presenting degenerate ringel-hall algebras of type b

12
SCIENCE CHINA Mathematics . ARTICLES . May 2012 Vol. 55 No. 5: 949–960 doi: 10.1007/s11425-011-4317-3 c Science China Press and Springer-Verlag Berlin Heidelberg 2011 math.scichina.com www.springerlink.com Presenting degenerate Ringel-Hall algebras of type B FAN LiJuan 1 & ZHAO ZhongHua 2, 1 Department of Mathematics, Beijing Normal University, Beijing 100875, China; 2 Department of Mathematics and Computer Science, School of Science, Beijing University of Chemical Technology, Beijing 100029, China Email: [email protected], [email protected] Received April 27, 2010; accepted June 28, 2011; published online October 30, 2011 Abstract We give a presentation for the degenerate Ringel-Hall algebras of type B by studying the corre- sponding generic extension monoid algebras. As an application, it is shown that the degenerate Ringel-Hall algebras of type B admit multiplicative bases. Keywords degenerate Ringel-Hall algebras, generic extension monoid algebras, Auslander-Reiten quivers, Frobenius morphism MSC(2010) 17B37, 16G20 Citation: Fan L J, Zhao Z H. Presenting degenerate Ringel-Hall algebras of type B. Sci China Math, 2012, 55(5): 949–960, doi: 10.1007/s11425-011-4317-3 1 Introduction In [12], Ringel gave a nice realization of the ±-part of quantum enveloping algebras of finite type by the so-called Hall algebra approach. By the existence of Hall polynomials of a Dynkin quiver Q with automorphism σ, Ringel introduced the generic Hall algebras H q (Q, σ), now known as generic Ringel- Hall algebras. By specializing q to 0, one can obtain a well-defined algebra H 0 (Q, σ), which is called the degenerate Ringel–Hall algebra. On the other hand, when doing calculations in Hall algebras, one often has the extension of two other representations. In case of a Dynkin quiver Q, the generic extension of any two representations of Q always exists (see [1]). In [9], Reineke first showed that the multiplication by taking generic extensions is associative, which leads to a monoid structure on the set of isoclasses (i.e., isomorphims classes) of reprentations. Furthermore, Reineke also showed that the monoid ring of generic extensions can be realized as the degenerate Ringel-Hall algebras. The structure of the degenerate Ringel-Hall algebras of Dynkin quivers has been studied in [5, 9–11, 16]. In this paper, we will investigate the structure of degenerate Ringel-Hall algebra H 0 (Q, σ) of a Dynkin quiver Q of type A with the automorphism σ such that the corresponding valued quiver is of type B (see Figure 1 below). Our approach is to use the generic extension of representations of quiver Q studied in [1] and to use the theory developed in [2] and further in [4] and its application to generic extensions. Using the Frobenius morphism F for a Dynkin quiver Q with automorphism σ (see [2]), it is known that the generic extension of any two F -stable representations always exists. Thus, taking the generic extension also defines a monoid structure on the set of isoclasses of F -stable representation of Q, denoted by M Q,σ . Corresponding author

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Page 1: Presenting degenerate Ringel-Hall algebras of type B

SCIENCE CHINAMathematics

. ARTICLES . May 2012 Vol. 55 No. 5: 949–960

doi: 10.1007/s11425-011-4317-3

c© Science China Press and Springer-Verlag Berlin Heidelberg 2011 math.scichina.com www.springerlink.com

Presenting degenerate Ringel-Hall algebras of type B

FAN LiJuan1 & ZHAO ZhongHua2,∗

1Department of Mathematics, Beijing Normal University, Beijing 100875, China;2Department of Mathematics and Computer Science, School of Science,

Beijing University of Chemical Technology, Beijing 100029, China

Email: [email protected], [email protected]

Received April 27, 2010; accepted June 28, 2011; published online October 30, 2011

Abstract We give a presentation for the degenerate Ringel-Hall algebras of type B by studying the corre-

sponding generic extension monoid algebras. As an application, it is shown that the degenerate Ringel-Hall

algebras of type B admit multiplicative bases.

Keywords degenerate Ringel-Hall algebras, generic extension monoid algebras, Auslander-Reiten quivers,

Frobenius morphism

MSC(2010) 17B37, 16G20

Citation: Fan L J, Zhao Z H. Presenting degenerate Ringel-Hall algebras of type B. Sci China Math, 2012, 55(5):

949–960, doi: 10.1007/s11425-011-4317-3

1 Introduction

In [12], Ringel gave a nice realization of the ±-part of quantum enveloping algebras of finite type by

the so-called Hall algebra approach. By the existence of Hall polynomials of a Dynkin quiver Q with

automorphism σ, Ringel introduced the generic Hall algebras Hq(Q, σ), now known as generic Ringel-

Hall algebras. By specializing q to 0, one can obtain a well-defined algebra H0(Q, σ), which is called the

degenerate Ringel–Hall algebra. On the other hand, when doing calculations in Hall algebras, one often

has the extension of two other representations. In case of a Dynkin quiver Q, the generic extension of

any two representations of Q always exists (see [1]). In [9], Reineke first showed that the multiplication

by taking generic extensions is associative, which leads to a monoid structure on the set of isoclasses

(i.e., isomorphims classes) of reprentations. Furthermore, Reineke also showed that the monoid ring of

generic extensions can be realized as the degenerate Ringel-Hall algebras. The structure of the degenerate

Ringel-Hall algebras of Dynkin quivers has been studied in [5, 9–11,16].

In this paper, we will investigate the structure of degenerate Ringel-Hall algebra H0(Q, σ) of a Dynkin

quiver Q of type A with the automorphism σ such that the corresponding valued quiver is of type B (see

Figure 1 below). Our approach is to use the generic extension of representations of quiver Q studied in [1]

and to use the theory developed in [2] and further in [4] and its application to generic extensions. Using

the Frobenius morphism F for a Dynkin quiver Q with automorphism σ (see [2]), it is known that the

generic extension of any two F -stable representations always exists. Thus, taking the generic extension

also defines a monoid structure on the set of isoclasses of F -stable representation of Q, denoted by MQ,σ.

∗Corresponding author

Page 2: Presenting degenerate Ringel-Hall algebras of type B

950 Fan L J et al. Sci China Math May 2012 Vol. 55 No. 5

For the purpose of this paper, we first give a presentation of the monoid algebra ZMQ,σ for the (Q, σ)

mentioned above. Then following the treatment as in [3], we show that ZMQ,σ is in fact isomorphic to

H0(Q, σ). Thus, we finally obtain a presentation of degenerate Ringel-Hall algebras of type B. Although

Reineke [10] has also given a presentation of this type, there is a gap in his proof. (For details, see

Remark 4.3).

2 The degenerate Ringel-Hall algebra H0(Q,σ)

Let (Q, σ) be a quiver Q with automorphism σ. The associated valued quiver Γ = Γ(Q, σ) is defined as

follows. Its vertex set Γ0 and arrow set Γ1 are simply the sets of σ-orbits in Q0 and Q1, respectively.

(For ρ ∈ Γ1, its tail resp. head is the σ-orbit of tails resp. heads of arrows in ρ.) The valuation of Γ is

given by

di = |{vertices in σ-orbit i}|, for i ∈ Γ0,

mρ = |{arrows in σ-orbit ρ}|, for ρ ∈ Γ1.

Let Fq be the finite field of q elements and K = Fq be the algebraic closure of Fq.

Definition 2.1 (See [2, 4]). Let M be a vector-space over K. An Fq-linear isomorphism F : M → M

is called a Frobenius map if it satisfies

(a) F (λm) = λqF (m) for all m ∈ M and λ ∈ K,

(b) for any m ∈ M , Fn(m) = m for some n > 0.

Let C be a K-algebra with identity 1. We do not assume generally that C is finite-dimensional. A

map FC : C → C is called a Frobenius morphism on C if it is a Frobenius map on the K-space C, and it

is also an Fq-algebra isomorphism sending 1 to 1.

Let A := KQ be the path algebra of Q over K. Following [2], σ induces a Frobenius morphism

F = FQ,σ = FQ,σ;q : A → A,∑

s

xsps �→∑

s

xqsσ(ps),

where∑

s xsps is a K-linear combination of paths ps and σ(ps) = σ(ρt) · · ·σ(ρ1) if ps = ρt · · · ρ1 for

arrows ρ1, . . . , ρt in Q1. Then the fixed point algebra

A(q) = AF = {a ∈ A | F (a) = a}

is an Fq-algebra associated with (Q, σ).

Definition 2.2 (See [2,4]). Let (Q, σ) be a quiver with automorphism σ. A representation V = (Vi, φρ)

of Q is called F -stable (or equivalently, a F -stable A-module) if there is a Frobenius map FV :⊕

i∈Q0Vi →⊕

i∈Q0Vi satisfying FV (Vi) = Vσ(i) for all i ∈ Q0 and FV φρ = φσ(ρ)FV for each arrow ρ ∈ Q1.

Lemma 2.3 (See [2, 4]). There is a one-to-one correspondence between isoclasses of indecomposable

A(q)-modules and isoclasses of indecomposable F -stable A-modules.

For an F -stable representation V = (Vi, φi), let

dimV =∑

i∈Γ0

(dim Vi), i ∈ NΓ0 and dimV =∑

i∈Γ0

dimVi

denote dimension vector and the dimension of V , respectively.

An F -stable representation is called indecomposable if it is nonzero and not isomorphic to a direct sum

of two non-zero F -stable representations.

From now on, we assume that (Q, σ) is a Dynkin quiver Q with automorphism σ. By regarding A(q) as

the tensor algebra as certain species (see [2, Proposition 6.3]), Dlab and Ringel [6] have shown that there

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Fan L J et al. Sci China Math May 2012 Vol. 55 No. 5 951

is a bijection from the isoclasses of indecomposable A(q)-modules to the set Φ+ = Φ+(Q, σ) of positive

roots in the root system associated with the valued quiver Γ = Γ(Q, σ). For each α ∈ Φ+, let Mq(α)

denote the corresponding indecomposable A(q)-module, thus dimMq(α) = α. By the Krull-Schmidt

theorem, every A(q)-module M is isomorphic to

Mq(λ) :=⊕

α∈Φ+

λ(α)Mq(α),

for some function λ : Φ+ → N. Thus, the isoclasses of A(q)-modules are indexed by the set

P = P(Q, σ) := {λ : Φ+ → N} = NΦ+

,

which is independent of q. By Lemma 2.3, the isoclasses of F -stable KQ-modules are also indexed by P.

Clearly, for each i ∈ Γ0, there is a complete simple A(q)-module Si corresponding to i.

For given A(q)-modules M,N1, . . . , Nm, the number of the filtrations

M = M0 ⊇ M1 ⊇ · · · ⊇ Mm−1 ⊇ Mm = 0

satisfying Mt−1/Mt∼= Nt, for all 1 � t � m, is finite. Denote this number by FM

N1,...,Nm. By [13],

FMN1,...,Nm

is a polynomial in q when q varies. More precisely, for λ, μ, ν ∈ P, there exists a polynomial

ϕλμ,ν(q) ∈ Z[q] (the polynomial ring over Z in one indeterminate q) such that ϕλ

μ,ν(q) = FMq(λ)

Mq(μ),Mq(ν)

holds for each prime power q �= 1.

The generic Ringel–Hall algebra H = Hq(Q, σ) is the free module over Z[q] with basis {uλ|λ ∈ P} and

multiplication defined by

uμuν =∑

λ∈P

ϕλμ,ν(q)uλ.

It is an N|Γ0|-graded algebra

H =⊕

e∈N|Γ0|

He,

where He is spanned by all μα, α ∈ Pe = {β ∈ P | dimMq(β) = e}.For each λ ∈ P, set

Mq(λ)K := Mq(λ) ⊗Fq K,

which is the F -stable KQ-module corresponding to λ.

Let W be the set of word in Γ0, i.e., W consists of the sequences w = i1i2 · · · im. Let ℘(w) be the

element in P defined by

[Mq(℘(w))K] = [Si1 ] ∗ · · · ∗ [Sim ].

Thus, there is a map

℘ : W −→ P, w �−→ ℘(w),

which is independent of the field K and ℘ is surjective, therefore, it induces a partitionW =⋃

λ∈P ℘−1(λ).

For each λ ∈ P, fix a word wλ ∈ ℘−1(λ). We call the set {wλ | λ ∈ P} a section of W over P. Such a

section is called distinguished if all wλ are distinguished words which are defined below.

For w = i1i2 · · · im ∈ W and λ ∈ P, let ϕλw(q) denote the Hall polynomial ϕλ

μ1,...,μm(q) defined above,

where μr = αir , 1 � r � m. For each word w, we can uniquely express w in the tight form w = jc11 · · · jcnn ,

where c1, . . . , cn are positive integers and jr �= jr+1, for 1 � r < n. Define uw = ui1ui2 · · ·uim and

u(w) = uc1j1uc2j2 · · ·ucnjn . Since u(m)i = um

i /[[m]]!, where [[m]]! = [[1]][[2]] · · · [[m]] with [[i]] = qi−1q−1 for all

m � 1, we have

u(w) =1

∏tr=1[[cr]]

!uw.

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952 Fan L J et al. Sci China Math May 2012 Vol. 55 No. 5

We denote by γλw(q) the Hall polynomial ϕλ

μ1,...,μn(q), where μr ∈ P with Mq(μr) ∼= crSjr . The word w

is called distinguished if γ℘(w)w (q) = 1. By [4], there always exist distinguished words in each fibre ℘−1(λ)

of ℘.

Now, by specializing q to 0, we obtain the Z-algebra H0(Q, σ), called the degenerate Ringel–Hall algebra

associated with Γ = Γ(Q, σ). In other words,

H0(Q, σ) = Hq(Q, σ)⊗Z[q] Z,

where Z is viewed as a Z[q]-module with the action of q being zero. By abuse of the notation, we also

write uλ = uλ ⊗ 1. Thus, the set {uλ | λ ∈ P} is a Z-basis of H0(Q, σ). Denote by ui = u[Si] ⊗ 1 in

H0(Q, σ) for i ∈ Γ0.

Lemma 2.4. As a Z-algebra, H0(Q, σ) is generated by ui, i ∈ Γ0.

Proof. By [4, Theorem 11.9], for each λ ∈ P, we fix a distinguished word wλ = i1i2 · · · im ∈ ℘−1(λ)

with tight form wλ = jc11 · · · jcnn . Then, by [4, Theorem 11.15], the distinguished section {u(wλ) | λ ∈ P}forms a basis for Hq(Q, σ). Since the value [[m]]0 of [[m]] at q = 0 is 1 for all m � 0, we have u(wλ) = uwλ

in H0(Q, σ). It follows that {uwλ}λ∈P is a Z-basis for H0. Hence, H0(Q, σ) is generated by ui.

The present paper mainly deals with the following quiver Q (of type A2n−1)

1

2

2′

3

3′

(n−1)

(n−1)′

n

n′

Q:

Figure 1

with the automorphism σ defined by σ(1) = 1, σ(i) = i′, σ(i′) = i for i = 2, · · · , n. Then the associated

valued quiver with the valuation ε1 = 1, ε2 = · · · = εn = 2 has the form

� �

1 2

(1,2) (1,1)

n

(1,1)

Γ(Q, σ) :

Figure 2

which is of type Bn.

For simplicity, write I = {1, 2, . . . , n} = Γ0(Bn), H0(Bn) = H0(Q, σ). It is easily observed that the

following relations hold in H0(Bn):

(R1) u2u1u2 = u1u22;

(R2) u21u2u1 = u3

1u2;

(R3) u1u2u21u2 = u3

1u22;

(R4) u2iui+1 = uiui+1ui, uiu

2i+1 = ui+1uiui+1, if 2 � i � n− 1;

(R5) uiuj = ujui, if |i− j| � 2.

The main result of this paper is the following theorem.

Theorem 2.5. The degenerate Ringel–Hall algebra H0(Bn) is generated by ui, i ∈ I, with defining

relations (R1)–(R5).

The proof of this theorem is reduced to proving a similar presentation for the corresponding generic

extension monoid algebra which will be discussed in the next section.

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Fan L J et al. Sci China Math May 2012 Vol. 55 No. 5 953

3 Relation with generic extension monoids and multiplicative bases

Throughout this section, we always assume that (Q, σ) is a Dynkin quiver Q with automorphism σ and

keep the notations introduced in Section 2. For KQ-modules M and N , the generic extension M ∗ N

of M by N was defined in [1] as the unique (up to isomorphism) element in Ext1KQ(M,N) having

endomorphism algebra of minimal dimension. As shown in [4, Proposition 1.30(2)], the star operation ∗defines the structure of a monoid on the set MQ = MQ,K of isoclasses of KQ-modules.

Proposition 3.1 (See [4, Proposition 11.1]). If M and N are F -stable KQ-modules, then M ∗ N is

also F -stable.

By this proposition, the set of isoclasses [M ] of F -stable KQ -modules, together with the operation

[M ] ∗ [N ] = [M ∗N ], defines a submonoid MQ,σ of MQ with the unit element [0].

Since all the indecomposable A(q)-modules are indexed by the set P = P(Q, σ) := {λ | Φ+ → N} =

NΦ+

, we give an enumeration on Φ+ defined by β1, β2, . . . , βN such that, for all prime powers q,

HomA(q)(Mq(βs),Mq(βt)) �= 0 =⇒ s � t.

Moreover, in this case, Ext1A(q)(Mq(βs),Mq(βt)) �= 0 implies s > t. Thus, we give an enumeration on

indecomposable A(q)-modules and set Mq(β1) ≺ Mq(β2) ≺ · · · ≺ Mq(βN ). By definition of the generic

extension, if Ext1A(q)(M,N) = 0, then M ∗ N ∼= M ⊕ N . Consequently, we have the following known

result:

Lemma 3.2 (See [9, Proposition 3.3]). Each element [Mq(λ)K] in MQ,σ with λ ∈ P can be written as

[Mq(λ)K] = [Mq(β1)K]∗λβ1 ∗ · · · ∗ [Mq(βN )K]∗λβN .

Moreover, these elements form a Z-basis of ZMQ,σ.

Proposition 3.3 (See [9, Lemma 3.3]). The monoid algebra ZMQ,σ of F -stable KQ-modules is gen-

erated by [Si], i ∈ Γ0. Moreover, for i, j ∈ Γ0, the following holds in ZMQ,σ:

[Si]∗p ∗ [Sj ]

∗q ∗ [Si]∗r ∗ [Sj ]

∗s = [Si]∗(p+r) ∗ [Sj ]

∗(q+s),

where the quadruples (p, q, r, s) are taken from the set Ei,j defined as follows:

(1) In the case aij = 0 = aji,

Eij = {(0, 1, 1, 0)};

(2) In the case aij = −1 = aji,

Eij = {(0, 1, 1, 1), (1, 1, 1, 0)};

(3) In the case aij = −1, aji = −2 and there is an arrow i → j,

Eij = {(0, 1, 1, 2), (1, 2, 1, 1), (1, 1, 1, 0)};

(4) In the case aij = −2, aji = −1 and there is an arrow i → j,

Eij = {(0, 1, 1, 1), (1, 1, 2, 1), (2, 1, 1, 0)};

(5) In the case aij = −1, aji = −3 and there is an arrow i → j,

Eij = {(0, 1, 1, 3), (1, 3, 1, 2), (1, 2, 1, 2), (1, 2, 2, 3), (2, 3, 1, 1), (1, 1, 1, 0)};

(6) In the case aij = −3, aji = −1 and there is an arrow i → j,

Eij = {(0, 1, 1, 1), (1, 1, 3, 2), (3, 2, 2, 1), (2, 1, 2, 1), (2, 3, 1, 1), (1, 1, 1, 0)}.

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954 Fan L J et al. Sci China Math May 2012 Vol. 55 No. 5

Note that all the aij in the above proposition are the entries in the Cartan matrix corresponding to

(Q, σ).

For a dimension vector d =∑

i∈Γ0dii ∈ NΓ0, we consider the affine space

Rd =∏

α:i→j

HomK(Kdi ,Kdj ).

Then the group Gd :=∏

i∈Γ0GLdi(K) acts on Rd by conjugation, i.e., by

(gi) · (xρ)ρ = (gjxρg−1i )ρ:i→j .

The orbits of Gd correspond bijectively to the isoclassess of representations of Γ of dimension vector

d. We denote by OM the orbit corresponding to the isoclass [M ]. Since there are only finitely many

Gd-orbits in Rd, there exists a dense one, whose corresponding representation is denoted by Ed.

Lemma 3.4 (See [10, Lemma 4.9]). Let i1, . . . , in be an enumeration of Γ0 such that k < l if there is

an arrow from ik to il. Then for all d =∑n

k=1 dkαik ∈ Φ+, we have in MQ,σ

[Ed] = [Si1 ]∗d1 ∗ · · · ∗ [Sin ]

∗dn .

We now return to the quiver Q with automorphism σ given in Figure 1 in Section 2. For simplicity,

write M = MQ,σ. Then by Proposition 3.3, the following relations hold in ZM:

(G1) [S2] ∗ [S1] ∗ [S2] = [S1] ∗ [S2]∗2;

(G2) [S1]∗2 ∗ [S2] ∗ [S1] = [S1]

∗3 ∗ [S2];

(G3) [S1] ∗ [S2] ∗ [S1]∗2 ∗ [S2] = [S1]

∗3 ∗ [S2]∗2;

(G4) [Si]∗2 ∗ [Si+1] = [Si] ∗ [Si+1] ∗ [Si], [Si] ∗ [Si+1]

∗2 = [Si+1] ∗ [Si] ∗ [Si+1], if 2 � i � n− 1;

(G5) [Si] ∗ [Sj ] = [Sj ] ∗ [Si], if |i− j| � 2.

Indeed, we have the following result whose proof will be given in Section 4.

Theorem 3.5. The monoid algebra ZM has a presentation with generators [Si], i ∈ I and relations

(G1)–(G5).

As the degenerate Ringel-Hall algebra, there is a natural grading on ZM in terms of dimension vectors:

ZM =⊕

e∈Nn

ZMe,

where ZMe is spanned by all [Mq(α)K], α ∈ Pe.

Corollary 3.6. There is a graded Z-algebra isomorphism

Φ : ZM −→ H0(Bn), [Si] �−→ ui, i ∈ I.

Proof. Since ui in H0(Bn) satisfies relations (R1)–(R5) given in Section 2, there is a surjective Z-algebra

homomorphism

Φ : ZM −→ H0(Bn), [Si] �−→ ui, i ∈ I.

Since {[Mq(λ)K] | λ ∈ P} and {uλ | λ ∈ P} are bases for ZM and H0(Bn), respectively, we obtain that

Φ is an isomorphism.

Remark 3.7. The above corollary together with Theorem 3.5 proves Theorem 2.5.

In the rest of this section, as an application of Corollary 3.6, we study a multiplicative basis of H0(Bn).

Recall that for M,N ∈ KQ-mod, we say that M degenerates to N (or N is a degeneration of M), written

as M �deg N , if dim M = dim N and

dimHomKQ(X,M) � dimHomKQ(X,N),

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Fan L J et al. Sci China Math May 2012 Vol. 55 No. 5 955

for all X ∈ KQ-mod, which is independent of the field K. See [1, 4].

In view of the above discussion, we define a partially ordering on P by putting

μ � λ⇐⇒Mq(μ)K �dg Mq(λ)K, λ, μ ∈ P.

Equivalently, μ � λ if and only if dimMq(λ)K = dimMq(μ)K, and for all X ∈ KQ-mod,

dimK HomKQ(X,Mq(μ)K) � dimK HomKQ(X,Mq(μ)K).

Note that this ordering is independent of the field K.

The application of Corollary 3.6 gives a multiplicative basis of H0(Bn). Precisely speaking, for each

π ∈ P, define θπ = Φ([Mq(π)K]). Then the set {θπ | π ∈ P} is a basis of H0(Bn). It is clear that

θπθπ′ = θππ′ . In other words, {θπ | π ∈ P} is a multiplicative basis of H0(Bn). For each π ∈ P, write

θπ =∑

λ∈P

aπ,λuλ.

We now determine the coefficients aπ,λ.

Proposition 3.8. For π, λ ∈ P, aπ,λ �= 0 implies λ � π, and moreover, aπ,π = 1.

Proof. Similar to the proof of Lemma 2.4, we fix a distinguished word w = i1i2 · · · im ∈ ℘−1(π) with

tight form w = jc11 · · · jcnn . Then

uw = ui1ui2 · · ·uim =

cn∏

r=1

[[cr]]!

λ�℘(w)

γλw(q)uλ.

In H0(Q, σ), we have uw =∑

λ�℘(w) γλw(0)uλ. Thus,

θπ = Φ[Mq(π)K] = Φ([Si1 ] ∗ · · · ∗ [Sim ])

= Φ([Si1 ]) · · ·Φ([Sim ]) = ui1ui2 · · ·uim

= uπ +∑

λ<π

γλw(0)uλ.

Hence, aπ,π = 1, and aπ,λ = 0 unless λ � π. Moreover, aπ,λ = γλw(0) for λ � π.

4 Proof of Theorem 3.5

This section is devoted to proving Theorem 3.5. Let S be the free Z-algebra with generators si, i ∈ I.

Consider the ideal J generated by the following elements, for i, j ∈ I,

(J1) s2s1s2 − s1s22;

(J2) s21s2s1 − s31s2;

(J3) s1s2s21s2 − s31s

22;

(J4) s2i si+1 − sisi+1si, sis2i+1 − si+1sisi+1, if 2 � i � n− 1;

(J5) sisj − sjsi, if |i− j| � 2.

Thus, the surjective monoid algebra homomorphism η : S → ZM, si �→ [Si], i ∈ I, induces a surjective

algebra homomorphism

η : S/J −→ ZM, si + J �−→ [Si], ∀ i ∈ I.

To complete the proof of Theorem 3.5, it suffices to show that η is injective.

Before proving the injectivity of η, we need some notation. Set ei = si + J for each i ∈ I. Giving an

F -stable KQ-module M with dimension vector dim M :=∑n

i=1(dimMi), i ∈ NΓ0, we define a monomial

in S/J by

m(M) = e1dimM1e2

dimM2 · · · endimMn .

Page 8: Presenting degenerate Ringel-Hall algebras of type B

956 Fan L J et al. Sci China Math May 2012 Vol. 55 No. 5

Note that I = {1, 2, . . . , n} are vertices of Γ(Q, σ) as labeled in Figure 2.

Let Q be the quiver with automorphism σ described in Figure 1. It is known that the Auslander-Reiten

quiver of KQ is as follows:

[Pn−1]

[Pn]

[Pn−2]

[P2]

[P1]

[τ−1Pn] [τ−2Pn]

[τ−1Pn−1]

[τ−1P1]

[τ−1P2]

[τ−1P3]

[τ−n+1Pn]

[τ−n+1P1]

[τ−n+1P2]

[τ−n+1P3]

[τ−1P3′ ]

[τ−1P2′ ][P2′ ]

[P(n−1)′ ]

[τ−n+1P2′ ]

[τ−n+1P3′ ]

[P(n−2)′ ]

[τ−1P(n−1)′ ]

[τ−n+1Pn′ ][τ−1Pn′ ][Pn′ ] [τ−2Pn′ ]

Figure 3

where Pi is the indecomposable projective KQ-module corresponding to vertex i and τ is the Auslander-

Reiten translation.

Using the Frobenius morphism F = FQ,σ = FQ,σ;q introduced in Section 2, it is easy to see that P1

is F -stable and all other Pi have F -period 2 with P[1]i = Pi′ . By folding the Auslander–Reiten quiver of

KQ, we obtain the Auslander-Reiten quiver of A(q) = (KQ)F :

[Xn−1]

[Xn]

[Xn−2]

[X2]

[X1]

[τ−1Xn] [τ−2Xn]

[τ−1Xn−1]

[τ−1X1]

[τ−1X2]

[τ−1X3]

[τ−n+1Xn]

[τ−n+1X1]

[τ−n+1X2]

[τ−n+1X3]

.

Figure 4

Here X1 = PF1 , Xi = (Pi ⊕ P ′

i )F

for i � 2, and τ = τAF is the Auslander–Reiten translation of A(q);

see [4] for more details.

Now we give an enumeration of indecomposable A(q)-modules in Figure 4 as follows:

Xn ≺ · · · ≺ X1 ≺ τ−1Xn ≺ · · · ≺ τ−1X1 ≺ · · · ≺ τ−n+1X2 ≺ τ−n+1X1. (4.1)

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Fan L J et al. Sci China Math May 2012 Vol. 55 No. 5 957

Set ⎧⎪⎪⎨

⎪⎪⎩

Xij = τ−(n−j)Xn+i−j for 1 < i � j � n,

Y1s = τ−(n−s)Xn for 1 � s � n,

Zlm = τ−(n−l)Xn−(m−l) for 1 � l < m � n.

Then

dimXij = (0, 0, . . . , 0, 1, . . . , 1︸ ︷︷ ︸i to j

, 0, . . . , 0) and m(Xij) = eiei+1 · · · ej ;

dimY1s = (1, 1, . . . , 1︸ ︷︷ ︸1 to s

, 0, . . . , 0) and m(Y1s) = e1e2 · · · es;

dimZlm = (2, 2, . . . , 2︸ ︷︷ ︸1 to l

, 1, 1, . . . , 1︸ ︷︷ ︸(l+1) to m

, 0, . . . , 0) and m(Zlm) = e12e2

2 · · · el2el+1 · · · em.

Proposition 4.1. For all the indecomposable A(q)-modules Xij , Y1s, Zlm, we have the following equal-

ities in S/J:

m(Xst)m(Xij) =

⎧⎪⎪⎨

⎪⎪⎩

m(Xsj), if s < t+ 1 = i � j;

m(Xsj)m(Xit), if s � i < t+ 1 � j;

m(Xij)m(Xst). otherwise ;

m(Y1s)m(Xij) =

⎧⎪⎪⎨

⎪⎪⎩

m(Xis)m(Y1j), if 1 < i < s+ 1 � j;

m(Y1j), if i = s+ 1 � j;

m(Xij)m(Y1s). otherwise ;

m(Y1s)m(Zlm) = m(Y1l)m(Zsm), if s � l;

m(Y1s)m(Y1t) = m(Zst), if s < t;

m(Zlm)m(Xij) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

m(Xil)m(Zjm), if i < l + 1 � j < m;

m(Xil)m(Zmj), if i < l + 1 � j, j > m;

m(Xil)m(Y1j)m(Y1j), if i < l + 1 � j = m;

m(Zjm), if i = l + 1 � j < m;

m(Zmj), if i = l + 1 � j, j > m;

m(Y1j)m(Y1j), if i = l + 1 � j, j = m;

m(Xim)m(Zlj), if l + 1 < i < m+ 1 � j;

m(Zlj), if l + 1 < i = m+ 1 � j;

m(Xij)m(Zlm), otherwise, i .e., i � j � l < m;

m(Zlm)m(Y1s) = m(Y1m)m(Zls), if m � s;

m(Zst)m(Zlm) =

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

m(Zlm)m(Zst), if s = l < t < m;

m(Y1t)m(Y1t)m(Zsm), if s < l = t < m;

m(Zlt)m(Zsm), if s < l < t < m;

m(Zlm)m(Zst), if s < l < m � t;

m(Ztl)m(Zsm), if s < t < l < m.

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958 Fan L J et al. Sci China Math May 2012 Vol. 55 No. 5

Proof. All the formulas can be proved in a similar way. Below we only prove the cases which are

relatively complicated.

Case 1. m(Y1s)m(Xij) = m(Xis)m((Y1j) with 1 < i < s+ 1 � j:

m(Y1s)m(Xij) = e1e2 · · · e(i−1)eie(i+1) · · · es · eie(i+1) . . . es · · · ej= e1e2 · · · e(i−1)eie(i+1)ei · · · es · e(i+1) · · · es · · · ej (by (J5))

= e1e2 · · · eie(i−1)eie(i+1) · · · es · e(i+1) · · · es · · · ej (by (J4))

= ei · e1e2 . . . e(i−1)eie(i+1) . . . es · e(i+1) . . . es . . . ej (by (J5))

= · · · = ei · · · e(s−2) · e1e2 . . . e(s−2)e(s−1)es · e(s−1)ese(s+1) . . . ej

= ei · · · e(s−2) · e1e2 · · · e(s−1)e(s−2)e(s−1)e2se(s+1) · · · ej (by (J4))

= ei · · · e(s−2)e(s−1) · e1e2 · · · e(s−2)e(s−1)e2se(s+1) · · · ej (by (J5))

= ei · · · e(s−2)e(s−1) · e1e2 · · · e(s−2)ese(s−1)ese(s+1) · · · ej (by (J4))

= ei · · · e(s−2)e(s−1)es · e1e2 · · · e(s−2)e(s−1)ese(s+1) · · · ej (by (J5))

= m(Xis)m(Y1j).

Case 2. m(Zlm)m(Xij) = m(Xil)m(Y1j)m(Y1j) with i < l + 1 � j = m:

m(Zlm)m(Xij)

= e21e22 · · · e2i e2(i+1)e

2(i+2) · · · e2l e(l+1) · · · ej · eie(i+1) · · · el · · · ej

= e21e22 · · · e2i e2(i+1)eie

2(i+2) · · · e2l e(l+1) · · · ej · e(i+1) · · · el · · · ej (by (J5))

= e21e22 · · · e3i e2(i+1)e

2(i+2) · · · e2l e(l+1) · · · ej · e(i+1) · · · el · · · ej (by (J3))

= · · · = e21e22 · · · e3i e3(i+1) · · · e3l e(l+1)e(l+2)e(l+3) · · · ej · e(l+1)e(l+2)e(l+3) · · · ej

= e21e22 · · · e3i e3(i+1) · · · e3l e(l+1)e(l+2)e(l+1)e(l+3) · · · ej · e(l+2)e(l+3) · · · ej (by (J5))

= e21e22 · · · e3i e3(i+1) · · · e3l e2(l+1)e(l+2)e(l+3) · · · ej · e(l+2)e(l+3) · · · ej (by (J5))

= · · · = e21e22 · · · e2(l−1)e

3i e

3(i+1) · · · e3l e2(l+1)e

2(l+2) · · · e2j ,

m(Xil)m(Y1j)m(Y1j)

= eie(i+1) · · · e(l−1)el · e1e2 · · · e(j−2)e(j−1)ej · e1e2 · · · e(j−2)e(j−1)ej

= eie(i+1) · · · e(l−1)el · e1e2 · · · e(j−2)e(j−1) · e1e2 · · · e(j−2)eje(j−1)ej (by (J5))

= eie(i+1) · · · e(l−1)el · e1e2 · · · e(j−2)e(j−1) · e1e2 · · · e(j−2)e(j−1)e2j (by (J4))

= · · · = eie(i+1) · · · e(l−1)el · e21e22 · · · e2(l−2)e2(l−1)e

2l e

2(l+1) · · · e2j

= eie(i+1) · · · e(l−1) · e21e22 · · · e2(l−2)ele2(l−1)e

2l e

2(l+1) · · · e2j (by (J5))

= eie(i+1) · · · e(l−1) · e21e22 · · · e2(l−2)e2(l−1)e

3l e

2(l+1) · · · e2j (by (J4))

for i > 2, then

= · · · = e21e22 · · · e2(i−1)e

3i e

3(i+1) · · · e3(l−1)e

3l e

2(l+1)e

2(l+2) · · · e2j

otherwise i = 2, then

= · · · = e2 · e21e22e33 · · · e3(l−1)e3l e

2(l+1)e

2(l+2) · · · e2j

= e21e32e

33 · · · e3(l−1)e

3l e

2(l+1)e

2(l+2) · · · e2j (by (J1)) .

Case 3. m(Zlm)m(Xij) = m(Zjm) with i = l + 1 � j < m:

m(Zlm)m(Xij) = e21e22 · · · e2l e(l+1)e(l+2)e(l+3) · · · ej · · · em · e(l+1)e(l+2)e(l+3) · · · ej

= e21e22 · · · e2l e(l+1)e(l+2)e(l+1)e(l+3) · · · ej · · · em · e(l+2)e(l+3) · · · ej (by (J5))

= e21e22 · · · e2l e2(l+1)e(l+2)e(l+3) · · · ej · · · em · e(l+2)e(l+3) · · · ej (by (J4))

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Fan L J et al. Sci China Math May 2012 Vol. 55 No. 5 959

= · · · = e21e22 · · · e2l e2(l+1)e

2(l+2) · · · e2j · · · em

= m(Zjm).

Case 4. m(Zlm)m(Xij) = m(Y1j)m(Y1j) with i = l + 1 � j = m:

m(Zlm)m(Xij) = e21e22 · · · e2l e(l+1) · · · e(j−1)ej · e(l+1) · · · e(j−2)e(j−1)ej

= e21e22 · · · e2l e(l+1) · · · e(j−1) · e(l+1) · · · e(j−2)eje(j−1)ej (by (J5))

= e21e22 · · · e2l e(l+1) · · · e(j−1) · e(l+1) · · · e(j−2)e(j−1)e

2j (by (J4))

= · · · = e21e22 · · · e2l e2(l+1) · · · e2(j−2)e

2(j−1)e

2j

= e21e22 · · · e2l e2(l+1) · · · e2(j−2)e(j−1)eje(j−1)ej (by (J4))

= e21e22 · · · e2l e2(l+1) · · · e(j−2)e(j−1)e(j−2)eje(j−1)ej (by (J4))

= e21e22 · · · e2l e2(l+1) · · · e(j−2)e(j−1)eje(j−2)e(j−1)ej (by (J5))

= · · · = e1e2 · · · e(j−2)e(j−1)eje1e2 · · · e(j−2)e(j−1)ej

= m(Y1j)m(Y1j).

Case 5. m(Zst)m(Zlm) = m(Y1t)m(Y1t)m(Zsm) with s < l = t < m:

m(Zst)m(Zlm)

= e21e22 · · · e2se(s+1) · · · e(t−1)et · e21e22 · · · e2se2(s+1) · · · e2(t−2)e

2(t−1)e

2t e(t+1) · · · em

= e21e22 · · · e2se(s+1) · · · e(t−1) · e21e22 · · · e2se2(s+1) · · · e2(t−2)ete

2(t−1)e

2t e(t+1) · · · em (by (J5))

= e21e22 · · · e2se(s+1) · · · e(t−1) · e21e22 · · · e2se2(s+1) · · · e2(t−2)e

2(t−1)e

3t e(t+1) · · · em (by (J4))

= · · · = e21e22 · · · e2s · e21e22 · · · e2(s−2)e

2(s−1)e

2se

3(s+1)e

3(s+2) · · · e3(t−1)e

3t e(t+1) · · · em

= e21e22 · · · e2s−1 · e21e22 · · · e2(s−2)e

2se

2(s−1)e

2se

3(s+1)e

3(s+2) · · · e3(t−1)e

3t e(t+1) · · · em (by (J5))

= e21e22 · · · e2s−1 · e21e22 · · · e2(s−2)e

2(s−1)e

4se

3(s+1)e

3(s+2) · · · e3(t−1)e

3t e(t+1) · · · em (by (J4))

= · · · = e21e22 · e21e22e43e44 · · · e4(s−2)e

4(s−1)e

4se

3(s+1) · · · e3(t−1)e

3t e(t+1) · · · em

= e41e42e

43e

44 · · · e4(s−2)e

4(s−1)e

4se

3(s+1) · · · e3(t−1)e

3t e(t+1) · · · em (by (J1), (J2), (J3)) ,

m(Y1t)m(Y1t)m(Zsm)

= e1e2 · · · es · · · e(t−1)et · e1e2 · · · es · · · e(t−1)et · e21e22 · · · e2se(s+1) · · · e(t−1)ete(t+1) · · · em= e1e2 · · · es · · · e(t−1) · e1e2 · · · es · · · ete(t−1)et · e21e22 · · · e2se(s+1) · · · e(t−1)ete(t+1) · · · em= · · · = e21e

22 · · · e2se2(s+1) · · · e2(t−1)e

2t · e21e22 · · · e2se(s+1) · · · et−2e(t−1)ete(t+1) · · · em

= e21e22 · · · e2se2(s+1) · · · e2(t−1) · e21e22 · · · e2se(s+1) · · · et−2e

2t e(t−1)ete(t+1) · · · em (by (J5))

= e21e22 · · · e2se2(s+1) · · · e2(t−1) · e21e22 · · · e2se(s+1) · · · e(t−2)e(t−1)e

3t e(t+1) · · · em (by (J4))

= · · · = e21e22 · · · e2se2(s+1) · e21e22 · · · e2(s−1)e

2se(s+1)e

3(s+2) · · · e(t−1)e

3te(t+1) · · · em

= e21e22e

21e

23e

22e

24e

23 · · · e2se2(s−1)e

2(s+1)e

2se(s+1)e

3(s+2) · · · e3(t−1)e

3t e(t+1) · · · em (by (J5))

= e21e22e

21e

23e

22e

24e

23 · · · e2se2(s−1)ese

3(s+1)ese

3(s+2) · · · e3(t−1)e

3te(t+1) · · · em (by (J4))

= · · · = e21e22e

21e2e

33e2e

34e3 · · · e3se(s−1)e

3(s+1)ese

3(s+2) · · · e3(t−1)e

3t e(t+1) · · · em

= e41e32e

33e2e

34e3 · · · e3se(s−1)e

3(s+1)ese

3(s+2) · · · e3(t−1)e

3t e(t+1) · · · em (by (J1),(J2),(J3))

= e41e42e

33e

34e3 · · · e3se(s−1)e

3(s+1)ese

3(s+2) · · · e3(t−1)e

3te(t+1) · · · em (by (J4))

= · · · = e41e42e

43e

44 . . . e

4(s−1)e

4se

3(s+1)e

3(s+2) · · · (e3(t−1)e

3t e(t+1) · · · em. �

Let V1, . . . , Vμ be all the non-isomorphic indecomposable A(q)-modules, where μ = |Φ+| = n2. We

assume that they are enumerated by V1 ≺ V2 ≺ · · · ≺ Vμ as given in (4.1), i.e., V1 = Xn, V2 =

Xn−1, . . . , Vμ = τ−n+1X1. Repeatedly applying Proposition 4.1 gives the following result.

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960 Fan L J et al. Sci China Math May 2012 Vol. 55 No. 5

Corollary 4.2. With the above notation, for 1 � i < j � μ, there exist 1 � j1 � j2 � · · · � jm � μ

such that

m(Vj)m(Vi) = m(Vj1)m(Vj2 ) · · ·m(Vjm).

Proof of Theorem 3.5. It remains to show that η : S/J → ZM is injective. Given a monomial w =

ei1 · · · eim , we have

w = ei1 · · · eim = m(Si1) · · ·m(Sim).

Applying Corollary 4.2 repeatedly, we eventually get

w = m(V1)n1 · · ·m(Vμ)

nμ ,

for some n1, . . . , nμ � 0. Hence, all the monomials m(V1)n1 · · ·m(Vμ)

nμ with n1, . . . , nμ � 0 span S/J.

On the other hand, Lemma 3.4 implies that for n1, . . . , nμ � 0,

η(m(V1)

n1 · · ·m(Vμ)nμ

)= [V1]

∗n1 ∗ · · · ∗ [Vμ]∗nμ .

By Lemma 3.2, the elements [V1]∗n1 ∗· · ·∗ [Vμ]

∗nμ with n1, · · · , nμ � 0 form a basis of ZM. Consequently,

η is injective. �Remark 4.3. In [10], Reineke has given a presentation for the generic extension monoid algebras of

all Dynkin quiver Q with automorphism σ in terms of the so-called σ-symmetric representations (which

indeed coincide with F -stable representations studied here). However, the proof of the “straightening

rule” (Proposition 5.4) contains a gap. Theorem 3.5 says that the main result in [10] holds for type B.

It would be interesting to know whether the above approach works for all other types.

Acknowledgements The authors would like to thank the referees for many helpful suggestions and im-

provements, and express their sincere gratitude to Professor Deng BangMing for many valuable and suggestive

discussions.

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