Categories of Representations of Cyclic Posets

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,weconstructanR-linear Frobenius category φ(X). I will go over the definition of a Frobenius category and indicate why our constructionF satisfies each condition. By a well-known result of Happel, the stable category φ(X)willbeatriangulated category over K.Ineachexampleinthechartbelow, (CX)willbeaclustercategory: Cφ cyclic poset automorphism cluster category comments

Xφ (X) Cφ

Zn φ(i)=i +1 (An 3)2-CY 1 < 2 <

S1 id not 2-CY but has clusters C continuous cluster category Y [1] ∼= Y

S1 Z id ˜ not 2-CY (Y [1] = Y ) ∗ C ∼ φ(x, i)=(x, i +1) ˜ 2-CY containsC Zm Z φ(i, j)=(i +1,j) m-cluster category (m +1)-CY ∗ φ(m, j)=(1,j+1) oftypeA ∞

3 3-cluster category (1)/3Z Z φ3(x, i)=(x, i +1) 4-CY P ∗ of type A ∞ Iwillgooversomeoftheeasierexamplesofthisconstruction.CYmeansCalabi-Yau.

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0.2. Cyclic poset. is same as periodic poset X˜.i.e. poset automorphism σ : X˜ X˜ so that x<σxfor all x. Also: ∃ → ( x, y X˜) x σjy for some j Z. • ∀ ∈ ≤ ∈ ˜ (1) Zn: X = Z, σ(x)=x + n (n ﬁxed). (2) X˜ = (1) (from Schmidmeier’s lecture), σ :goupthreesteps. P (3) X˜ Z means X˜ Z with lexicographic order (from van Roosmalen). ∗ × Let X =setofσ orbits. How to describe cyclic poset structure just in terms of X? (1) Choose representativex ˜ X˜ for each orbit x X. (2) ( x, y X)letb = b(x, y∈)beminimalsothat˜∈x σby˜. (3) Let∀ c =∈δb: ≤ c(xyz):=b(xy)+b(yz) b(xz) − Then c : X3 N is independent of the choice of representativesx ˜. → X is in cyclic order iﬀ c(xyz) 1. In that case: • ≤ 0ifxyzincyclicorder c(xyz)= 1otherwise

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Proposition 0.2.1. The cyclic poset structure on a set X is uniquely determined by the function c : X3 N which is an arbitrary reduced cocycle ( reduced means c(xxy)=0=c(xyy). cocycle→ means δc =0.) 0.3. Representations. Deﬁnition 0.3.1. A representation M of (X, c)overR is (1) An R-module M for each x X˜ so that M = M . x ∈ x σx (2) An R-linear map My Mx for x

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0.4. Frobenius category. Deﬁnition 0.4.1. Let (X)denotethecategoryofallpairs(P, d)whereP (X) and d : P P so thatFd2 = t (mult by t). Morphism f :(P, d) (Q, d)aremaps∈P f : P Q→so that df = fd. · → → Theorem 0.4.2. In all examples on page 1, (X) is Krull-Schmidt with indecom- posable objects: F

β 0 β M(x, y):= Px Py, : Px Py ⊕ α 0 α with αβ = t, βα = t. · · Lemma 0.4.3. The functor G : (X) (X) given by P →F t 0 t · GP := P P, : P P ⊕ 10 id is both left and right adjoint to the forgetful functor F : (X) (X). F →P

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Theorem 0.4.4. For any cyclic poset X, (X) is a Frobenius category where a sequence F (A, d) (B,d) (C, d) → → is defined to be exact in (X) if A B C is (split) exact in (X). GP are the projective injective objects.F f = g iff→f →g = ds + sd for some s :PP Q. − → 0.5. Twisted version. An automorphism φ of X is admissible if: x φ(x) φ2(x) σx ≤ ≤ ≤ for all x X˜.Thenweget ∈ η ξ P x φP = P x P x −→ x φ(x) −→ x giving natural transformations η ξ P P φP P P −→ −→ Definition 0.5.1. Let φ(X)bethefullsubcategoryof (X)ofall(P, d)whered factors through η : P F φP . F P → Theorem 0.5.2. (X) is a Frobenius category with projective-injective objects Fφ ξP 0 ξP GφP := P φP, : P η φP ⊕ ηP 0 P

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0.6. m-cluster categories. Theorem 0.6.1. Let m 3, let X be a cyclically ordered set (equivalently, c(xyz) ≥ ≤ 1), and φ is an admissible automorphism of X Z so that φm(x, i)=(x, i +1)then ∗ φ(X Z) is (m +1)-Calabi-Yau. F ∗ On Zm Z let φ(i, j)=(i +1,j)fori

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Example 0.6.4. (m =5).Exampleofamaximalcompatiblesetof6-rigidobjectsin φ(Z5 Z). M(x, y)isarcfromx to y (horizontal if standard, vertical if nonstandard). CompatibleC ∗ arcs do not cross. There is 8-gon in center. Other regions have 7 sides. 11111000 00 1 1 1 1 1 − − − − − E D C B A E D C B A E D C B A

7-gon X1

Y1 7-gon Y2 7-gon 8-gon Y1 Y2

X1 7-gon

A B C D E A B C D E A B C D E 1 1 1 1 10 0 0 0 0 1 1 1 1 1 − − − − − Standard: X1 = M(A0,B1)(horizontal). Y1 = M(C1,E 1), Y2 = M(A 1,D1)arenonstandardbut(m +1)-rigid(vertical). − − Notation: (1,j)=Aj, (2,j)=Bj, etc.

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Department of Mathematics, Brandeis University, Waltham, MA 02454 E-mail address: [email protected]