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. 123 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. 123 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 . 123 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 . 123 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 . 123 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 ), 123 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). 123 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. References 1. Buchberger, B.: An algorithm for finding a basis for the residue class ring of a zero-dimensional ideal. Ph.D. Thesis, University of Innsbruck (1965) 2. Bergman, G.M.: The diamond lemma for ring theory. Adv. Math. 29, 178–218 (1978) 3. Shirshov, A.I.: Some algorithmic problems for Lie algebras. Sib. Math. J. 3, 292–296 (1962) 4. Wolf, S.: The Hall Algebra and the Composition Monoid. arXiv:0907.1106 5. Reineke, M.: Generic extensions and multiplicative bases of quantum groups at q = 0. Represent. Theory 5, 147–163 (2001) 6. Bongartz, K.: On degenerations and extensions of finite dimensional modules. Adv. Math. 121, 245–287 (1996) 7. Deng, B., Du, J.: Frobenius morphisms and representations of algebras. Trans. Am. 358(8), 3591–3622 (2006) 8. Ringel, C.M.: The composition algebra of a cyclic quiver. Proc. Lond. Math. Soc. 66, 507–537 (1993) 123 Gröbner-Shirshov basis for degenerate Ringel-Hall algebra… 9. Fan, L., Zhao, Z.: Presenting degenerate Ringel-Hall algebras of type B. Sci. China Math. 55(5), 949–960 (2012) 10. Deng, B., Du, J., Xiao, J.: Generic extensions and canonical bases for cyclic quivers. Can. J. Math. 59(6), 1260–1283 (2007) 11. Reineke, M.: Feigin’s map and monomial bases for quantized enveloping algebras. Math. Z. 237, 639–667 (2001) 12. Reineke, M.: The Quantic Monoid and Degenerate Quantized Enveloping Algebras. arXiv:math/0206095v1 13. Ringel, C.M.: Representations of K -species and bimodules. J. Algebra 41, 269–302 (1976) 14. Bokut, L.A.: Imbeddings into simple associative algebras. Algebra Log. 15, 117–142 (1976) 15. Deng, B.M., Du, J., Parshal, B., Wang, J.P.: Finite Dimensional Algebra and Quantum Groups, Mathematical Surveys and Monographs, vol. 150. American Mathematical Society, Providence (2008) 16. Yunus, G., Obul, A.: Göbner-Shirshov basis of quantum group of type D4 . Chin. Ann. Math. 32B(4), 581–592 (2011) 17. Kang, S., Lee, K.: Gröbner-Shirshov bases for representation theory. J. Korean Math. Soc. 37, 55–72 (2000) 123

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