GENERIC EXTENSIONS AND COMPOSITION MONOIDS OF CYCLIC QUIVERS
BANGMING DENG, JIE DU AND ALEXANDRE MAH
Abstract. We provide a combinatorial construction of generic extensions of two nilpotent representations of a cyclic quiver. In the case of the cyclic quiver with two vertices, we determine a set of minimal defining relations for the associated composition monoid.
1. Introduction An advantage in the study of representations of a quiver Q is the geometric ap- proach. In this approach, representations V of Q with a fixed dimension vector d determine bijectively points p of a representation variety R(d) on which an alge- braic group G(d) acts and the isoclass (=isomorphism class) of V determines the G(d)-orbit OV of p. There are several important applications of the geometric ap- proach. For example, through this geometric correspondence, one sees easily that a quiver Q of finite representation type (i.e., Q has finitely many isomorphism classes of indecomposable representations) must be a Dynkin quiver, while the geometric construction of the canonical basis for the quantized enveloping algebra associated with Q rests on a more advanced knowledge of topology of singular spaces and in- tersection cohomology. Another interesting example is the generic extension M ∗ N of a representations M by a representation N of Q. This representation M ∗ N is defined as the representation whose corresponding orbit OM∗N is the unique dense orbit of the extension variety E(OM , ON ) which consists of points p such that the corresponding representation V (p) is an extension of M by N. In other words, M ∗N is the unique (up to isomorphism) maximal element among all extensions of M by N with respect to the degeneration order. Like the existence of the algebraic construction for canonical bases, generic exten- sions for a Dynkin or cyclic quiver can also be described algebraically in terms of generators and relations for the associated generic extension monoid algebra, which is isomorphic to the corresponding degenerate Ringel–Hall algebra. See [17, 12] for the Dynkin quiver case and [4, 24] for the cyclic quiver case. Since the composition monoid is a proper submonoid of the generic extension monoid in the cyclic quiver case, it would be interesting to determine the algebraic structure of the composition monoids of cyclic quivers.
Date: September 7, 2012. 2000 Mathematics Subject Classification. 17B37, 16G20. Supported partially by the Australian Research Council, the Natural Science Foundation of China, and the Doctoral Program of Chinese Higher Education. The research was carried out while Deng was visiting the University of New South Wales. The hospitality and support of UNSW are gratefully acknowledged. 1 2 BANGMINGDENG,JIEDUANDALEXANDREMAH
In this paper, we first give a combinatorial construction of the generic extension of two nilpotent representations of an arbitrary cyclic quiver based on some results in [3, 6]. Next we focus our study on the composition monoid for the cyclic quiver with two vertices. In this case, we obtain a set of minimal defining relations for the composition algebra (see Theorem 4.6). The proof relies on the fact that each fibre of the natural map from the monoid Ω of words to the composition monoid Mc of generic extensions contains a unique compact word. We expect to treat the general case, which is much more complicated, in a forthcoming paper. It is interesting to note that there are infinitely many defining relations for the degenerate composition algebra although the (generic) composition algebra itself has only finitely many defining relations. This reflects a phenomenon—an affine phenomenon—which does not occur in the Dynkin quiver case. Moreover, these in- finitely many relations displayed in Theorem 4.6 are the specialization at q = 0 of the so-called higher order fundamental relations, which are similar to the higher order quantum Serre relations discussed in [16, Ch. 7], in the Ringel–Hall algebra. Throughout the paper, let △ = △n denote the cyclic quiver with n vertices: n b
b b b b b 1 2 3 n−2 n−1 where n > 1. In what follows, we will identify the vertex set I of △ with Z/nZ = {1¯,..., n¯}, but we will write 1,... ,n instead of 1¯,..., n¯.
2. The generic extension monoid of △n In this section we recall from [3] the definition of the generic extension monoid M for the cyclic quiver △ = △n and present a combinatorial description of generic extensions of two nilpotent representations of △, as well as a combinatorial criterion for the degeneration order. 0 0 Let k be a field. By Rep △ = Rep k△ we denote the category of finite dimensional nilpotent representations V =(Vi, fi)i∈I of △ over k, where Vi are finite dimensional k-vector spaces and fi : Vi → Vi+1 are k-linear such that the composition fn ··· f2f1 : n V1 → V1 is nilpotent. The vector dim V = (dim kVi)i ∈ N is called the dimension vector of V . The sum i∈I dim kVi is called the dimension of V . Finally, let M = 0 M(△) denote the set of isoclasses of representations in Rep △ and let Md be the subset of isoclasses withP dimension vector d. Also, for each M ∈ Rep 0△, we denote by [M] the isoclass of M, and for m > 1, write mM = M ⊕···⊕ M . m It is well known that up to isomorphism,| {z there} are n simple nilpotent represen- tations Si (all having dimension 1) corresponding bijectively to the vertices of △. Moreover, for each i ∈ I and each integer l > 1, there is a unique indecomposable 0 object Si[l] in Rep △ with top Si and dimension l, and the Si[l] form a complete set of indecomposable objects in Rep 0△. GENERICEXTENSIONSANDCOMPOSITIONMONOIDS 3
In order to parameterize the isoclasses of nilpotent representations, we need some notation. By a partition we mean a sequence p =(p1,p2,... ) of non-negative integers such that p1 > p2 > ··· and pm =0 for m ≫ 0.
The zero partition (0, 0,... ) is simply denoted by (0) and write |p| = i pi. We will identify two partitions which only differ by adding some zeros at the end. For P a partition p = (p1,p2,... ), let p = (p1, p2,... ) denote the conjugate (or dual) of p, i.e., for each s > 1, eps =e|{te| pt > s}|. Let P denote the set of all partitions and Π = Pn the set of n-tuples of partitions (or n-partitions). e (1) (n) (i) (i) (i) Given π = (π ,...,π ) ∈ Π with π = (π1 , π2 ,... ) for 1 6 i 6 n, we define a representation1 in Rep 0△ (i) M(π)= Mk(π)= Si[πt ]. i∈I, t>1 M Applying Krull–Schmidt(–Remak) theorem gives a bijection f : Π −→ M, π 7−→ [M(π)]. (2.0.1) n Note that this bijection is independent of the field k. For a =(a1,... ,an) ∈ N , let a a πa = ((1 1 ),..., (1 n )) ∈ Π, where (1ai )=(1,..., 1) for 1 6 i 6 n. Then
ai | {z } M(πa)= a1S1 ⊕···⊕ anSn =: Sa. Following [1, 17, 3], given two objects M, N in Rep 0△, there exists a unique (up to isomorphism) extension G of M by N such that dim End k△(G) is minimal (or equivalently, when k is algebraically closed, the corresponding orbit OG has a maximal dimension), where k△ denotes the path algebra of △. The extension G is called the generic extension of M by N, denoted by M ∗ N. Thus, if we define [M] ∗ [N] = [M ∗ N], then it is known from [17, 3] that ∗ is associative and (M, ∗) is a monoid with identity [0], the isoclass of zero object in Rep 0△, called the generic extension monoid of △. The submonoid of M generated by [Si], for i ∈ I, is called the composition monoid of △, denoted by Mc = Mc(△). We note that for each i ∈ I and m > 1, m [Si] = [Si] ∗···∗ [Si] = [mSi]. m (1) (n) > An n-partition π = (π ,...,π )|∈ Π is{z called} aperiodic if for each l 1, there is (i) a i ∈ I such that πl = 0. By Π we denote the set of all aperiodic n-partitions. Then, by [3, Th. 4.1], a Mc = {[M(π) | π ∈ Π }.
1The representation M(π) was labeled by π := (π(1),..., π(n)) in [22] and [3]. They are also labeled by multisegments in [11] and [6]. This new labeling is convenient for a combinatorial description. e e e 4 BANGMINGDENG,JIEDUANDALEXANDREMAH
Thus, a representation of the form M(π) for π ∈ Πa is also called aperiodic. For π,λ ∈ Π, define π ∗ λ ∈ Π by M(π ∗ λ) ∼= M(π) ∗ M(λ). In the following we are going to give a combinatorial description of π ∗ λ. For a partition p =(p1,p2,... ) ∈ P and a positive integer m > 1, set L R tm(p)=(p1,...,pm) and tm(p)=(pm+1,pm+2,... ). L R We also set t0 (p) = (0) and t0 (p)= p by convention. Furthermore, given two partitions λ = (λ1,λ2,... ) and µ = (µ1,µ2,... ), we for- mally define the addition of λ and µ as
λ + µ =(λ1 + µ1,λ2 + µ2,... ).
Let λ ∪ µ denote the partition formed by the integers λ1,λ2,...,µ1,µ2,... arranging in decreasing order. For example, if λ = (3, 2, 1, 1) and µ = (5, 3, 2), then λ + µ = (8, 5, 3, 1) and λ ∪ µ = (5, 3, 3, 2, 2, 1, 1).
The generic extension of a simple representation Si by an arbitrary M(π) can be easily computed as follows: (j) (i+1) (i+1) Si ∗ M(π)= Sj[πt ] ⊕ Si[π1 + 1] ⊕ Si+1[πt ]; j∈I\{i+1}, t>1 t>2 (2.0.2) M M see [3, Prop. 3.7]. Thus, we have the following. Lemma 2.1. For i ∈ I and π ∈ Π, if we define i ∗ π =: (λ(1),... ,λ(n)) by setting
(i) (i) (i+1) (i+1) R (i+1) (j) (j) λ = π ∪ (π1 + 1), λ = t1 (π ), and λ = π for j =6 i, i +1, then Si ∗ M(π) ∼= M(i ∗ π). Note that there is a dual version of the above lemma which describes the generic extension M(π) ∗ Si; see [4, Rem. 2.2]. n For a sequence a =(a1,...,an) ∈ N , define a map (1) (n) θa : Π −→ Π, π 7−→ θa(π) := (λ ,... ,λ )
(i) R (i) L (i+1) ai where λ = tai−1 (π ) ∪ tai (π ) + (1 ) for all i ∈ I (and so a0 = an and π(n+1) = π(1)). In particular, if n = 1 (i.e., △ is a loop), then θ (p)= p + (1a) for all a partition p and a ∈ N. Example 2.2. Suppose n = 3. Let a = (3, 1, 2) and π = ((4,, 3, 2, 1), (5, 2, 2), (3, 1, 1, 1, 1)). Then θa(π) = ((6, 3, 3, 2, 1), (4), (5, 4, 1, 1, 1, 1)). Lemma 2.3. For each λ ∈ Π and a ∈ Nn,
πa ∗ λ = θa(λ), i.e., Sa ∗ M(λ) ∼= M(θa(λ)). GENERICEXTENSIONSANDCOMPOSITIONMONOIDS 5
Proof. For each i ∈ I and partition p, write
Mi(p)= Si[pt]. t>1 M (i) 1 Then M(λ) = ⊕i∈I Mi, where Mi = Mi(λ ), and since Ext k△(Si, Mj) = 0 unless j = i + 1, a fact which is already used in (2.0.2), ∼ ∼ Sa ∗ M(λ) = (aiSi) ∗ Mi+1 = (Si ∗···∗ Si) ∗ Mi+1. i∈I i∈I M M ai Thus, applying repeatedly Lemma 2.1 yields | {z }
L (i+1) ai R (i+1) Si ∗···∗ Si ∗Mi+1 = Mi(tai (λ )+(1 )) ⊕ Mi+1(tai (λ )),
ai giving the required| {z formula.} For each π =(π(1),... ,π(n)) ∈ Π and s > 1, define as (1) (n) n π =(πs ,..., πs ) ∈ N . Then, by the construction of M(π), for all s > 1, e e s−1 s ∼ as rad M(π)/rad M(π) = S π , where rad 0M(π) = M(π), rad M(π) denotes the radical (i.e., the intersection of all maximal submodules) of M(π), and rad sM(π) = rad(rad s−1M(π)) for s> 1. as Theorem 2.4. For π,λ ∈ Π, let m > 1 be such that π =0 for s > m. Then
a1 a2 am π ∗ λ = θ π θ π ··· θ π (λ). Proof. We simply write M for M(π). Then
m s−1 s ∼ as 6 6 rad M =0 and rad M/rad M = S π for 1 s m. By [6, Cor. 3.2], M ∼= (M/rad M) ∗ (rad M/rad 2M) ∗···∗ (rad m−2M/rad m−1M) ∗ (rad m−1M)
∼ a1 a2 m−1 am = S π ∗ S π ∗···∗ Saπ ∗ S π . Applying Lemma 2.3 together with an inductive argument gives that
∼ a1 a2 am M(π ∗ λ)= M(π) ∗ M(λ) = (S π ∗ S π ∗···∗ S π ) ∗ M(λ)
∼ a1 a2 m−1 am = (S π ∗ S π ∗···∗ Saπ ) ∗ M(θ π (λ))
∼ a1 a2 am = M(θ π θ π ··· θ π (λ)). Hence,
a1 a2 am π ∗ λ = θ π θ π ··· θ π (λ). 6 BANGMINGDENG,JIEDUANDALEXANDREMAH
Examples 2.5. (1) If n = 1 and λ,µ ∈ P, then e λ ∗ µ = λ + (1µs )= λ + µ. s>1 X This fact has been proved in [13]. (2) Let n = 3 and π = ((3, 2, 1), (2, 2), (3, 1)) and λ = ((4, 3, 2, 2), (5, 2, 1), (3, 1, 1, 1)). Then a1 a2 a3 a4 π = (3, 2, 2), π = (2, 2, 1), π = (1, 0, 1), π = (0, 0, 0). Applying the above theorem gives that
a1 a2 a3 π ∗ λ = θ π θ π θ π (λ) = ((7, 5, 2, 2, 2, 1), (8, 2), (4, 4, 1, 1)). By [17, 3], the generic extension M ∗N of two nilpotent representations of △ can be alternatively characterized as the unique maximal element among all the extensions of M by N with respect to the degeneration order 6dg defined as follows. Given 0 M, N ∈ Rep △, we define M 6dg N if dim M = dim N and 0 dim kHomk△(N,X) > dim kHomk△(M,X), for all X ∈ Rep △. This indeed defines a partial order relation on M. Moreover, if k is algebraically closed, then M 6dg N if and only if OM ⊆ ON , the closure of ON ; see [25, Th. 2] or [3, Lem. 3.2]. We say that N degenerates to M (or M is a degeneration of N). Since the degeneration order 6dg is independent of the ground field k, we define
π 6dg λ ⇐⇒ M(π) 6dg M(λ), where π,λ ∈ Π.
We now give a combinatorial description for the degeneration order 6dg. Lemma 2.6. Let π =(π(1),... ,π(n)) ∈ Π. Then, for each i ∈ I and l > 1, l (i+l−s) dim kHomk△(M(π),Si[l]) = πs , s=1 X where the superscript i + l − s should be considered ase an element in I = Z/nZ.
Proof. We prove the equality by induction on l. If l = 1, then Si[l]= Si and (j) > (j) ∼ k, if j = i and πt 1; Homk△(Sj[πt ],Si) = (0, otherwise. Hence, (i) (i) dim Homk△(M(π),Si)= |{t | πt > 1}| = π1 .
Now let l > 1 and suppose that the equality holds for all Sj[l − 1] with j ∈ I. Let X be the subset of Homk△(M(π),Si[l]) consisting of all non-surjectivee morphisms f : M(π) → Si[l]. Since Si+1[l − 1] is the unique maximal submodule of Si[l], each f ∈ X satisfies Im(f) ⊆ Si+1[l − 1]. Hence, X is a subspace of Homk△(M(π),Si[l]) and ∼ X = Homk△(M(π),Si+1[l − 1]). GENERICEXTENSIONSANDCOMPOSITIONMONOIDS 7
By the induction hypothesis, we have
l−1 (i+l−s) dim kX = dim kHomk△(M(π),Si+1[l − 1]) = πs . s=1 X (j) e For each j ∈ I, write Mj(π)= ⊕t>1Sj[πt ]. Then M(π)= ⊕j∈I Mj(π) and
Homk△(M(π),Si[l]) = Homk△(Mj(π),Si[l]). j∈I M (j) If j =6 i, then any morphism f : Sj[πt ] → Si[l] can not be surjective. This implies that ∼ Homk△(M(π),Si[l])/X = Homk△(Mi(π),Si[l])/Xi, where Xi is the subspace of Homk△(Mi(π),Si[l]) consisting of all non-surjective mor- phisms g : Mi(π) → Si[l], i.e., Im(g) ⊆ Si+1[l − 1]. For each t > 1, there exists a (i) (i) surjective morphism Si[πt ] → Si[l] if and only if πt > l. We obtain that (i) (i) dim kHomk△(Mi(π),Si[l])/Xi = |{t | πt > l}| = πl . Consequently, e dim kHomk△(M(π),Si[l]) = dim kX + dim kHomk△(Mi(π),Si[l])/Xi l−1 l (i+l−s) (i) (i+l−s) = πs + πl = πs . s=1 s=1 X X This finishes the proof. e e e
The above lemma together with the definition gives rise to the following description −1 n of 6dg. Recall the bijection (2.0.1) and let Πd = f (Md) for any d ∈ N .
(1) (n) (1) (n) Proposition 2.7. Let π = (π ,...,π ),λ = (λ ,...,λ ) ∈ Πd. Then π 6dg λ if and only if for all i ∈ I and l > 1,
l l (i+l−s) (i+l−s) πs > λs . s=1 s=1 X X e In the case of n = 1, we have bye the above proposition that for λ =(λ1,λ2,... ),µ = (µ1,µ2,... ) ∈ P,
l l
λ 6dg µ ⇐⇒ |λ| = |µ| and λs > µs, for all l > 1 s=1 s=1 X X l e l e ⇐⇒ |λ| = |µ| and λs 6 µs, for all l > 1; s=1 s=1 X X see [14]. Thus, in this case, the degeneration order 6dg coincides with the dominant order ¢. 8 BANGMINGDENG,JIEDUANDALEXANDREMAH
3. Degenerate composition algebras of △n We now recall from [20, 22] the definition of (generic) Ringel–Hall and composition algebras of △n and make a comparison of the monoid algebra ZM and its subalgebra ZMc with the degenerate Ringel–Hall algebra H0 and the degenerate composition algebra C0 of △n. We also deduce some relations in the degenerate composition algebra C0. 0 Let k be a finite field. Then, for given representations M, N1,...,Nm in Rep k(△n), M let FN1,...,Nm denote the number of the filtrations
M = M0 ⊇ M1 ⊇···⊇ Mm−1 ⊇ Mm =0 ∼ M satisfying Mt−1/Mt = Nt, for all 1 ≤ t ≤ m. By [22] and [7], FN1,...,Nm is a polynomial π Z in |k|, i.e., for π,µ1,...,µm in Π, there is a polynomial ϕµ1,... ,µm (q) ∈ [q] (the polynomial ring over Z in indeterminate q) such that for any finite field k,
ϕπ (|k|)= F Mk(π) . µ1,...,µm Mk(µ1),...,Mk(µm)
The (generic) Ringel–Hall algebra Hq = Hq(△n) of △n is the free Z[q]-module with basis {uπ = u[M(π)] | π ∈ Π} and multiplication given by
π uµuν = ϕµ,ν(q)uπ. π∈Π X
The Z[q]-subalgebra Cq = Cq(△n) of Hq generated by u[mSi], for i ∈ I and m > 1, is called the composition algebra of △n.
It is known from [20, 22] that the ui = u[Si] satisfy the so-called fundamental relations
uiuj = ujui if i, j ∈ I and j =6 i ± 1, 2 2 ui ui+1 − (q + 1)uiui+1ui + qui+1ui =0, (3.0.1) 2 2 uiui+1 − (q + 1)ui+1uiui+1 + qui+1ui =0, i ∈ I for n > 3, and 3 2 2 2 2 3 qu1u2 − (q + q + 1)u1u2u1 +(q + q + 1)u1u2u1 − qu2u1 =0, 3 2 2 2 2 3 qu2u1 − (q + q + 1)u2u1u2 +(q + q + 1)u2u1u2 − qu1u2 =0 (3.0.2) for n = 2.
Remark 3.1. By [22, §8.7, Th. 3], the algebra Cq ⊗Z[q] Q(q) is generated by ui (i ∈ I) with the defining relations (3.0.1) for n > 3 and (3.0.2) for n = 2. Moreover, a −1 twisted form of Cq over the Laurent polynomial ring Z[v, v ] is isomorphic to the integral form of the positive part of quantum affine sln; see [21, 6]. The structure of Hq ⊗Z[q] Q(q) has been studied in [23, 8, 10] via a description of its center. By specializing q to 0, we obtain the degenerate Ringel–Hall and composition alge- bras of △n
H0 = H0(△n)= Hq ⊗Z[q] Z and C0 = C0(△n)= Cq ⊗Z[q] Z, GENERICEXTENSIONSANDCOMPOSITIONMONOIDS 9 where Z is view as a Z[q]-module with the action of q being zero. Since we have in C0, m u[mSi] = ui for all i ∈ I and m > 1, it follows that C0 is generated by ui, i ∈ I. By [4, Cor. 2.5] and [24, Th. 2.8], there is a Z-algebra isomorphism H0 → ZM which induces an isomorphism
ξ : C0 −→ ZMc, ui 7−→ [Si] for i ∈ I. (3.1.1)
A presentation for ZM (i.e., H0) with infinitely many generators has been obtained in [4]. Now specializing the relations in (3.0.1) and (3.0.2) at q = 0 gives the relations in C0(△n): 2 2 uiuj = ujui, ui ui+1 = uiui+1ui, and uiui+1 = ui+1uiui+1 (i, j ∈ I, j =6 i ± 1) (3.1.2) for n > 3, and 2 2 2 2 u1u2u1 = u1u2u1 and u2u1u2 = u2u1u2 (3.1.3) for n = 2. We now derive some further relations in C0(△n). For each t > 1, define qr − 1 [[t]]! = [[1]][[2]] ··· [[t]] with [[r]] = , q − 1 and set [[0]]! = 1 by convention. For 0 6 r 6 t, we set t [[t]]! = ∈ Z[q]. r [[r]]![[t − r]]! Applying [2, Th. 4.1] to the cyclic quiver △n gives the relations in Cq(△n): t − − r (r t)(r t+1) t r m t−r (−1) q 2 u u u = 0 and r i i+1 i r=0 (3.1.4) Xt − − r (r+m t)(r+m t+1) n r m t−r (−1) q 2 u u u =0 r i+1 i i+1 r=0 X hh ii with i ∈ I, m > 1, and t > m +1 for n > 3, and t − − r (r+m t)(r+m t+1) n r m t−r (−1) q 2 u u u = 0 (3.1.5) r i i+1 i r=0 X hh ii with i = 1, 2, m > 1, and t > 2m +1 for n = 2. These relations are the so-called higher order fundamental relations which are a variant of the higher order quantum Serre relations studied in [16, Ch. 7]; see [2, Cor. 4.2]. The specialization at q =0 of (3.1.4) and (3.1.5) gives rise to the following relations in C0(△n): t−1 m t m t−m−1 m m+1 t−m m m ui ui+1ui = uiui+1 and ui+1 ui ui+1 = ui+1 ui ui+1 (3.1.6) with i ∈ I, m > 1, and t > m +1 for n > 3, and t−m−1 m m+1 t−m m m ui ui+1ui = ui ui+1ui (3.1.7) 10 BANGMINGDENG,JIEDUANDALEXANDREMAH with i =1, 2, m > 1, and t > 2m +1 for n = 2. For n > 3, it is easy to check that the relations in (3.1.6) can be deduced from those in (3.1.2). In case n = 2, setting t =2m + 1 in (3.1.7) gives the relations m m m+1 m+1 m m ui ui+1ui = ui ui+1ui (i =1, 2, m > 1) (3.1.8) in C0(△2). Since, for t> 2m + 1, t−m−1 m m+1 n−m m m t−2m−1 m m m+1 m+1 m m ui ui+1ui − ui ui+1ui = ui (ui ui+1ui − ui ui+1ui ), it follows that the relations in (3.1.7) and those in (3.1.8) are equivalent. We will show in the next section that (3.1.8) forms a set of minimal defining rela- tions for C0(△2). It seems that for n > 3, the relations in (3.1.2) (hence, including those in (3.1.6)) are far from enough to provide a presentation for C0(△n). Remark 3.2. By the isomorphism ξ in (3.1.1), the relations given in (3.1.6) and (3.1.7) can be easily deduced from Lemma 2.1 as we have in Mc(△n) t−1 m t m [Si] ∗ [Si+1] ∗ [Si] = [mSi[2] ⊕ (t − m)Si] = [Si] ∗ [Si+1] and t−m−1 m m+1 t−m m m [Si+1] ∗ [Si] ∗ [Si+1] = [mSi[2] ⊕ (t − m)Si+1] = [Si+1] ∗ [Si] ∗ [Si+1] , for n > 3, and t−m−1 m m+1 t−m m m [Si] ∗ [Si+1] ∗ [Si] = [mSi[3] ⊕ (t − 2m)Si] = [Si] ∗ [Si+1] ∗ [Si] , for n = 2. This evidences that working with Mc(△n) instead of C0(△n) is a right strategy.
4. A presentation for the composition monoid of △2
This section focuses on the cyclic quiver △2, which is further depicted as ⇆, and is devoted to giving a presentation for the monoid algebra ZMc of ⇆ and, thus, also for C0(⇆). Throughout this section, I = Z/2Z = {1, 2} is the vertex set of ⇆. Let Ω be the set of all words in alphabet {s1,s2} and let A denote the monoid algebra ZΩ over Z. Hence, the set Ω forms a Z-basis for A. For a word w = si1 ··· sit ∈ Ω, we call t the length of w, denoted by ℓ(w). Since the generic extension monoid Mc of ⇆ is generated by [S1] and [S2], there is a surjective Z-algebra isomorphism
φ : A −→ ZMc, si 7−→ [Si], i ∈ I. By Remark 3.2, for each i ∈ I and m > 1, we have m m m+1 m m m+1 φ(si si+1si ) = [Si] ∗ [Si+1] ∗ [Si] m+1 m m m+1 m m = [Si] ∗ [Si+1] ∗ [Si] = φ(si si+1si ). Thus, all the elements m m m+1 m+1 m m xi,m := si si+1si − si si+1si , for all i ∈ I and m > 1, lie in the kernel Ker φ of φ. Let J be the ideal of A = ZΩ generated by xi,m for i ∈ I and m > 1. Then J ⊆ Ker φ. We are going to show that Ker φ = J . In other words, the xi,m form a set of defining relations for ZMc. Moreover, we will prove that these relations are indeed minimal. GENERIC EXTENSIONSAND COMPOSITIONMONOIDS 11
For notational simplicity, in the following we sometimes write M for the isoclass [M]. Thus, if M1 and M2 are two isomorphic representations, we often write M1 = M2 ∼ instead of M1 = M2. ′ Let w and w be two words in Ω. If there are w0,w1 ∈ Ω, i ∈ I, and m > 1 such that ′ m m m+1 ′ m+1 m m w − w = w0xi,mw1 (i.e.,w = w0si si+1si w1 and w = w0si si+1si w1), we say that w can be transformed to w′, denoted by w w′. Further, we say that two words w and w′ are equivalent, or simply w ∼ w′, if there is a sequence of words ′ w = w0,w1,...,wt = w such that for each 0 6 i 6 t − 1,
wi wi+1 or wi+1 wi. Obviously, if w ∼ w′, then w − w′ ∈J and, thus, φ(w)= φ(w′). l+1 Given a sequence a =(a0, a1 ... ,al) ∈ N and i ∈ I, define a word
a0 a1 al wi,a = si si+1 ··· si+l ∈ Ω, where each subscript i + j should be understood as an element in I = Z/2Z. If all the ai are positive, then wi,a is said to be in tight form. Clearly, each word in Ω can be written in tight form.
a0 a1 al Definition 4.1. A word w = si si+1 ...si+l (written in tight form) is said to be (left) compact if, for each 0 6 t 6 l − 2, either at < at+1 or at+1 > at+2. Equivalently, there is 0 6 m 6 l satisfying
a0 < a1 < ··· < am and am+1 > am+2 > ··· > al. The following four lemmas will be needed in the proof of the main result of the section.
a0 a1 al Lemma 4.2. Let w = si si+1 ··· si+l be a word in tight form. If w is compact, then a0−1 a1−1 al−1 si si+1 ...si+l is again compact. Proof. This is straightforward from the definition. Lemma 4.3. Each word in Ω is equivalent to a compact word.
a0 a1 ap ′ b0 b1 bq Proof. Given two words w = si si+1 ··· si+p and w = sj sj+1 ··· sj+q in tight form, define w ≺ w′ if either
p < q and a0 = b0, a1 = b1,...,ap = bp or there is 0 6 t 6 min{p, q} such that
a0 = b0, a1 = b1,...,at−1 = bt−1, and at < bt. Clearly, ≺ is transitive and w ≺ w′ implies w =6 w′. Moreover, if w w′ then w ≺ w′. a0 a1 ap Now let w = si si+1 ··· si+p be an arbitrary word in Ω written in tight form. If w is compact, there is nothing to prove. Suppose that w is not compact. Then, by definition, there is 0 6 t 6 p − 2 such that
at > at+1 and at+1 < at+2. 12 BANGMINGDENG,JIEDUANDALEXANDREMAH
Write m = at+1. Then
a0 ai+t−1 at−m m m m+1 at+2−m−1 at+3 ap ′ m m m+1 ′′ w = si ··· si+t−1 st st st+1st+2 st+2 st+3 ··· sp = w st st+1st+2 w ,
′ a0 ai+t−1 at−m ′′ at+2−m−1 at+3 ap where w = si ··· si+t−1 st and w = st+2 st+3 ··· sp . Set ′ m+1 m m ′′ w1 = w st st+1st+2w .
Then w w1, since t = t + 2 in I, and so w ≺ w1. If w1 is compact, we are done. Otherwise, continuing the above process, we obtain a chain
w := w0 ≺ w1 ≺ w2 ≺··· such that wd wd+1 for all d > 0. Since the words w1,w2,... all have the same length ℓ(w) and are pairwise distinct, the above chain must stop in finite steps, say
w = w0 ≺ w1 ≺···≺ wh.
This implies that wh is compact and w ∼ wh, as required.
a0 a1 al Lemma 4.4. Let w = si si+1 ··· si+l be a word in tight form. Then
a0−1 a1−1 al−1 φ(w)= Si[l + 1] ⊕ φ(si si+1 ...si+l ), and all indecomposable summands of φ(w) have dimension 6 l +1. Proof. We proceed by induction on l. The assertion is clearly true by (2.0.2) for l =0, 1. Now let l> 1 and assume that the assertion holds for l − 1. a0 a1 al ′ a1 al ′ Let w = si si+1 ··· si+l be a word in tight form. Write w = si+1 ··· si+l. Then w is again in tight form. Hence, by the induction hypothesis,
′ a1−1 a2−1 al−1 φ(w )= Si+1[l] ⊕ φ(si+1 si+2 ...si+l ) and all the indecomposable summands of φ(w′) have dimension 6 l. By the associativity of generic extensions and the fact that a0Si = ((a0 − 1)Si) ∗ Si, ′ ′ φ(w)=(a0Si) ∗ φ(w )=((a0 − 1)Si) ∗ (Si ∗ φ(w )).
′ Since Si+1[l] is the indecomposable summand of φ(w ) of maximal dimension with top Si+1, we have by (2.0.2) that
′ a1−1 a2−1 al−1 Si ∗ φ(w )= Si[l + 1] ⊕ φ(si+1 si+2 ...si+l ). Consequently, we obtain that
a1−1 a2−1 al−1 φ(w)=((a0 − 1)Si) ∗ Si[l + 1] ⊕ φ(si+1 si+2 ...si+l ) a1−1 a2−1 al−1 = Si[l + 1] ⊕ ((a 0 − 1)Si) ∗ φ(si+1 si+2 ...si+l ) a0−1 a1−1 al−1 = Si[l + 1] ⊕ φ(si si+1 ...si+l ) 1 since Ext k△2 (Si,Si[l + 1]) = 0. This finishes the proof.
−1 Lemma 4.5. For each M ∈Mc, φ (M) contains exactly one compact word. GENERIC EXTENSIONSAND COMPOSITIONMONOIDS 13
Proof. We proceed by induction on the dimension d of M. If d = 0 or 1, the statement is obviously true. Now let d > 1 and suppose that the statement holds for modules in Mc of dimension b b b ′ +1 0+1 1+1 l ′ w = si si+1 ··· si+l′ si+l +1 ··· si+l. If j = i + 1, we put b ′ +1 b0+1 l ′ w = sisi+1 ··· si+l′+1si+l +2 ··· si+l. It is easy to see that in both cases, w is compact and ′ φ(w)= Si[l + 1] ⊕ φ(w ). Therefore, φ−1(M) contains a compact word. It remains to prove the uniqueness. Let w,w′ be two compact words in φ−1(M) a0 a1 al ′ b0 b1 bm with tight form w = si si+1 ··· si+l and w = sj sj+1 ··· sj+m. Then, by Lemma 4.4, ′ ′ φ(w)= Si[l + 1] ⊕ φ(w1) and φ(w )= Sj[m + 1] ⊕ φ(w1), a0−1 a1−1 al−1 ′ b0−1 b1−1 bm−1 where w1 = si si+1 ··· si+l and w1 = sj sj+1 ··· sj+m . Since φ(w) = M = φ(w′), we must have l +1= m + 1, i.e., l = m. The fact that M is aperiodic forces ′ that i = j. Then applying Krull–Schimdt Theorem gives φ(w1)= φ(w1). By Lemma ′ ′ 4.2, both w1 and w1 are compact. Thus, w1 = w1 by the induction hypothesis since ′ ′ φ(w1)= φ(w1) has dimension d − l − 1 ′ Given a word w = si1 si2 ··· sit in Ω, a word w of the form sic sic+1 ··· sid is called a subword (or segment) of w, where 1 6 c 6 d 6 t. If, moreover, w′ =6 w, then w′ is said to be a proper subword. We are now ready to prove the main theorem of the paper. Theorem 4.6. The generic extension monoid algebra ZMc(⇆) is generated by [S1] and [S2] with relations m m m+1 m+1 m m [S1] ∗ [S2] ∗ [S1] = [S1] ∗ [S2] ∗ [S1] and m m m+1 m+1 m m (4.6.1) [S2] ∗ [S1] ∗ [S2] = [S2] ∗ [S1] ∗ [S2] , for all m > 1. Moreover, these relations form a set of minimal defining relations. 14 BANGMINGDENG,JIEDUANDALEXANDREMAH Proof. Since J = hxi,m | i = 1, 2, m > 1i ⊆ Ker φ, the surjective homomorphism φ : A = ZΩ → ZMc induces a surjective homomorphism ψ : A/J −→ ZMc, w¯ := w + J 7−→ φ(w). On the one hand, by Lemma 4.3, for every word w ∈ Ω, there is a compact word w′ such that w ∼ w′. Then w − w′ ∈ J , i.e.,w ¯ =w ¯′. Thus, if Ωc denotes the set of all compact words, then B := {w¯ | w ∈ Ωc} forms a spanning set for A/J . On the other hand, by Lemma 4.5, each fiber of φ contains exactly one compact word. This c shows that {ψ(¯w) = φ(w) | w ∈ Ω } is a Z-basis for ZMc. Consequently, B is a Z-linearly independent set and, hence, is a Z-basis for A/J . We conclude that ψ is an isomorphism. Since ψ factors through the natural homomorphism A/Ker φ → ZMc induced by φ, it follows that A/J ∼= A/Ker φ ∼= ZMc. Therefore, Ker φ = J is generated by xi,m for i ∈ I and m > 1, that is, the relations in (4.6.1) are defining relations for ZMc. We now show that the relations in (4.6.1) are minimal, or equivalently, the ideal J can not be generated by any proper subset of {xi,m | i ∈ I, m > 1}. It suffices to show that for each fixed i0 ∈ I and m0 > 1, xi0,m0 does not lie in the ideal Ji0,m0 of A generated by all the elements xi,m with (i, m) =(6 i0, m0). For each pair (i, m), write + m m m+1 − m+1 m m xi,m = si si+1si and xi,m = si si+1si . + − Then xi,m = xi,m − xi,m. It is easily seen that any proper subword (or segment) of ± ± xi,m can not have the form xj,n for some j ∈ I and n > 1. Suppose xi0,m0 ∈ Ji0,m0 . This would imply that in ZΩ, t + − ′ ′′ xi0,m0 = xi0,m0 − xi0,m0 = alwlxil,ml wl , l=1 ′ ′′ X where al ∈ Z, il ∈ I, ml > 1, wl,wl ∈ Ω, and (il, ml) =(6 i0, m0) for 1 6 l 6 t. Hence, ± ± 6 6 xil,ml must be a proper subword of xi0,m0 for some 1 l t. This is a contradiction. Thus, xi0,m0 6∈ Ji0,m0 , as required. The above theorem together with the isomorphism C0 → ZMc,ui 7→ [Si] gives the following result. Corollary 4.7. The degenerate composition algebra C0(⇆) is generated by u1 and u2 with a set of minimal defining relations m m m+1 m+1 m m m m m+1 m+1 m m u1 u2 u1 = u1 u2 u1 and u2 u1 u2 = u2 u1 u2 , for all m > 1. Remark 4.8. In [9], the degenerate composition algebra of the Kronecker quiver ⇉: i b b j has been studied. Namely, let Uq(⇉) be the Q(q)-algebra generated by i and j subject to the relations i3j − (q2 + q + 1)i2ji + q(q2 + q + 1)iji2 − q3ji3 = 0 and (4.8.1) ij3 − (q2 + q + 1)jij2 + q(q2 + q + 1)j2ij − q3j3i =0. GENERIC EXTENSIONSAND COMPOSITIONMONOIDS 15 The generic composition algebra Cq(⇉) of ⇉ is defined to be the Q[q]-subalgebra of Uq(⇉) generated by the divided powers in jn i(n) := and j(n) := , for n > 1. [[n]]! [[n]]! It is shown in [9, Prop. 7] that C0(⇉)= Cq(⇉) ⊗Q[q] Q is generated by i and j with minimal defining relations: (i) imjm+1im+1jm+2 = i2m+1j2m+3 for m > 0; (ii) in+2jn+1in+1jn = i2n+3j2n+1 for n > 0; (iii) isjsitjt = itjtisjs for s, t > 1 and s =6 t. By comparing (i)–(iii) with Corollary 4.7, it seems that there is no canonical isomor- phism from C0(⇉) to C0(⇆) ⊗Z Q. However, both the twisted versions of Cq(⇉) and Cq(⇆) are isomorphic, giving a realization of the positive part of quantum affine sl2. 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