. Symmetric quasihereditary algebras are never directed, since their projective indecomposable objects have isomorphic head and socle.

Remark 31 It is not necessarily the case that a Ringel self-dual hica is a highest weight category with respect to two opposing orderings. Examples of length 5 are found amongst module categories of rhombal algebras [3].

Let X(λ) denote the kernel of the surjective homomorphism Δ(λ) ↠ λ, for λ ∈ Λ.

Definition 32. The Δ-quiver of A is the quiver with vertices indexed by Λ, and with arrows λ → μ corresponding to simple composition factors μ in the top of X(λ).

Lemma 33. Components of the Δ-quiver of length 2 are directed lines. Components of the Δ-quiver of length 3 are directed topsy-turvy quivers.

Proof. The length 2 case is easy. In length 3, we have a permutation ϕ ⟲ Λ which takes λ to the socle of Δ(λ). We prove that F(λ) = P(λ^{ϕ}) via a sequence of correspondences: arrows emanating from λ in the Δ-quiver are in correspondence with simple composition factors μ in the top of X(λ); simple composition factors μ in the top of X(λ) are in correspondence with extensions of λ by μ such that λ > μ; since Δ^{↑}(λ) = ∇^{↓}(λ^{ϕ}), whilst ↑ and ↓ are opposites, extensions of λ by μ such that λ > μ are in one-one correspondence with extensions of μ by λ^{ϕ} such that μ > λ^{ϕ}; extensions of μ by λ^{ϕ} such that μ > λ^{ϕ} are in correspondence with simple composition factors λ^{ϕ} in the top of X(μ); simple composition factors λ^{ϕ} in the top of X(μ) are in one-one correspondence with arrows into λ^{ϕ} in the Δ-quiver. Since standard modules are directed, the Δ-quivers are also directed (ie they generate a poset). □

Lemma 34. A has a Δ-subalgebra B.

Proof. We want to show that _{B}Δ ≅ _{B}B. Let us write A = F Q/I as the path algebra of Q modulo relations, where Q is the Ext^{1}-quiver of A. If there is a positive arrow x → y in Q, that is to say an arrow x → y in Q such that x > y, then x and y lie in the same component of the Δ-quiver. Since all standard modules are directed, the connected component of the quiver generated by these arrows are the components of the Δ-quiver. Let B be the subalgebra of A generated by arrows x → y in Q such that x > y. Since all standard modules are directed, composing the natural maps Be_{λ} → Ae_{λ} → Δ(λ) gives us a surjection Be_{λ} ↠ Δ(λ). To establish this composition map is an isomorphism, we have to worry about its kernel, which must lie in Rad^{2}(B)e_{λ}, which is the socle of B since A has length 4. Assume there is a simple S in the kernel. Then S would have to be a factor of Δ(μ) in a Δ-filtration of Ae_{λ}; restrictions imply S would lie in the socle of some Δ(μ) of length 2, where μ > λ (otherwise if Δ(μ) has length 3 then λ lies in the socle of Δ(μ) so S > λ, λ > S since S appears in Be_{λ}, contradiction). Since S lies in Rad^{2}(B)e_{λ}, we have positive arrows λ → ν → S, for some ν, so S must lie in Δ(ν), and there is an arrow λ → ν in the Δ-component of λ. There are now two possibilities. Either Δ(λ) has length 3, implying S lies in a Δquiver component of length 2 (for μ), and a Δ-quiver component of length 3 (for λ)- contradiction. Else Δ(λ) has length 2, which implies we have a Δ-quiver component of length 2 containing the quiver μ → S ← ν - contradiction (the structure of any length 2 Δ-quiver component is an oriented line by Lemma 33). We conclude that the map B ↠ Δ must in fact have zero kernel, ie B is a Δ-subalgebra.

□

Let B be the Δ-subalgebra of A.

Lemma 35. Suppose B has length 3. Then the algebra homomorphism B → A splits.

Proof. Let I denote the ideal of A which is a sum of spaces AaA where a is a negative arrow in the quiver Q of A. Then the kernel J of the A-module homomorphism A → _{A}Δ is contained in I, since A has length 4 and Δs have length 3, implying J is generated in the top of the radical of A. Also, J contains I since I is generated as a vector space by products of 1, 2, or 3 arrows in the quiver, at least one of which lies in I, and these products all lie in J since all Δs are directed. Thus the kernel of A → _{A}Δ is equal to I. By symmetry, the homomorphism of right A-modules A → Δ_{A} also has kernel I. Therefore B ⊕ I → A is an isomorphism of B-B-bimodules, and the algebra homomorphisms

B → A → A/I

compose to give an algebra isomorphism B ≅ A/I. Therefore the homomorphism B → A splits as required. □

Lemma 36. B is self-injective.

Proof. We write ^{↑}B for the _{A}Δ-subalgebra taken with respect to the ↑ ordering, and B^{↓} the Δ_{A}-subalgebra taken with respect to the ↓ ordering. We know that

B = ^{↑}B ≅ ⨁_{x∈Λ}_{A}Δ^{↑}(x) ≅ ⨁_{x∈Λ}_{A}∇^{↓}(x) ≅ ⨁_{x∈Λ}(Δ_{A}^{↓}(x))^{∗} ≅ (B^{↓})^{∗},

where B^{↓} is also a Δ-subalgebra of A. Thus ^{↑}B ≅ (B^{↓})^{∗} as A-modules, and therefore as ^{↑}B-modules. To prove ^{↑}B is self-injective we must show that ^{↑}B ≅ B^{↓}. Indeed,^{↑}B is defined to be the subalgebra generated by left positive ↑-arrows, whilst B^{↓} is defined to be the subalgebra generated by right positive ↓-arrows. Passing from the left regular action of an algebra on itself to the right regular action reverses arrow orientation. Therefore left positive ↑-arrows are equal to right negative ↑- arrows which are equal to right positive ↓-arrows. Thus ^{↑}B ≅ B^{↓} as required. □

Lemma 37. If B has Loewy length 3, then A is isomorphic to T (B), the trivial extension algebra of B by its dual. If B has Loewy length 2, then A is isomorphic to U(B, M) where M is a self-dual B-B-bimodule.

Proof. We may assume B = B^{↑} has Loewy length 3, in which case B^{↓} has Loewy length 2. We have a surjection of algebras A ↠ B which splits, via an algebra embedding B ↪ A. Dually, we have an embedding of A-A-bimodules B^{∗} ↪ A^{∗}. Since A ≅ A^{∗} as bimodules, we have a homomorphism of A-A-bimodules B^{∗} ↪ A. Taking the sum of our two embeddings gives us a homomorphism of B-B-bimodules,

B ⊕ B^{∗} → A.

This homomorphism is a bimodule isomorphism, because every projective indecomposable A-module has a canonical Δ-filtration featuring precisely two Δ(λ)s, one of which is a summand of B, and the other of which is a summand of B^{∗}. We can thus identify the image of B^{∗} in A with the kernel of the algebra homomorphism A → B. The image of B^{∗} in A multiplies to zero, because the map B^{∗} → A is a homomorphism of A-A-bimodules, on which the kernel of the surjection A ↠ B acts trivially. The image of B in A multiplies via according to multiplication in B. In other words, the map T (B) = B ⊕ B^{∗} → A is an algebra isomorphism, as required. The algebra A has a ℤ_{+}^{2}-grading whose first component comes from the radical grading on B^{↑}, and whose second component comes from the trivial extension grading, with B^{↑} in degree 0 and its dual in degree 1. In other words, the degree (∗, 0) part of A is B^{↑}. We can then identify the degree (0, ∗) part of A with B^{↓}, which is self-injective of Loewy length 2. The degree (2, ∗) part of A is then isomorphic to B^{↓∗}, and we define M to be the degree (1, ∗) part of A. The isomorphism A ≅ A^{∗} exchanges the B^{↓}-B^{↓}-bimodules B^{↓} and B^{↓∗}, whilst it defines an isomorphism M ≅ M^{∗}. This way, we obtain the algebra isomorphism A ≅ U(B^{↓}, M). □

Let 𝔅𝔦𝔭 denote the 2-category whose objects are bipartite graphs; whose arrows Γ → Γ′ are given by sequences (γ_{1}, ..., γ_{n}) of distinct vertices of Γ, such that Γ′ = Γ\{γ_{1}, ..., γ_{n}}; whose 2-arrows are given by permutations of such sequences. The following result is a refinement of Theorem 1.

Theorem 38. The correspondence Γ ↦ C_{Γ} extends to a 2-functor

𝔅𝔦𝔭 → ℌ𝔦𝔠𝔞_{4}

which is essentially bijective on objects.

Proof. The correspondence Γ ↦ A_{Γ} -mod is essentially bijective on objects, by Lemmas 16, 34, 36, and 37. We have to associate functors and natural transformations in ℌ𝔦𝔠𝔞_{4} to arrows and 2-arrows in 𝔅𝔦𝔭. Suppose γ ∈ Γ is a vertex of a bipartite graph, and Γ′ = Γ\γ. We have an isomorphism A_{Γ′} ≅ e_{Γ′}A_{Γ}e_{Γ′}, and therefore an exact functor

F_{γ} = e_{Γ′}A_{Γ}⊗_{A_{Γ}} : A_{Γ} -mod → A_{Γ′} -mod

which sends the irreducible corresponding to a vertex v to the irreducible corresponding to a vertex v, if v ≠ γ and to zero if v = γ. To a sequence (γ_{1}, ..., γ_{n}) we associate the composition functor F_{γ_{n}}...F_{γ_{1}}. There are natural isomorphisms between various functors corresponding to isomorphisms of bimodules. □

Let B^{↑} = F Q^{↑}/R^{↑}, B^{↓} = F Q^{↓}/R^{↓} be minimal presentations of B^{↑} and B^{↓} by quiver and relations. Let Q be the union of Q^{↑} and Q^{↓} in which we identify the vertices of these quivers if they represent the same irreducible A-module. Let R be the union of R^{↑}, R^{↓} and R^{↕}. Let R^{↕} denote the set of relations which insist that squares in Q involving two arrows of Q^{↑} and two arrows of Q^{↓} commute.

Lemma 39. A = F Q/R is a minimal presentation of A by quiver and relations.

Proof. We have a surjective map F Q ↠ A. It is not difficult to see this must factor through a map F Q/R ↠ A. We now want to bound the dimension of a projective of F Q/R. Without loss of generality assum that B = B^{↑} has Loewy length 3 and B^{↓} therefore has Loewy length 2. So Q^{↑} is a topsy-turvy quiver and Q^{↓} is linear. We claim that a spanning set of (F Q/R)e_{x} is given by abe_{x} where b ∈ B and a is either an idempotent or an arrow from Q^{↓}. Without a doubt a spanning set is given by the union of all elements of the form a_{1}b_{1} ⋯ a_{r}b_{r}e_{x} where a_{i} are either idempotents or arrows in Q^{↓} and b_{i} ∈ B. However, if we have an arrow a in Q^{↓} (say with source y and target ϕ^{−1}(y)) and and arrow b ∈ Q^{↑} starting in ϕ^{−1}(y) , the product bae_{y} = be_{ϕ^{−1}y}ae_{y} equals a′b′e_{y} where b′ = ϕ(b) and a′ is the unique arrow starting at the end vertex of b′e_{y}. Indeed, Q^{↑} being topsy turvy implies the existence of b′ and in Q^{↓} there is an arrow from x to ϕ^{−1}x for every x. So denoting by z the end vertex of b, there is a square

\xymatrix{y \ar[r]^{a} \ar[d]^{ϕ(b)} & ϕ^{−1}(y) \ar[d]^{b} \\ z \ar[r]^{a′} & ϕ^{−1}(z)} .

By the required relations this has to commute and we obtain bae_{y} = a′b′e_{y}. Hence the path a_{1}b_{1} ⋯ a_{r}b_{r}e_{x} is equivalent modulo R to a path a′_{1} ⋯ a′_{r}b′_{1} ⋯ b′_{r} = a′_{1} ⋯ a′_{r}b′. However, by the relations in B^{↓}, any product of arrows in Q^{↓} is zero, so we obtain the claim that (F Q/R)e_{x} is spanned by abe_{x} where b ∈ B and a is either an idempotent or an arrow from Q^{↓}. This implies that dim(F Q/R)e_{x} ≤ 2 dim Be_{x} = dim(B + B^{∗})e_{x} = dim Ae_{x}, the equality dim Be_{x} = dim(B + B^{∗})e_{x} coming from the fact that B is self-injective. Combining the above surjection F Q/R ↠ A and this inequality, we obtain the statment of the lemma. □

7. Koszulity

For an algebra C we denote by C^{!} the quadratic dual of C.

Theorem 40. The following are equivalent:

1. Γ is not a simply laced Dynkin graph.

2. Z_{Γ} is Koszul.

3. B_{Γ} is Koszul.

4. A_{Γ} is Koszul.

5. A_{Γ}^{!} -mod is Calabi-Yau of dimension 3.

The proof of this result is the length of the section.

1 is equivalent to 2, by a theorem of Martínez-Villa [10].

2 is equivalent to 3, since B_{Γ} -mod^{⊕2} is equivalent to Z_{Γ} -gr by Lemma 18. The implication 3 ⇒ 4 follows from the following lemma, in case A = A_{Γ}, and B = B_{Γ}.

Lemma 41. If B is a self-injective Koszul algebra of length n, the trivial extension algebra A = B ⊕ B^{∗}⟨n⟩ is Koszul.

Proof. Since B is selfinjective, we have an isomorphism B ≅ B^{∗} of B-modules. The algebra A is a trivial extension A = B ⊕ B^{∗}, and we thus have a map A → A of B-modules extending to a map of A-modules whose kernel is B^{∗} and whose cokernel is B. Stringing these together gives us a projective resolution

... → A → A → B

of B as a left A-module. Since B is self-injective and radical graded, every injective B-module has length n, and consequently this is a linear resolution of B as a left A-module. Taking summands, we find that every projective B-module has a linear resolution as a left A-module. If B is Koszul, then B^{0} has a linear resolution by projective B-modules. Thus B^{0} is quasi-isomorphic to a linear complex of projective B-modules. Since projective B-modules are quasi-isomorphic to a linear complex of projective A-modules, we deduce B^{0} is isomorphic to a linear complex of projective A-modules. That is to say, A^{0} = B^{0} has a linear resolution by projective A-modules. In other words, A is Koszul. □

The implication 4 ⇒ 3 follows from the following lemma, in case A = A_{Γ}, and B = B_{Γ}.

Lemma 42. If B is a radical-graded selfinjective algebra of length n, such that A = B ⊕ B^{∗}⟨n⟩ is Koszul, then B is Koszul.

Proof. We have a ℤ_{+} × ℤ_{+}-grading on A in which B lies in degree (?, 0), the dual of B lies in degree (?, 1), and in which the inherent ℤ_{+}-grading on B is the radical grading. This corresponds to the action of a two-dimensional torus 𝕋 on A. Thus 𝕋 acts on A^{1} and we have an exact sequence

0 → R → A^{1} ⊗_{A^{0}} A^{1} → A^{2} → 0

of 𝕋-modules, where R denotes the relations for A and A^{j} refers to the jth component in the total grading, whose dual

0 ← A^{!2} ← A^{!1} ⊗_{A^{!0}} A^{!1} ← R^{!} ← 0

is also an exact sequence of 𝕋-modules. Since A^{!} is quadratic by definition, with relations R^{!}, we have an action of 𝕋 on A^{!}, which gives a ℤ_{+} × ℤ_{+}-grading on A^{!}. We have a linear resolution of W = A^{(0,0)}, given by the Koszul complex

A ⊗_{W} A^{!∗}

of A ([1], 2.8). Here A^{!∗} denotes the graded dual of A^{!}. The differential on the Koszul complex respects the ℤ_{+} × ℤ_{+}-grading on A and A^{!} (see [1], 2.6). In other words, it sends terms involving arrows in A^{(0,1)} or A^{!(0,1)∗} to terms involving arrows in A^{(0,1)} or A^{!(0,1)∗}, and terms not involving arrows in A^{(0,1)} or A^{!(0,1)∗} to terms not involving arrows in A^{(0,1)} or A^{!(0,1)∗}. Consequently the subcomplex A^{(?,0)} ⊗_{R} A^{!(−,0)∗}) is a direct summand of the Koszul complex regarded as a complex of B-modules. Taking this component gives us a linear resolution of R = B^{0} as a B-module. Therefore B is Koszul. □

If C is a graded algebra and C -mod is Calabi-Yau of dimension n, then Ext_{C}^{∗}(C^{0}, C^{0}) is a super-symmetric algebra concentrated in degrees 0, 1, ..., n, by Van den Bergh A.5.2 [2]. We have a converse which applies for Koszul algebras:

Theorem 43. Suppose K is a Koszul algebra such that K^{!} is super-symmetric of length n + 1, then K -mod is Calabi-Yau of dimension n.

Proof. There is an equivalence between derived categories of graded modules for K^{!} and K via the Koszul complex. Since K^{!} is locally finite dimensional, this restricts an equivalence of bounded derived categories, by a theorem of Beilinson, Ginzburg, and Soergel ([1], Theorem 2.12.6). Under this equivalence, simple K^{!}- modules correspond to projective indecomposable K-modules. Since K^{!} is locally finite-dimensional The equivalence therefore restricts to an equivalence between D^{b}(K^{!} -gr) and D^{b}(K -grperf). Also under this equivalence, injective K^{!}-modules correspond to simple K-modules, whilst shifts ⟨i⟩ in D^{b}(K^{!} -gr) correspond to shifts in degree ⟨−i⟩[−i] in D^{b}(K -grperf). This homological shift in degree means that the Calabi-Yau-n property for K -mod is equivalent to the super-Calabi-Yau-0 property for K^{!} -perf, thanks to Van den Bergh’s calculation A.5.2 [2]. To prove the super-Calabi-Yau-0 property for K^{!} -perf, it is enough to check that K^{!} is a supersymmetric algebra (cf [16], Theorem 3.1). □

Assume 4. Then the Koszul dual A^{!} of A is Calabi-Yau of dimension 3. The Koszul dual of a supersymmetric algebra of length n + 1 is Calabi-Yau of dimenion n by Theorem 43. The trivial extension algebra A = B + B^{∗}⟨3⟩ is super-symmetric with B concentrated in degrees 0, 1, 2, with B^{∗}⟨3⟩ concentrated in degrees 3, 2, 1, and with bilinear form pairing B^{i} and (B^{i})^{∗}⟨3⟩ via

< β, b >= β(b) < b, β >= (−1)^{i(3−i)}β(b),

for b ∈ B^{i}, β ∈ (B^{i})^{∗}. Thus 4 implies 5.

Assume 5. Since A^{!} is Calabi-Yau of dimension 3, its relations are the derivatives of a superpotential, and its degree 0 part has a 4-term resolution, its Jacobi resolution [2]. The superpotential must be cubic, since A^{!} is quadratic. This implies further that the Jacobi resolution of A^{!0} is linear, so A^{!} must be Koszul. Thus 5 implies 4.

We have now shown that 1 ⇔ 2 ⇔ 3 ⇔ 4 ⇔ 5, completing the proof of Theorem 40.

Remark 44 If Γ is a bipartite graph, then an orientation of Γ gives rise to a ℤ_{+}^{3}-grading on A_{Γ}. If every vertex of Γ is attached to at least two vertices, then this leads to a ℤ_{+}^{3}-grading of the Calabi-Yau algebra A^{!} of dimension 3, which can otherwise be thought of as the action of a 3-dimensional torus on A^{!}. The algebra A^{!} has homological dimension 3, and admits the action of a 3-dimensional torus. It thus belongs to the realm of 3-dimensional noncommutative toric geometry.

Example 45 If Γ is given by tiling of a bi-infinite line

...  •  •  •  •  •  •  •  ...

then the Calabi-Yau algebra of dimension 3 we obtain is familiar from toric geometry. It is the algebra associated to the brane tiling of the plane by hexagons [6]. Its quiver can be thought of as an orientation of the A_{2}-lattice (for a picture, see section 8, assumption 3). If we give Γ an alternating orientation,

[figure: ...; •; •; •; •; •; •; •; ...]

then in the resulting grading on A^{!}, the three copies of ℤ_{+} correspond to the three directions of arrows in the A_{2}-lattice.

8. Relaxing the assumptions

We have given a combinatorial classification of hicas of length ≤ 4 by bipartite graphs. Here we show that the relaxation of any of the homological assumptions on our categories would necessarily introduce further combinatorial complexity into the classification.

Assumption 1: highest weight structure. The existence of a highest weight structure on a category is a strong assumption, and the assumptions of Ringel self-duality and homogeneity of standard modules require the existence of a highest weight structure on the category. It is a cinch to give examples of indecomposable Calabi-Yau 0 categories of length 4 which are not highest weight categories, such as the module category of the local symmetric algebra F [x]/x^{4}.

Assumption 2: Calabi-Yau 0 property. The Calabi-Yau property is another strong homological restriction on a category. An example of a length 4 highest weight category which is indecomposable and Ringel self-dual, and whose standard modules are homogeneous, is the path algebra of the linear quiver

\xymatrix{... & • \ar[r] & • \ar[r] & • \ar[r] & • \ar[r] & • \ar[r] & • \ar[r] & • & ...}

modulo all relations of degree ≥ 4.

Assumption 3: homogeneity. The homogeneity restriction on a hica is fairly natural, since the known examples of highest weight Calabi-Yau 0 categories arising in group representation theory and the theory of tilings satisfy this restriction. However, some interesting combinatorics arise in length 4 if the condition is dropped. For example, let C_{Γ} be the hica associated to a bi-infinite line Γ, whose quiver is an orientation

\xymatrix{.. & • \ar[dr] &  & • \ar[ll] \ar[dr] & .. \\ • \ar[ur] \ar[dr] &  & • \ar[ll] \ar[ur] \ar[dr] &  & • \ar[ll] \\ .. & • \ar[ur] &  & • \ar[ll] \ar[ur] & ..}

of the A_{2} lattice, by example 45, and by construction comes with a (horizontal) projection π onto Γ. There are two natural ways to obtain highest weight indecomposable Calabi-Yau 0 categories which are not homogeneous from C_{Γ}. The first is by choosing a section of π, that is a path in the A_{2} lattice which projects homeomorphically onto Γ via π. The elements of Λ to the right of the path form a coideal in the poset. Truncating C_{Γ} at this coideal gives us a highest weight category of length 4 which is CY-0, but not homogeneous (cf. [4], 3.5(b)). Such a truncation is not Ringel self-dual. Beneath is a portion of such a truncated poset, whose left edge defines an orientation of Γ. We use dotted arrows to represent directions in a partial order on the vertices of the lattice, rather than solid arrows which represent arrows in a quiver:

\xymatrix{• \ar@{.>}[dr] &  & • \ar@{.>}[dr] &  & • \ar@{.>}[dr] &  & • \ar@{.>}[dr] &  & .. \\  & • \ar@{.>}[dr] \ar@{.>}[ur] &  & • \ar@{.>}[dr] \ar@{.>}[ur] &  & • \ar@{.>}[dr] \ar@{.>}[ur] &  & • \\  &  & • \ar@{.>}[dr] \ar@{.>}[ur] &  & • \ar@{.>}[dr] \ar@{.>}[ur] &  & • \ar@{.>}[dr] \ar@{.>}[ur] &  & .. \\  & • \ar@{.>}[dr] \ar@{.>}[ur] &  & • \ar@{.>}[dr] \ar@{.>}[ur] &  & • \ar@{.>}[dr] \ar@{.>}[ur] &  & • \\  &  & • \ar@{.>}[ur] &  & • \ar@{.>}[ur] &  & • \ar@{.>}[ur] &  & ..}

The second way to obtain an inhomogeneous highest weight indecomposable Calabi-Yau 0 category from C_{Γ} is by merely altering the ordering of the vertices. Certain orderings of the vertices of the A_{2} lattice give A_{Γ} -mod is an inhomogeneous highest weight category, which is Ringel self-dual. Here is an example of a portion of such a partial ordering:

[figure: •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •]

Assumption 4: length ≤ 4. We have studied hicas of length 4, since 4 is the shortest length in which a nontrivial classification is possible. There are two kinds of hicas of length 5: those whose left and right standard modules have length 4 and 2, and those whose left and right standard modules have length 3 and 3. The category of graded modules over a radical-graded symmetric algebra of length 4 is equivalent to the category of modules over a directed self-injective algebras of length 4. Trivial extensions of directed self-injective algebras of length 4 by their duals give examples of hicas of length 5 whose left and right standard modules have length 4 and 2. Michael Peach’s rhombal algebras give examples of hicas of length 5 associated to rhombic tilings of the plane whose left and right standard modules have length 3 and 3. We would be interested to learn more about hicas of length 5.

9. Tilting

There are natural self-equivalences of the derived categories of C_{Γ}, which are obtained from a standard tilting procedure for symmetric algebras:

Lemma 46. Suppose A is a symmetric algebra, and suppose that the endomorphism ring e_{λ}Ae_{λ} is isomorphic to the dual numbers F [d]/d^{2}. Then we have an exact selfequivalence ψ_{λ} of the derived category of A given by tensoring with the two-term complex

Ae_{λ} ⊗_{F} e_{λ}A → A,

whose arrow is the multiplication map.

Proof. This functor is obviously exact. It fixes all simple modules with the exception of the simple top λ of Ae_{λ}, which it sends to Ω(λ). The module Ω(λ) has simple socle λ since A is symmetric, and other composition factors different from λ since e_{λ}Ae_{λ} is isomorphic to the dual numbers. Therefore collection of simples μ ≠ λ, along with Ω generate D^{b}(A -mod), and ψ_{λ} is an equivalence. □

The self-equivalence ψ_{λ} is called a spherical twist, because the cohomology ring of the sphere can be identified with the dual numbers (cf. [18]). One way to obtain self-equivalences of D^{b}(C_{Γ}) from spherical twists is by lifting self-equivalences of the derived category of the zigzag algebra Z_{Γ}, whose projective indecomposable modules for the algebra Z_{Γ} all have an endomorphism ring isomorphic to the algebra of dual numbers. A second way to obtain self-equivalences of D^{b}(C_{Γ}) is to apply spherical twists directly to C_{Γ}, whose projective indecomposable objects also have endomorphism rings isomorphic to the dual numbers. Let us consider the first case. The projective indecomposable modules for the algebra Z_{Γ} all have an endomorphism ring isomorphic to the algebra of dual numbers. Standard tilts generate an action of a 2-category T_{Γ} on D^{b}(Z_{Γ} -gr) which lifts to an action of T_{Γ} on D^{b}(C_{Γ}), by a result of Rickard [15, Thm 3.1]. A second way to obtain self-equivalences of D^{b}(C_{Γ}) is to apply Seidel-Thomas twists directly to A_{Γ}, whose projective indecomposable modules have endomorphism rings isomorphic to the dual numbers. Standard tilts generate the action of a 2-category U_{Γ} on D^{b}(C_{Γ}), whose combinatorics is rather different from that of T_{Γ}.

Example 47 When Γ is a bi-infinite line, we have an action of the braid 2-category BC_{∞} on a bi-infinite line on the derived category of Z_{Γ}, by a theorem of Seidel and Thomas [9]. The action of BC_{∞} on D^{b}(Z_{Γ} -mod) lifts to an action of BC_{∞} on C_{Γ}. Arrows in BC_{∞} are braids with an infinite number of strands, and 2-arrows are braid cobordisms, such as Reidemeister moves pictured as follows:

[figure: •; •; •; •; •; •; •; ⇝; •; •; •; •; •; •; •]

In this picture, a rhombus represents a pair of braids running parallel to the two sides and crossing in the middle.

[figure: •; •; •; •]

We have a natural 2-functor BC_{∞} → T_{Γ}. The Reidemeister move depicted above therefore corresponds in a natural way to a 2-arrow in T_{Γ}. However, this Reidemeister move also corresponds naturally to an arrow in S_{Γ}. Let us explain how. Suppose we remove edges of the A_{2}-lattice to give a rhombic tiling T of the plane, whose edges lie in the quiver Q of A_{Γ}. We have a grading of A_{Γ} which places arrows in Q which are edges of T in degree 0 and arrows in Q which are not edges of T in degree 1. Let us denote by D_{T} the degree 0 part of A taken with respect to this tiling. The algebra A_{Γ} is a trivial extension of D_{T} by D_{T}^{∗}. If T′ is obtained from T by a Reidemeister move centred on the vertex λ,

\xymatrix{ & • &  & • \ar[ll] \ar[dr] &  &  &  & • \ar[dr] &  & • \ar[ll] \ar[dr] \\ • \ar[ur] \ar[dr] &  & • \ar[ll] \ar[ur] \ar[dr] &  & • & ⇝ & • \ar[ur] \ar[dr] &  & • &  & • \ar[ll] \\  & • &  & • \ar[ll] \ar[ur] &  &  &  & • \ar[ur] &  & • \ar[ll] \ar[ur]}

then D_{T′} is derived equivalent to D_{T}, because the complex of D_{T} -modules given by the sum of D_{T} e_{λ} ⊗ e_{λ}D_{T} → D_{T} and D_{T} e_{λ} → 0 is a tilting complex whose derived endomorphism ring is isomorphic to D_{T′}. This derived equivalence between D_{T} and D_{T′} lifts to an equivalence of trivial extensions, that is to say a self-equivalence of D^{b}(A_{Γ} -mod) = D^{b}(C_{Γ}); this self-equivalence of D^{b}(C_{Γ}) is precisely the spherical twist ψ_{λ}.

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