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Rend. Circ. Mat. Palermo, II. Ser
DOI 10.1007/s12215-017-0321-y
Gröbner-Shirshov basis for degenerate Ringel-Hall
algebra of type D4
Rabigul Tuniyaz1 · Abdukadir Obul2
Received: 15 January 2017 / Accepted: 29 August 2017
© Springer-Verlag Italia S.r.l. 2017
Abstract In this paper, we first give a presentation of the degenerate Ringel-Hall algebra
H0 (D4 ) by using the Frobenius morphism and the multiplication formulas of the generic
extension monoid algebra. Then, by using the relations which are computed to give this
presentation, we construct the Gröbner-Shirshov basis for this algebra.
Keywords Presentation · Gröbner-Shirshov basis · Frobenius map · Degenerate Ringel-Hall
algebras · Multiplication formula
Mathematics Subject Classification 16S15 · 17A01 · 16G50 · 20G42
1 Introduction
Through the works of Buchberger [1], Bergman [2] and Shirshov [3], the Gröbner-Shirshov
basis theory has become a powerful tool for the solution of reduction problems in algebras and
provide a computational approach for the study of structures of algebras. By the existence of
Hall polynomials of Dynkin quivers Q with automorphism σ , Ringel introduced the generic
Ringel-Hall algebra Hq (Q, σ ). The degenerate Ringel-Hall algebra is the specialization of
Ringel-Hall algebra at q = 0 and in [4] Reineke give a remarkable basis which is closed under
multiplication. On the other hand, when doing calculation in Hall algebras, one often has the
extension of two other representations. In case of a Dynkin quiver Q, the generic extension
Supported by National Natural Science Foundation of China (Grant No. 11361056).
B
Abdukadir Obul
abdu@vip.sina.com
Rabigul Tuniyaz
rabigul802@sina.com
1
Institute of Science and Technology(Akesu Campus), Xinjiang University, Akesu, Xinjiang
843000, China
2
College of Mathematics and System Sciences, Xinjiang University, Ürümqi, Xinjiang 830046, China
123
R. Tuniyaz, A. Obul
of any two representations of Q always exists (see [5]). In [4], Reineke first showed that the
multiplication by taking generic extension is associative, which leads to a monoid structure
on the set of isomorphism classes of representations. Furthermore, Reineke also showed that
the monoid ring of generic extensions can be realized as the degenerate Ringel-Hall algebra.
The structure of the degenerate Ringel-Hall algebras of Dynkin quivers has been studied in
[4,6–9]. In this paper, we will investigate the structure of degenerate Ringel-Hall algebra
Hq (Q, σ ) of a Dynkin quiver Q of type D4 . Our approach is to use the generic extensions
of representations of quiver Q studied in [5] and to use the theory developed in [10] and
further in [11] and its application to generic extensions. Using the Frobenius morphism F for
a Dynkin quiver Q with automorphism σ (see [10]), 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.
For the purpose of this paper, we first give a presentation of degenerate Ringel-Hall algebra
of type D4 . Then, by using the relations which are computed to give this presentation, we
construct a Gröbner-Shirshov basis for the degenerate Ringel-Hall algebra H0 (D4 ).
2 Some preliminaries
First, we recall some relevant notions and results about Gröbner-Shirshov bases theory from
[12].
Let X be a linearly ordered set, k a field, kX the free associative algebra generated by X
over k. Let X ∗ be the free monoid generated by X and for any word u ∈ X ∗ , we denote by
|u| the length of the word u, that is the number of the letters contained in u. Clearly, |x| = 1
for any x ∈ X.
Let I be a well ordered set. We order the words u and v in X ∗ as follows:
(i) If |uv| = 2, then u = xi > v = x j if and only if i > j, where i, j ∈ I .
(ii) If |uv| > 2, then u > v if and only if one of the following cases holds:
(a) |u| > |v|,
(b) If |u| = |v|, then u >lex v,
where >lex is the lexicographical ordering on X ∗ . It is clear that > is a monomial
ordering on X ∗ in the following sense:
(a) > is well ordering.
(b) For any u, v ∈ X ∗ , we have
u > v ⇒ ω1 uω2 > ω1 vω2 , for all ω1 , ω2 ∈ X ∗ .
Such an ordering is called deg-lex ordering. Then, any polynomial f ∈ kX has the
leading word f . We call f monic if the coefficient of f is 1.
Let f, g ∈ kX be two monic polynomials and ω ∈ X ∗ . If ω = f b = ag for some
a, b ∈ X ∗ such that deg( f ) + deg(g) < deg(ω), then ( f, g)ω = f b − ag is called the
intersection composition of f, g relative to ω. If ω = f = agb for some a, b ∈ X ∗ , then
( f, g)ω = f − agb is called the inclusion composition of f, g relative to ω.
Let R ⊂ kX be a monic set. A composition ( f, g)ω is called trivial modulo (R, ω) if
( f, g)ω =
αi ai ti bi ,
where each αi ∈ k, ti ∈ R, ai , bi ∈ X ∗ and ai t i bi < ω.
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Gröbner-Shirshov basis for degenerate Ringel-Hall algebra…
R is called a Gröbner-Shirshov basis if any composition of polynomials from R is trivial
modulo R.
A standard result about the Gröbner-Shirshov bases theory is the following lemma (see
[12]).
Lemma 2.1 (Composition-Diamond Lemma) Let k be a field, A = kX |R = kX /I d(R)
and “<” a monomial order on X ∗ ,where I d(R) is the ideal of kX generated by R. Then
the following statements are equivalent:
(a) R is a Gröbner-Shirshov basis;
(b) f ∈ I d(R) ⇒ f = atb for some t ∈ R and a, b ∈ X ∗ ;
(c) I rr (R) = {u ∈ X ∗ | u = atb, t ∈ R, a, b ∈ X ∗ } is a k−linear basis of the factor
algebra A = kX |R.
Next, we recall some relevant notions and results about Frobenius morphisms from [10,11].
Let (Q, σ ) be a quiver Q with automorphism σ . The associated valued quiver =
(Q, σ ) is defined as follows. Its vertex set 0 and arrow set 1 are simply the sets of σ orbits in Q 0 and Q 1 , respectively. For ρ ∈ Q 1 , its tail (resp., head) is the σ -orbit of tails
(resp., heads) of arrows in ρ. The valuation of is given by
di = | {vertices in σ -orbit i} | for i ∈ 0 ,
m ρ = | {arrows in σ -orbit ρ} | for ρ ∈ 1 .
Let Fq be a finite field with q elements and K = F̄q the algebraic closure of Fq .
Definition 2.2 [10,11] 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) = λq F(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 := K Q be the path algebra of Q
over K. Then σ induces a Frobenius
morphism
q
F = F Q,σ = F Q,σ,q : A −→ A given by s xs ps −→ s xs σ ( ps ), where s xs ps is a
K-linear combination of paths ps and σ ( ps ) = σ (ρt ) · · · σ (ρ1 ), if ps = ρt · · · ρ1 for arrows
ρ1 , . . . , ρt ∈ Q 1 . Then, the fixed-point algebra
A(q) = A(Q, σ ; q) := AF = {a ∈ A | F(a) = a}
is an Fq -algebra associated with (Q, σ ).
Definition 2.3 [10] Let (Q, σ ) be a quiver with automorphism σ . A representation V =
(Vi , φρ ) of Q is called F-stable (or
equivalently, an F-stable A-module) if there is a Frobenius
map F = FV : i∈Q 0 Vi −→ i∈Q 0 Vi satisfying FV (Vi ) = Vσi for all i ∈ Q 0 such that
FV φρ = φσ (ρ) FV for each arrow ρ ∈ Q 1 .
For an F-stable representation V = (Vi , φρ ), let dimV = i∈0 (dimVi )i ∈ N0 and
dimV = i∈0 dimVi denote the 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.
Lemma 2.4 [7] There is a one-to-one correspondence between isoclasses of indecomposable
A(q)-modules and isoclasses of indecomposable F-stable A-modules.
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R. Tuniyaz, A. Obul
Now, we recall some relevant notions and results about degenerate Ringel-Hall algebra
from [7].
From now on, we assume that (Q, σ ) is a Dynkin quiver Q with automorphism σ . Ringel
(see [13]) proved that there is a bijection from the isoclasses of indecomposable A(q)-modules
to the set + = + (Q, σ ) of positive roots in the root system associated with the valued
quiver = (Q, σ ). For each α ∈ + , 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) :=
λ(α)Mq (α),
α∈+
for some function λ : + −→ N. Thus, the isoclasses of A(q)-modules are indexed by the
set
+
B = B(Q, σ ) =: {λ | λ : + −→ N} = N ,
which is independent of q. By Lemma 2.4, the isoclasses of F-stable K Q-modules are also
indexed by B. Clearly, for each i ∈ 0 , there is a complete simple A(q)-module Si corresponding to i.
For M, N1 , . . . , Nt ∈A(q)-mod, let FNM1 ,...,Nt be the number of filtrations
M = M0 ⊇ M1 ⊇ · · · ⊇ Mt−1 ⊇ Mt = 0,
such that Mi−1 /Mi ∼
= Ni for all 1 ≤ i ≤ t is finite. By [14], FNM1 ,...,Nt is a polynomial in
q when q varies. More precisely, for λ, μ, ν ∈ B = B(Q, σ ), there exists a polynomial
λ (q) ∈ Z[q] (the polynomial ring over Z in one indeterminate q) such that
ϕμ,ν
Mq (λ)
λ
ϕμ,ν
(qk ) = FMq k(μ) ,Mq
k
k (ν)
holds for any finite field k with qk elements.
The generic Ringel-Hall algebra H = Hq (Q, σ ) is the free module over Z[q] with basis
{u λ | λ ∈ B} and multiplication defined by
λ
uμuν =
ϕμ,ν
(q)u λ .
λ∈B
It is an
N|0 | -graded
algebra
H=
He ,
e∈N|0 |
where He is spanned by all μα , α ∈ Be := {β ∈ B | dimMq (β) = e}.
For each λ ∈ B, set Mq (λ)K := Mq (λ) ⊗Fq K, which is the F-stable K Q-module
corresponding to λ.
Now by specializing q to 0, we obtain the Z-algebra H0 (Q, σ ), called the degenerate
Ringel-Hall algebra associated with = (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 by zero. By abuse of notations, we
also write u λ = u λ ⊗1. Thus, the set {u λ | λ ∈ B} is a Z-basis of H0 (Q, σ ). Let u i = u [Si ] ⊗1
in H0 (Q, σ ) for i ∈ 0 .
Lemma 2.5 [15] As a Z-algebra, H0 (Q, σ ) is generated by u i , i ∈ 0 .
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Gröbner-Shirshov basis for degenerate Ringel-Hall algebra…
Finally, we recall some relevant notions and results about monoid algebra from [4].
For K Q-modules M and N , the generic extension M ∗ N of M by N was defined in [5] as
the unique (up to isomorphism) element in Ext1K Q (M, N ) having endomorphism algebra of
minimal dimension. As shown in [4], the star operation ∗ defines the structure of a monoid
on the set M Q = M Q,K of isoclasses of K Q-modules.
Proposition 2.6 [11] If M and N are two F-stable K Q-modules, then M ∗ N is also F-stable.
By this proposition, the set of isoclasses [M] of F-stable K Q-modules, together with the
operation [M] ∗ [N ] = [M ∗ N ], defines a submonoid M Q,σ of M Q with the unit element
[0].
Since all the indecomposable A(q)-modules are indexed by the set B, we give an enumeration on + defined by β1 , β2 , . . . , β N such that for all prime powers q,
Hom A(q) (Mq (βs ), Mq (βt )) = 0 implies 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 (β N ).
By the definition of the generic extension, if Ext1A(q) (M, N ) = 0, then M ∗ N ∼
= M ⊕ N.
Consequently, we have the following known result (see [4]):
Lemma 2.7 Each element [Mq (λ)K ] in M Q,σ with λ ∈ B can be written as
[Mq (λ)K ] = [Mq (β1 )K ]∗λβ1 ∗ · · · ∗ [Mq (β N )K ]∗λβ N .
Moreover, these elements form a Z-basis of ZM Q,σ .
For a dimension vector d =
di i ∈ N0 , we consider the affine space
HomK (Kdi , Kd j ).
Rd =
i∈0
α:i→ j
Then, the group G d :=
i∈0
GLdi (K) acts on Rd by conjugation, i.e., by
(gi ) · (xρ )ρ = (g j xρ gi−1 )ρ:i→ j .
The orbits of G d correspond bijectively to the isoclasses of representations of of the dimension vector d. Denote by O M the orbit corresponding to the isoclass [M]. Since there are only
finitely many G d -orbits in Rd , there exists a dense one, whose corresponding representation
is denoted by E d .
be an enumeration of 0 such that k < l, if there is an arrow
Lemma 2.8 [14] Let i 1 , . . . , i n from i k to il . Then, for all d = nk=1 dk αik ∈ + , we have [E d ] = [Si1 ]∗di ∗ · · · ∗ [Sin ]∗dn
in M Q, σ .
Like the Ringel-Hall algebras, there is a natural grading on the monoid algebra ZM Q,σ
in terms of dimension vectors:
ZMe ,
ZM Q,σ =
e∈N|0 |
where ZMe is spanned by all [Mq (α)K ], α ∈ Be .
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R. Tuniyaz, A. Obul
3 Presentation of degenerate Ringel-Hall algebra H0 ( D4 )
In the following, we consider the quiver:
• 2
D4 :
1
•
• 3
(1)
• 4
It is easy to see that the following relations hold in H0 (D4 ):
(J1) u 2 u 3 = u 3 u 2 , u 2 u 4 = u 4 u 2 , u 3 u 4 = u 4 u 3 ,
(J2) u 21 u i = u 1 u i u 1 , for i = {2, 3, 4},
(J3) u 1 u i2 = u i u 1 u i , for i = {2, 3, 4}.
We consider the corresponding monoid algebra ZM D4 ,id . By [8], the following relations
hold in ZM D4 ,id :
(J 1) [S2 ] ∗ [S3 ] = [S3 ] ∗ [S2 ]; [S2 ] ∗ [S4 ] = [S4 ] ∗ [S2 ]; [S3 ] ∗ [S4 ] = [S4 ] ∗ [S3 ];
(J 2) [S1 ]∗2 [Si ] = [S1 ] ∗ [Si ] ∗ [S1 ], for i = {2, 3, 4};
(J 3) [S1 ][Si ]∗2 = [Si ] ∗ [S1 ] ∗ [Si ], for i = {2, 3, 4}.
In the following we prove that the set {[S1 ], [S2 ], [S3 ], [S4 ]} and the relations (J 1)−(J 3)
between them give a presentation of the monoid algebra ZM D4 ,id .
Proposition 3.1 The monoid algebra ZM D4 ,id has a presentation with generators [Si ](1 ≤
i ≤ 4) and relations (J 1) − (J 3).
Proof For convenience, set ZM = ZM D4 ,id . Let S be the free Z-algebra with generators
si (1 ≤ i ≤ 4). Consider the ideal J generated by the following elements
(J 1) s2 s3 − s3 s2 ; s2 s4 − s4 s2 ; s3 s4 − s4 s2 ;
(J 2) s12 si − s1 si s1 , for i = {2, 3, 4};
(J 3) s1 si2 − si s1 si , for i = {2, 3, 4}.
Then, there is a surjective monoid algebra homomorphism η : S −→ ZM given by si −→
[Si ] with 1 ≤ i ≤ 4. Because we have J i = 0 (1 ≤ i ≤ 3) in ZM, the map η induces a
surjective algebra homomorphism η̄ : S /J −→ ZM given by si + J −→ [Si ] (1 ≤ i ≤ 4).
To complete the proof, it suffices to show that η̄ is injective. For this, we will do some
preparations.
Set f i = si + J (1 ≤ i ≤ 4). Given a K D4 -module M with dimension vector dimM :=
(a, b, c, d), we define a monomial in S /J by
n(M) = fa1 fb2 fc3 fd4 .
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Gröbner-Shirshov basis for degenerate Ringel-Hall algebra…
It is known that the Auslander-Reiten quiver for K D4 is as follows:
[M12 ]
[M11 ]
[M13 ]
[M22 ]
[M21 ]
[M14 ]
[M23 ]
[M32 ]
[M31 ]
[M24 ]
[M33 ]
(2)
[M34 ]
where [Mi j ] (1 ≤ i ≤ 3, 1 ≤ j ≤ 4) are the isoclasses of indecomposable K D4 -modules and
[M11 ], [M32 ], [M33 ], [M34 ] are the isoclasses of indecomposable projective K D4 -modules
corresponding to vertices i = 1, 2, 3, 4 (see [16] for details). Moreover, the dimension vectors
of Mi j (1 ≤ i ≤ 3, 1 ≤ j ≤ 4) and associated monomials in S /J are given by
dimM11 = (1, 0, 0, 0) and n(M11 ) = f1 ,
dimM12 = (1, 1, 0, 0) and n(M12 ) = f1 f2 ,
dimM13 = (1, 0, 1, 0) and n(M13 ) = f1 f3 ,
dimM14 = (1, 0, 0, 1) and n(M14 ) = f1 f4 ,
dimM21 = (2, 1, 1, 1) and n(M21 ) = f21 f2 f3 f4 ,
dimM22 = (1, 0, 1, 1) and n(M22 ) = f1 f3 f4 ,
dimM23 = (1, 1, 0, 1) and n(M23 ) = f1 f2 f4 ,
dimM24 = (1, 1, 1, 0) and n(M24 ) = f1 f2 f3 ,
dimM31 = (1, 1, 1, 1) and n(M31 ) = f1 f2 f3 f4 ,
dimM32 = (0, 1, 0, 0) and n(M32 ) = f2 ,
dimM33 = (0, 0, 1, 0) and n(M33 ) = f3 ,
dimM34 = (0, 0, 0, 1) and n(M34 ) = f4 .
Now we give an enumeration of indecomposable A(q)-modules in figure (2):
Mi+14 ≺ Mi+13 ≺ Mi+12 ≺ Mi+11 ≺ Mi,4 ≺ Mi,3 ≺ Mi,2 ≺ Mi,1 .(∗)
Then, by using the relations (J1)–(J3), we get following result
Proposition 3.2 The following equalities hold in S /J.
n(Mi1 )n(Mi2 ) = n(Mi2 )n(Mi1 ),
n(Mi1 )n(Mi4 ) = n(Mi4 )n(Mi1 ),
n(Mi1 )n(Mi+1,2 ) = n(Mi3 )n(Mi4 ),
n(Mi1 )n(Mi+1,4 ) = n(Mi2 )n(Mi3 ),
n(M11 )n(M32 ) = n(M12 ),
n(M11 )n(M34 ) = n(M14 ),
n(Mi2 )n(Mi4 ) = n(Mi4 )n(Mi2 ),
n(Mi2 )n(Mi+1,2 ) = n(Mi+1,1 ),
n(Mi2 )n(Mi+1,4 ) = n(Mi+1,4 )n(Mi2 ),
n(M12 )n(M32 ) = n(M32 )n(M12 ),
n(M12 )n(M34 ) = n(M23 ),
n(Mi3 )n(Mi+11 ) = n(Mi+1,1 )n(Mi3 ),
n(Mi3 )n(Mi+1,3 ) = n(Mi+1,1 ),
n(Mi1 )n(Mi3 ) = n(Mi3 )n(Mi1 ),
n(Mi1 )n(Mi+1,1 ) = n(Mi2 )n(Mi3 )n(Mi4 ),
n(Mi1 )n(Mi+1,3 ) = n(Mi2 )n(Mi4 ),
n(M11 )n(M31 ) = n(M21 ),
n(M11 )n(M33 ) = n(M13 ),
n(Mi2 )n(Mi3 ) = n(Mi3 )n(Mi2 ),
n(Mi2 )n(Mi+1,1 ) = n(Mi+1,1 )n(Mi2 ),
n(Mi2 )n(Mi+1,3 ) = n(Mi+1,3 )n(Mi2 ),
n(M12 )n(M31 ) = n(M23 )n(M24 ),
n(M12 )n(M33 ) = n(M24 ),
n(Mi3 )n(Mi4 ) = n(Mi4 )n(Mi3 ),
n(Mi3 )n(Mi+1,2 ) = n(Mi+1,2 )n(Mi3 ),
n(Mi3 )n(Mi+1,4 ) = n(Mi+1,4 )n(Mi3 ),
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R. Tuniyaz, A. Obul
n(M13 )n(M31 ) = n(M22 )n(M24 ),
n(M13 )n(M33 ) = n(M33 )n(M13 ),
n(Mi4 )n(Mi+1,1 ) = n(Mi+1,1 )n(Mi4 ),
n(Mi4 )n(Mi+1,3 ) = n(Mi+1,3 )n(Mi4 ),
n(M14 )n(M31 ) = n(M22 )n(M23 ),
n(M14 )n(M33 ) = n(M22 ),
n(M13 )n(M32 ) = n(M24 ),
n(M13 )n(M34 ) = n(M22 ),
n(Mi4 )n(Mi+1,2 ) = n(Mi+1,2 )n(Mi4 ),
n(Mi4 )n(Mi+1,4 ) = n(Mi+1,1 ),
n(M14 )n(M32 ) = n(M23 ),
n(M14 )n(M34 ) = n(M34 )n(M14 ),
where Mi, j (1 ≤ i ≤ 6, 1 ≤ j ≤ 6) are indecomposable representations.
Note Let us denote the set of equalities in Proposition 3.2 by S .
Proof Since all the equalities can be treated in a similar way, we only prove two of them as
examples:
n(M14 )n(M21 ) = f1 f4 · f21 f2 f3 f4 = f1 f4 · f1 f2 f1 f3 f4 = f1 f2 · f1 f4 f1 f3 f4
= f1 f2 · f1 f3 f1 f4 f4 = f1 f2 · f1 f3 f4 f1 f4 = f21 f2 · f3 f4 f1 f4
= n(M21 )n(M14 ).
n(M24 )n(M31 ) = f1 f2 f3 · f1 f2 f3 f4 = f1 f2 f3 · f1 f2 f4 · f3 = f1 f2 f4 · f1 f2 f23
= f1 f2 f4 · f1 f23 f2 = f1 f2 f4 f3 · f1 f3 f2 = f1 f2 f3 f4 · f1 f2 f3
= n(M31 )n(M24 ).
From above, we get
n(Mi4 )n(Mi+11 ) = n(Mi+11 )n(Mi4 ).
Let V1 , . . . , V12 be all the non-isomorphic indecomposable A(q)-modules. We assume that
they are enumerated by V1 ≺ · · · ≺ V12 as given in (∗). Repeatedly applying relations in S we get the following result:
Corollary 3.3 For 1 ≤ i < j ≤ 12, there exist 1 ≤ j1 ≤ j2 ≤ · · · ≤ jm ≤ 12 such that
n(V j )n(Vi ) = n(V j1 )n(V j2 ) · · · n(V jm ).
Now we are ready to prove the injectivity of
η̄ : S /J −→ ZM, si + J −→ [Si ] (1 ≤ i ≤ 4).
Given a monomial ω = fi1 · · · fim (1 ≤ i 1 ≤ i m ≤ 4), we have ω = fi1 · · · fim =
n(Si1 ) · · · n(Sim ). Applying above result repeatedly, we finally get ω = n(V1 )n 1 · · · n(V12 )n 12
for some n 1 , . . . , n 12 ≥ 0. Hence, all the monomials n(V1 )n 1 · · · n(V12 )n 12 with n 1 , . . . , n 12 ≥
0 span S /J. On the other hand, Lemma 2.8 implies that for n 1 , . . . , n 12 ≥ 0,
η̄(n(V1 )n 1 · · · n(V12 )n 12 ) = [V1 ]∗n 1 ∗ · · · ∗ [V12 ]∗n 12 .
By Lemma 2.7, the elements [V1 ]∗n 1 ∗ · · · ∗ [V12 ]∗n 12 with n 1 , . . . , n 12 ≥ 0 form a basis of
ZM D4 ,id . Consequently, the morphism η̄ is injective.
Note that we have following result
Proposition 3.4 There is a graded Z-algebra isomorphism
: ZM D4 ,id −→ H0 (D4 ), [Si ] −→ u i , (1 ≤ i ≤ 4).
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Gröbner-Shirshov basis for degenerate Ringel-Hall algebra…
Proof By Lemma 2.5 and Proposition 3.1, there is a surjective Z-algebra homomorphism
: ZM D4 ,id −→ H0 (D4 ) given by [Si ] −→ u i with 1 ≤ i ≤ 4. Since {[Mq (λ)K ] |λ ∈ B}
and {u λ | λ ∈ B} are bases for ZM D4 ,id and H0 (D4 ), respectively, we know that is an
isomorphism.
So we have theorem
Theorem 3.5 The generators u i (1 ≤ i ≤ 4) and the relations (J1)−(J3) give a presentation
of H0 (D4 ).
4 Gröbner-Shirshov basis for H0 ( D4 )
First, we define a degree lexicographic order ≺ as follows:
u ≺ v if and only if l(u) < l(v) or l(u) = l(v) and u <lex v,
where l(u) = |u|, then it is a monomial order (see [17]).
We already shown that H0 (D4 ) is an associative algebra over Z generated by C =
{u 1 , u 2 , u 3 , u 4 } with generating relations
⎧
u2u4 = u4u2,
u3u4 = u4u3,
⎪
⎨ u2u3 = u3u2,
J = u 1 u 22 = u 2 u 1 u 2 , u 1 u 23 = u 3 u 1 u 3 , u 1 u 24 = u 4 u 1 u 4 ,
⎪
⎩ 2
u 1 u 2 = u 1 u 2 u 1 , u 21 u 3 = u 1 u 3 u 1 , u 21 u 4 = u 1 u 4 u 1 .
By Propositions 3.1 and 3.2, if we apply the algebra isomorphism ◦ η to the relations in
S , then we get a new set J of relations in H0 (D4 )( u 1 u 2 u 3 u 4 ):
u2u3 = u3u2,
u2u4 = u4u2,
u3u4 = u4u3,
u 21 u 2 = u 1 u 2 u 1 ,
u 21 u 3 = u 1 u 3 u 1 ,
u 31 u 2 u 3 u 4 = u 1 u 2 u 1 u 3 u 1 u 4 ,
u 21 u 2 u 4 = u 1 u 2 u 1 u 4 ,
u 21 u 4 = u 1 u 4 u 1 ,
u1u2u1u3 = u1u3u1u2,
u1u2u1u4 = u1u4u1u2,
u1u3u1u4 = u1u4u1u3,
u 21 u 2 u 3 u 4 u 1 u 2 = u 1 u 2 u 21 u 2 u 3 u 4 ,
u1u2u1u2u4 = u1u2u4u1u2,
u 21 u 2 u 3 u 4 = u 1 u 2 u 1 u 3 u 4 ,
u 21 u 3 u 4 = u 1 u 3 u 1 u 4 ,
u 21 u 2 u 3 = u 1 u 2 u 1 u 3 ,
u 1 u 2 u 1 u 2 u 3 u 4 = u 1 u 2 u 3 u 1 u 2 u 4 , u 1 u 24 = u 4 u 1 u 4 ,
u 1 u 22 = u 2 u 1 u 2 ,
u 1 u 23 = u 3 u 1 u 3 ,
u 21 u 2 u 3 u 4 u 1 u 3 = u 1 u 3 u 21 u 2 u 3 u 4 ,
u 21 u 2 u 3 u 4 = u 1 u 3 u 1 u 2 u 4 ,
u1u3u1u3u4 = u1u3u4u1u3,
u1u2u3u1u3 = u1u3u1u2u3,
u 1 u 2 u 3 u 1 u 3 u 4 = u 1 u 3 u 1 u 2 u 3 u 4 , u 21 u 2 u 3 u 4 u 1 u 4 = u 1 u 4 u 21 u 2 u 3 u 4 ,
u1u3u4u1u4 = u1u4u1u3u4,
u 21 u 2 u 3 u 4
= u1u4u1u2u3,
u1u2u4u1u4 = u1u4u1u2u4,
u1u2u4u1u3u4 = u1u4u1u2u3u4,
123
R. Tuniyaz, A. Obul
u 21 u 2 u 3 u 4 u 1 u 3 u 4 = u 1 u 3 u 4 u 21 u 2 u 3 u 4 , u 21 u 2 u 3 u 4 u 1 u 2 u 4 = u 1 u 2 u 4 u 21 u 2 u 3 u 4 ,
u 21 u 2 u 3 u 4 u 1 u 2 u 3 = u 1 u 2 u 3 u 21 u 2 u 3 u 4 , u 21 u 2 u 3 u 4 u 1 u 2 u 3 u 4 = u 1 u 2 u 3 u 4 u 21 u 2 u 3 u 4 ,
u 21 u 2 u 3 u 4 u 2 = u 2 u 21 u 2 u 3 u 4 ,
u 21 u 2 u 3 u 4 u 4
=
u 4 u 21 u 2 u 3 u 4 ,
u 21 u 2 u 3 u 4 u 3 = u 3 u 21 u 2 u 3 u 4 ,
u1u2u4u1u3u4 = u1u2u4u1u3u4,
u1u2u3u1u3u4 = u1u3u4u1u2u3,
u1u2u3u4u1u3u4 = u1u3u4u1u2u3u4,
u1u2u3u4 = u1u3u4u2,
u1u3u4u3 = u3u1u3u4,
u1u2u3u1u2u4 = u1u2u4u1u2u3,
u1u2u3u4u1u2u4 = u1u2u4u1u2u3u4,
u1u2u4u2 = u2u1u2u4,
u 1 u 2 u 24 = u 4 u 1 u 2 u 4 ,
u1u2u3u1u2u3u4 = u1u2u3u4u1u2u3, u1u2u3u2 = u2u1u2u3,
u 1 u 2 u 23 = u 3 u 1 u 2 u 3 ,
u1u2u3u4u2 = u2u1u2u3u4,
u1u2u3u4u3 = u3u1u2u3u4,
u 1 u 2 u 3 u 24 = u 4 u 1 u 2 u 3 u 4 ,
u1u2u1u2u3 = u1u2u3u1u2,
u 1 u 3 u 24 = u 4 u 1 u 3 u 4 .
By a routine check of compositions between the elements of J ∪ J , we get following
new set J of relations in H0 (D4 ) :
u1u3u2u1u4u2 = u1u4u2u1u3u2,
u 1 u 3 u 22 = u 2 u 1 u 2 u 3 ,
u 1 u 4 u 3 u 2 1u 4 u 3 = 1u 4 u 3 u 1 u 4 u 3 u 2 ,
u1u4u1u4u2 = u1u4u2u1u4,
u1u3u4u1u4u2 = u1u4u2u1u4,
u1u3u4u1u4u2 = u1u4u2u1u3u4,
u 1 u 4 u 22
= u2u1u4u2,
u1u4u1u4u3 = u1u4u3u1u4,
u1u3u2u1u4u3 = u1u3u4u1u2u3,
u 1 u 4 u 23 = u 3 u 1 u 4 u 3 ,
u1u3u1u3u2 = u1u3u2u1u3,
u1u4u1u3u2u1u4 = u1u4u1u4u1u3u2,
u1u3u1u4u2u1u3u2 = u1u3u2u1u3u1u4u2, u1u4u2u1u4u3 = u1u4u3u1u4u2.
We set J = J ∪ J ∪ J . Then by the construction of the set J of relations in H0 (D4 ),
we get our main result in this paper:
Theorem 4.1 With notations above, J is a Gröbner-Shirshov bases for H0 (D4 ).
Acknowledgements We are very grateful for the referee for useful comments and pointing us many mistakes
in the original manuscript.
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