CATEGORIES OF REPRESENTATIONS OF CYCLIC POSETS
KIYOSHI IGUSA, BRANDEIS UNIVERSITY
0.1. Abstract. This is joint work with Gordana Todorov. Let R = K[[t]] where
K is any field. Given a “recurrent” cyclic poset X and “admissible automorphism”
φ, we construct an R-linear Frobenius category Fφ (X). I will go over the definition
of a Frobenius category and indicate why our construction satisfies each condition.
By a well-known result of Happel, the stable category Cφ (X) will be a triangulated
category over K. In each example in the chart below, Cφ (X) will be a cluster category:
cyclic poset
automorphism
cluster category
X
φ
Cφ (X)
Zn
1 < 2 < · · · < n < σ1
φ(i) = i + 1
C(An−3 )
2-CY
Z
( with cyclic order )
φ(i) = i + 1
C(A∞ )
infinity-gon
2-CY
S1
id
S1 ∗ Z
id
φ(x, i) = (x, i + 1)
Zm ∗ Z
φ(i, j) = (i + 1, j)
φ(m, j) = (1, j + 1)
P(1)/3Z ∗ Z
comments
C
not 2-CY but has clusters
continuous cluster category
Y [1] ∼
=Y
φ3 (x, i) = (x, i + 1)
C˜
C˜0
contains
m-cluster category
of type A∞
3-cluster category
of type A∞
not 2-CY (Y [1] ∼
= Y)
2-CY
(m + 1)-CY
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I will go over some of the easier examples of this construction. CY means Calabi-Yau.
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4-CY
1.1. Cyclic poset. is same as periodic poset X̃. i.e. ∃ poset automorphism σ :
X̃ → X̃ so that x < σx for all x. Also:
• (∀x, y ∈ X̃) x ≤ σ j y for some j ∈ Z.
(1) Zn : X̃ = Z, σ(x) = x + n (n fixed).
(2) X = S 1 . Then X̃ = R with σ(x) = x + 2π.
(3) X̃ ∗ Z means X̃ × Z with lexicographic order (from van Roosmalen).
Let X = set of σ orbits. How to describe cyclic poset structure just in terms of
X?
Following, van Roosmalen 1011.6077, p.10 and Drinfeld 0304064, p.5, (who refers
to Besser and Greyson), this structure is equivalent to an N-category structure on
X.
Definition 1.1.1. An N-category is a category X with the property that the additive
monoid N acts freely on every Hom set
N × X (x, y) → X (x, y)
so that composition satisfies:
nf ◦ mg = (n + m)f g : x → z
(Acting freely means Hom sets are disjoint unions of copies of N: X (x, y) =
`
Nfi .)
Proposition 1.1.2. A cyclic poset structure on a set X is the same as an N category
X with object set X so that every Hom set X (x, y) is freely generated by one morphism
fxy .
So, given three objects, x, y, z ∈ X, we have
(1.1)
fyz fxy = nfxz
for some n ∈ N.
1.2. Linearized cyclic poset. We write: X = (X, c).
d
Definition 1.2.1. For any field k, the (completed) linearization k
X of X is defined
to be the category with object set X and morphism sets
kd
X (x, y) = kX (x,y) ∼
= k[[t]]
composition is given by
(rfyz ) ◦ (sfxy ) = rstn fxz
for any r, s ∈ R := k[[t]] where n is given by (1.1).
This is an R-category: Hom sets are R-modules and composition is R-bilinear.
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Definition 1.2.2. A representation of X is defined to be an R-linear functor
d
M :k
X → R-mod
Definition 1.2.3. Let P(X ) be the category of all finitely generated projective
representations of X .
op
d
Proposition 1.2.4 (Yoneda). P(X) ∼
X
= add k
Let Px be the projective representation of X generated at the point x ∈ X.
1.3. Frobenius category.
Definition 1.3.1. Let F(X) denote the category of all pairs (P, e) where P ∈ P(X)
and e : P → P so that e2 = ·t (mult by t). Morphism f : (P, e) → (Q, e) are maps
f : P → Q so that ef = f e.
Lemma 1.3.2. The functor G : P(X) → F(X) given by
0 t
GP := P ⊕ P,
:
1 0
P
z
·t
id
<P
is both left and right adjoint to the forgetful functor F : F(X) → P(X).
Theorem 1.3.3. For any cyclic poset X, F(X) is a Frobenius category where a
sequence
(A, e) (B, e) (C, e)
is defined to be exact in F(X) if A B C is (split) exact in P(X). GP are the
projective injective objects.
Proof. We can easily verify each step in the definition of a Frobenius category, namely,
a Frobenius category is an exact category which has enough projectives so that all
projective objects are injective. (E.g., kG-mod for any finite group G.) An exact
category is an additive category with a collection of exact sequences
E = {A f B g C}
so that
(1) A = ker g and C = coker f
(2) 0 0 0 ∈ E
(3) Given A f1 B coker f1 , B f2 C coker f2 in E, then
f2 f1
A −−→ C coker f2 f1
∈E
(4) The pushout of any A B C ∈ E along any f : A → A0 is in E.
(5) Dually for pull backs along C 0 → C.
Finally, GP is projective since G is left adjoint to F and GP is injective because G
is right adjoint to F . So, F(X ) is Frobenius.
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1.4. Twisted version. An automorphism φ of X is admissible if:
x ≤ φ(x) ≤ φ2 (x) ≤ σx
d
for all x ∈ X̃. In k
X this gives
ηx
ξx
Px −→ φPx = Pφ(x) −
→ Px
giving natural transformations
η
ξ
P
P
P −→
φP −→
P
Definition 1.4.1. Let Fφ (X) be the full subcategory of F(X) of all (P, e) where e
factors through ηP : P → φP .
Theorem 1.4.2. Fφ (X) is a Frobenius category with projective-injective objects
0 ξP
Gφ P := P ⊕ φP,
:
ηP 0
P
ξP
z
ηP
8
φP
2. Cluster categories
Definition 2.0.3. The stable category F of a Frobenius category F has the same
set of objects as F with morphism sets:
F(A, B)
F(A, B) =
A → P → B, P proj-inj
Theorem 2.0.4 (Happel). The stable category of a Frobenius category is triangulated.
Definition 2.0.5. Let C(X) = F(X) and Cφ (X) = F φ (X).
Theorem 2.0.6. In all examples on page 1, F(X) is Krull-Schmidt R-category with
indecomposable objects:
0 β
M (x, y) := Px ⊕ Py ,
:
α 0
Px
y
β
α
9
Py
with x, y ∈ X.
Corollary 2.0.7. Cφ (X) is Krull-Schmidt
M (x, y) where y ∼
6= φx, φ−1 x.
k-category with indecomposable objects
Remark 2.0.8. Cluster categories were first constructed by Buan-Marsh-ReinekeReiten-Todorov (0402054) as orbit categories. This construction is an alternate
construction in type A.
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2.1. Example X = Zn , φ(i) = i + 1. The cyclic poset has n elements in a circle.
Indecomposable objects are M (x, y) where x, y are at least two steps apart (because
M (i, i + 1) is projective-injective). This is the well-known CCS model (0401316) for
the cluster category of type An−3 . (But they did not give the triangulated structure
of the category.)
2.2. Example X = Z, φ(i) = i + 1. Indecomposable objects are M (x, y) where
x, y ∈ Z, |y − x| ≥ 2. This is the ∞-gon (0902.4125).
2.3. Example S 1 with φ = id. The objects are M (x, y) where x, y are distinct
points on the circle. This is the continuous cluster category (1209.1879).
2.4. Example X = Z5 ∗ Z.
(
(x + 1, i) if 1 ≤ x < 5
φ(x, i) =
(1, i + 1) if x = 5
Then Cφ (X) is 6-CY.
Theorem 2.4.1. Maximal compatible sets of 6 rigid objects correspond to 2-periodic
partitions of the doubled ∞-gon into 7-gons (except for the one in the middle).
Example of a maximal compatible set of 6-rigid objects in Cφ (Z5 ∗ Z). M (x, y) is
arc from x to y (horizontal if standard, vertical if nonstandard). Compatible arcs do
not cross. There is 8-gon in center. Other regions have 7 sides.
1
1
1
1
1
0
0
0
0
0
−1
−1
−1
−1
−1
E
D
C
Bf
A
E
D
C
B
A
E
D
C
B
A
7-gon
X1
Y1
Y2 7-gon
8-gon
7-gon
Y1
Y2
X1
7-gon
A
B
C
D
E
A
B
C
D
E
A
−1
−1
−1
−1
−1
0
0
0
0
0
1
&
B
C
D
E
1
1
1
1
Standard: X1 = M (A0 , B1 ) (horizontal).
Y1 = M (C1 , E−1 ), Y2 = M (A−1 , D1 ) are nonstandard but (m + 1)-rigid (vertical).
Notation: (1, j) = Aj , (2, j) = Bj , etc.
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