# PBW Basis of the Modified Ringel–Hall Algebra of TypeAn via Gröbner–Shirshov Basis

Authors: Desheng Hu; Abdukadir Obul
Journal: Journal of Mathematics
 (2023)

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## Abstract

In the Ringel–Hall algebra of Dynkin type, the set of all commutator relations between the isoclasses of indecomposable representations forms a minimal Gröbner–Shirshov basis, and the set of the corresponding irreducible elements forms a PBW-type basis of the Ringel–Hall algebra. The aim of this paper is to generalize this result to the modified Ringel–Hall algebra of type. First, we compute a minimal Gröbner–Shirshov basis for the modified Ringel–Hall algebra of typeAn by proving all possible compositions between the commutator relations are trivial. Then, by taking the corresponding irreducible monomials, we construct a PBW-type basis for the modified Ringel–Hall algebra of type An.


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## Body

## 1. Introduction

The Gröbner–Shirshov bases theory for algebras was established by Buchberger [1], Shirshov [2], and Bergman [3], independently for commutative algebras, Lie algebras, and associative algebras, respectively, in order to solve the reduction problems in corresponding algebras. Later, in [4], Bokut and Malcolmson established the Gröbner–Shirshov bases theory for quantized enveloping algebras. Since then, the Gröbner–Shirshov bases theory has found wide applications in mathematics and other subjects and now has become an important computational approach to study the structures of algebras. For a good survey, see [5].Since Ringel [6, 7] introduced the Ringel–Hall algebra and, by using it, gave a realization of the positive part of the quantized enveloping algebra in terms of the representation theory of a finite-dimensional algebra over a finite field k, several efforts have been made to realize the whole quantized enveloping algebras. First, Xiao [8] gave a realization of the whole quantized enveloping algebras by constructing the Drinfeld double of the extended Ringel–Hall algebra of a hereditary algebra. Later, to give a pure categorical realization of the whole quantized enveloping algebra, people considered larger categories such as dg categories [9] and triangulated categories [10]. In [11], Bridgeland considered the category of 2-periodic complexes of projective modules over a finite-dimensional hereditary algebra A and defined an algebra (now) called the Bridgeland Hall algebra of A. He proved that Ringel–Hall algebra of A can be embedded into its Bridgeland Hall algebra, and then, the quantized enveloping algebra associated to A can also be embedded into the reduced Bridgeland Hall algebra of A, and so this provides a realization of the whole quantized enveloping algebra. In [11], Bridgeland also stated that the Drinfeld double of the extended Ringel–Hall algebra of a finite-dimensional hereditary algebra A is isomorphic to its Bridgeland Hall algebra but did not prove it. Later, Yanagita [12] and Zhang [13] proved the result in different ways. In [14], the authors defined the modified Ringel–Hall algebras and got two main results about it. One is the authors provided a new proof of the Green’s formula on Ringel–Hall numbers by using the associative multiplication of the modified Ringel–Hall algebras. The second one is the authors proved that in certain twisted cases, the derived Hall algebra can be embedded into the modified Ringel–Hall algebra.In [15], by using the composition-diamond lemma, the authors proved that in Ringel–Hall algebra of Dynkin type, the set of all commutator relations between the isoclasses of indecomposable representations forms a minimal Gröbner–Shirshov basis, and the set of the corresponding irreducible elements forms a PBW-type basis of the Ringel–Hall algebra, and as an application, the Gröbner–Shirshov bases for the cases E7 and Dn are constructed in detail. In this paper, we generalize this result to the modified Ringel–Hall algebra of type An. For this, first we construct a minimal Gröbner–Shirshov basis for the modified Ringel–Hall algebra by computing all commutator relations between the isoclasses of indecomposable representations of An and then prove that all possible compositions between the commutator relations are trivial. Then, as an application, we construct a PBW-type basis of the modified Ringel–Hall algebra of type An by taking the corresponding irreducible monomials.


## 2. Preliminaries

### 2.1. Gröbner–Shirshov Bases Theory

We recall some basic notions and results on Gröbner–Shirshov bases theory from [5].LetX be a well-ordered nonempty set, <X the well-ordering on X and X∗ the free semigroup generated by X and assuming that 1∈X∗. Let K be a field and denote by KX the free associative K-algebra generated by X. Then, the set X∗ is a K-linear basis of KX. For any w∈X∗, we denote by w the length of w, i.e., the number of elements x∈X contained in w. The well-ordering <X on the set X induces a deg-lex ordering ≺ on X∗, i.e., for any two elements v,w∈X∗, we define v≺w if and only if v<w or v=w and v<Xw. It is clear that ≺ is a monomial ordering, i.e., compatible with the multiplication of X∗. We denote by f¯ the leading monomial of a polynomial f∈KX with respect to the ordering ≺, and if the coefficient of f¯ is 1, then the polynomial f is called monic.For any two monic polynomialsf and g in KX, we define the composition as follows:(1)
If there area,b,w∈X∗ such that w=f¯b=ag¯, and f¯+g¯>w, then f,gw=fb−ag is called the composition of intersection of f and g with respect to w(2)
Ifw=f¯=ag¯b for some unique elements a and b in X∗, then f,gw=f−agb is called the composition of inclusion of f and g with respect to wThe composition of intersection and the composition of inclusion are referred to as the compositions.LetS⊆KX be a nonempty set of monic polynomials. For any f,g∈KX, if f,gw=∑αiaisibi, where αi∈K,ai,bi∈X∗,si∈S and aisibi¯≺w, then we say f,gw is trivial modulo S and write(1)f,gw≡0modS,w.For anyp,q∈KX, if p−q=∑αiaisibi, where αi∈K,ai,bi∈X∗,si∈S and aisibi¯≺w, then we write(2)p≡qmodS,w.We denote byIdS the ideal of KX generated by S.Definition 1.
If all the compositions among the polynomials inS are trivial modulo S, then we call S a Gröbner–Shirshov basis in KX of the ideal IdS. If there is no composition of inclusion, then S is called a minimal Gröbner–Shirshov basis.
The following lemma is the key result in Gröbner–Shirshov bases theory.Lemma 1 (Composition-Diamond lemma) (see [2, 3]).
LetS⊆KX be nonempty set of monic polynomials, ≺ the monomial ordering on X∗. The following statements are equivalent:(1)
S is a Gröbner–Shirshov basis(2)
If0≠f∈IdS, then there are s∈S and a,b∈X∗ such that f¯=as¯b(3)
IrrS=u∈X∗|u≠as¯b,s∈S,a,b∈X∗ is a K-linear basis of the factor algebra KX|S=KX/IdS


### 2.2. The Modified Ringel–Hall Algebra

Now, we recall some notions about the modified Ringel–Hall algebra from [14].Letk be a finite field with q elements, and we set υ=q, and A a finitary hereditary abelian k-linear category (for definition see [6]). We denote by IsoA and KA the set of isoclasses of objects in A and the Grothendieck group of A, respectively. For any object M in A, we denote by M the isoclass of M; that is M∈IsoA, and for a finite set S, we denote by S the cardinality of S. For each object M in A, the class of M in KA is denoted by M^, and the automorphism group of M is denoted by AutM=aM.For any three objectsA,B,C in A, we let gABC be the number of subobjects A′ of C such that A′≅A and C/A′≅B. For any two objects M,N∈A, we define(3)M,N=dimkHomAM,N−dimkExtA1M,N.Then, it induces a bilinear form(4)⋅,⋅:KA×KA⟶Z,known as the Euler form. We define the symmetric Euler form(5)⋅,⋅:KA×KA⟶Z,as α,β=α,β+β,α for all α,β∈KA.For four objectsA,B,M,N in IsoA, we let VM,B,A,N be the set of exact sequences(6)0⟶M⟶B⟶A⟶N⟶0.Then,VM,B,A,N is a finite set, and from [16], we have(7)γABMN≔VM,B,A,NAutA⋅AutB=AutMAutNAutAAutB∑L∈IsoAgMLBgLNAAutL.From [14], the modified Ringel–Hall algebra Dℳℋ is an associative algebra with 1 generated by(8)X=UA,n,Kα,n|A∈IsoA,α∈KA,n∈Z,subject to the relations(9)UA,nUB,n=∑C∈IsoAaAaB/aCgABCUC,n;Kα,nUA,n=qA^,αUA,nKα,n;Kα,nKβ,n=q−α,βKα+β,n;UA,nKα,n+1=qα,A^Kα,n+1UA,n;Kα,nUA,n+1=UA,n+1Kα,n;Kα,nKβ,n+1=qβ,αKβ,n+1Kα,n;UB,nUA,n+1=∑M,N∈IsoAγABMNaAaB/aMaNqB^−M^,M^KB^−M^,n+1UN,n+1UM,n;UA,mUB,n=UB,nUA,mifm−n⩾2;Kα,mUB,n=UB,nKα,mifm−n⩾2;Kα,mKβ,n=Kβ,nKα,mifm−n⩾2.


## 2.1. Gröbner–Shirshov Bases Theory

We recall some basic notions and results on Gröbner–Shirshov bases theory from [5].LetX be a well-ordered nonempty set, <X the well-ordering on X and X∗ the free semigroup generated by X and assuming that 1∈X∗. Let K be a field and denote by KX the free associative K-algebra generated by X. Then, the set X∗ is a K-linear basis of KX. For any w∈X∗, we denote by w the length of w, i.e., the number of elements x∈X contained in w. The well-ordering <X on the set X induces a deg-lex ordering ≺ on X∗, i.e., for any two elements v,w∈X∗, we define v≺w if and only if v<w or v=w and v<Xw. It is clear that ≺ is a monomial ordering, i.e., compatible with the multiplication of X∗. We denote by f¯ the leading monomial of a polynomial f∈KX with respect to the ordering ≺, and if the coefficient of f¯ is 1, then the polynomial f is called monic.For any two monic polynomialsf and g in KX, we define the composition as follows:(1)
If there area,b,w∈X∗ such that w=f¯b=ag¯, and f¯+g¯>w, then f,gw=fb−ag is called the composition of intersection of f and g with respect to w(2)
Ifw=f¯=ag¯b for some unique elements a and b in X∗, then f,gw=f−agb is called the composition of inclusion of f and g with respect to wThe composition of intersection and the composition of inclusion are referred to as the compositions.LetS⊆KX be a nonempty set of monic polynomials. For any f,g∈KX, if f,gw=∑αiaisibi, where αi∈K,ai,bi∈X∗,si∈S and aisibi¯≺w, then we say f,gw is trivial modulo S and write(1)f,gw≡0modS,w.For anyp,q∈KX, if p−q=∑αiaisibi, where αi∈K,ai,bi∈X∗,si∈S and aisibi¯≺w, then we write(2)p≡qmodS,w.We denote byIdS the ideal of KX generated by S.Definition 1.
If all the compositions among the polynomials inS are trivial modulo S, then we call S a Gröbner–Shirshov basis in KX of the ideal IdS. If there is no composition of inclusion, then S is called a minimal Gröbner–Shirshov basis.
The following lemma is the key result in Gröbner–Shirshov bases theory.Lemma 1 (Composition-Diamond lemma) (see [2, 3]).
LetS⊆KX be nonempty set of monic polynomials, ≺ the monomial ordering on X∗. The following statements are equivalent:(1)
S is a Gröbner–Shirshov basis(2)
If0≠f∈IdS, then there are s∈S and a,b∈X∗ such that f¯=as¯b(3)
IrrS=u∈X∗|u≠as¯b,s∈S,a,b∈X∗ is a K-linear basis of the factor algebra KX|S=KX/IdS


## 2.2. The Modified Ringel–Hall Algebra

Now, we recall some notions about the modified Ringel–Hall algebra from [14].Letk be a finite field with q elements, and we set υ=q, and A a finitary hereditary abelian k-linear category (for definition see [6]). We denote by IsoA and KA the set of isoclasses of objects in A and the Grothendieck group of A, respectively. For any object M in A, we denote by M the isoclass of M; that is M∈IsoA, and for a finite set S, we denote by S the cardinality of S. For each object M in A, the class of M in KA is denoted by M^, and the automorphism group of M is denoted by AutM=aM.For any three objectsA,B,C in A, we let gABC be the number of subobjects A′ of C such that A′≅A and C/A′≅B. For any two objects M,N∈A, we define(3)M,N=dimkHomAM,N−dimkExtA1M,N.Then, it induces a bilinear form(4)⋅,⋅:KA×KA⟶Z,known as the Euler form. We define the symmetric Euler form(5)⋅,⋅:KA×KA⟶Z,as α,β=α,β+β,α for all α,β∈KA.For four objectsA,B,M,N in IsoA, we let VM,B,A,N be the set of exact sequences(6)0⟶M⟶B⟶A⟶N⟶0.Then,VM,B,A,N is a finite set, and from [16], we have(7)γABMN≔VM,B,A,NAutA⋅AutB=AutMAutNAutAAutB∑L∈IsoAgMLBgLNAAutL.From [14], the modified Ringel–Hall algebra Dℳℋ is an associative algebra with 1 generated by(8)X=UA,n,Kα,n|A∈IsoA,α∈KA,n∈Z,subject to the relations(9)UA,nUB,n=∑C∈IsoAaAaB/aCgABCUC,n;Kα,nUA,n=qA^,αUA,nKα,n;Kα,nKβ,n=q−α,βKα+β,n;UA,nKα,n+1=qα,A^Kα,n+1UA,n;Kα,nUA,n+1=UA,n+1Kα,n;Kα,nKβ,n+1=qβ,αKβ,n+1Kα,n;UB,nUA,n+1=∑M,N∈IsoAγABMNaAaB/aMaNqB^−M^,M^KB^−M^,n+1UN,n+1UM,n;UA,mUB,n=UB,nUA,mifm−n⩾2;Kα,mUB,n=UB,nKα,mifm−n⩾2;Kα,mKβ,n=Kβ,nKα,mifm−n⩾2.


## 3. The Commutator Relations in the Modified Ringel–Hall Algebra

From now on, we assume thatA is the category of finite-dimensional representations of the quiver of type An. In this section, we compute all the commutator relations between the generators UA,m (where A is an isoclass of indecomposable representation of the quiver of type An) and Kαi,m (where αi is the standard unit vectors in ℕn).We choose the following orientation for the Dynkin graphQ of type An:We know that the Cartan matrixC and the Auslander–Reiten quiver Γ of Q are(10)C=2−10⋯000−12−1⋯000⋯⋯⋯⋯⋯⋯⋯000⋯−12−1000⋯0−12,andrespectively, whereMij is the following indecomposable representation  or, equivalently, is the factor representation Pi/Pj, where Pi1⩽i⩽n is the indecomposable projective representation corresponding to the vertex i.For convenience, from [17], we set(11)C1=ij,mk|i=m<j<k,C2=ij,mk|i<m<k<j,C3=ij,mk|i<m<j=k,C4=ij,mk|i<m<j<k,C5=ij,mk|i<j=m<k,C6=ij,mk|i<j<m<k.By [18, 19] and making some modifications if necessary, we get the relations(12)2.1UMrs,mUMij,m=υ−2UMij,mUMrs,m,ifij,rs∈C1∪C3;2.2UMrs,mUMij,m=UMij,mUMrs,m,ifij,rs∈C2∪C6;2.3UMrs,mUMij,m=υ−2UMij,mUMrs,m−1−υ−2UMis,mUMrj,m,ifij,rs∈C4;2.4UMrs,mUMij,m=UMij,mUMrs,m−υ2−1UMis,m,ifij,rs∈C5.By direct computation, we also have following formulas:(13)2.5UMrs,mUMij,m′=UMij,m′UMrs,m,m−m′⩾2;2.6Kαrs,mKαij,m=v2Kαij,mKαrs,m,ifij,rs∈C1∪C3∪C5;2.7Kαrs,mKαij,m=Kαij,mKαrs,m,ifij,rs∈C2∪C6;2.8Kαrs,mKαij,m=v4Kαij,mKαrs,m,ifij,rs∈C4;2.9Kαij,mKαrs,m+1=v2Kαrs,m+1Kαij,m,ifij,rs∈C1∪C3∪C4;2.10Kαij,mKαrs,m+1=Kαrs,m+1Kαij,m,ifij,rs∈C2∪C5∪C6;2.11Kαrs,mKαij,m+1=Kαij,m+1Kαrs,m,ifij,rs∈C1∪C2∪C3∪C6;2.12Kαrs,mKαij,m+1=v−2Kαij,m+1Kαrs,m,ifij,rs∈C4∪C5;2.13Kαrs,mKαij,m′=Kαij,m′Kαrs,m,m−m′⩾2;2.14Kαij,mKαij,m+1=v2Kαij,m+1Kαij,m,1⩽i<j⩽n,2.15UMij,mKαrs,m=Kαrs,mUMij,m,ifij,rs∈C1∪C2∪C3∪C6;2.16UMij,mKαrs,m=v2Kαrs,mUMij,m,ifij,rs∈C4∪C5;2.17UMrs,mKαij,m=v−2Kαij,mUMrs,m,ifij,rs∈C1∪C3∪C4;2.18UMrs,mKαij,m=Kαij,mUMrs,m,ifij,rs∈C2∪C5∪C6;2.19UMij,mKαij,m=v−2Kαij,mUMij,m,ifij,rs⩽i<j⩽n;2.20UMij,mKαrs,m+1=v2Kαrs,m+1UMij,m,ifij,rs∈C1∪C3∪C4;2.21UMij,mKαrs,m+1=Kαrs,m+1UMij,m,ifij,rs∈C2∪C5∪C6;2.22UMrs,mKαij,m+1=Kαij,m+1UMrs,m,ifij,rs∈C1∪C2∪C3∪C6;2.23UMrs,mKαij,m+1=v−2Kαij,m+1UMrs,m,ifij,rs∈C4∪C5;2.24UMij,mKαij,m+1=v2Kαij,m+1UMij,m,1⩽i<j⩽n;2.25UMij,m+1Kαrs,m=Kαrs,mUMij,m+1,ifij,rs∈C1∪C2∪C3∪C4∪C5∪C6;2.26UMrs,m+1Kαij,m=Kαij,mUMrs,m+1,ifij,rs∈C1∪C2∪C3∪C4∪C5∪C6;2.27UMij,m+1Kαij,m=Kαij,mUMij,m+1,1⩽i<j⩽n;2.28UMij,mKαrs,m′=Kαrs,m′UMij,m,m−m′⩾2;2.29UMrs,mKαij,m′=Kαij,m′UMrs,m,m−m′⩾2;2.30UMij,mKαij,m′=Kαij,m′UMij,m,m−m′⩾2.Finally, by using the Auslander–Reiten quiver previously, we compute the relations(14)UB,nUA,n+1=∑M,N∈IsoAγABMNaAaB/aMaNqB^−M^,M^KB^−M^,n+1UN,n+1UM,n.To computeγMijMrsMN, we consider the exact sequence(15)0⟶M⟶Mrs⟶ϕMij⟶N⟶0.Note that since the Cartan matrixC is symmetric here, so AutM=q−1, for each indecomposable module M.Ifij,rs∈C1, then dimkHomMrs,Mij=1, and we let f be the basis of dimkHomMrs,Mij. Then, from the Auslander–Reiten quiver, we know that f is a right almost split epimorphism, so N=cokerf=0, and the kernel of f is indecomposable. By direct computation, we know that kerf=M=Mjs. So, if ϕ=f, then(16)γMijMrsMjs0=AutMjsAutMijAutMrs∑LgMjsLMrsgL0MijAutL=AutMjsAutMijAutMijAutMrsgMjsMijMrs=gMjsMijMrs.From [19], we know that gMjsMrjMrs=1, so γMijMrsMjs0=1.Ifϕ=0, then M≅Mrs and N≅Mij. So(17)γMijMrsMrsMij=AutMrsAutMijAutMijAutMrs∑LgMrsLMrsgLMijMijAutL=1.Hence,(18)2.31UMrs,mUMij,m+1=UMij,m+1UMrs,m+1−v−2Kαrj,m+1UMjs,m,ifijrs∈C1.In a similar way, we get the relations(19)2.32UMij,mUMrs,m+1=UMrs,m+1UMij,m,ifijrs∈C1∪C2∪C3∪C4∪C5∪C6;2.33UMij,mUMij,m+1=UMij,m+1UMij,m+v2−1Kαij,m+1,if1⩽i<j⩽n;2.34UMrs,mUMij,m+1=UMij,m+1UMrs,m+v2−1Kαrs,m+1UMir,m+1,ifijrs∈C3;2.35UMrs,mUMij,m+1=UMij,m+1UMrs,m,ifijrs∈C2∪C5∪C6;2.36UMrs,mUMij,m+1=UMij,m+1UMrs,m+1−v−2Kαrj,m+1UMir,m+1UMjs,m,ifijrs∈C4.We set(20)s1=UMrs,mUMij,m−v−2UMij,mUMrs,m,ifij,rs∈C1∪C3,s2=UMrs,mUMij,m−UMij,mUMrs,m,ifij,rs∈C2∪C6,s3=UMrs,mUMij,m−v−2UMij,mUMrs,m+1−v−2UMis,mUMrj,m,ifij,rs∈C4,s4=UMrs,mUMij,m−UMij,mUMrs,m+v2−1UMis,m,ifij,rs∈C5,t5=UMij,mUMrs,m+1−UMrs,m+1UMij,m,ij,rs∈C1∪C2∪C3∪C4∪C5∪C6,t6=UMij,mUMij,m+1−UMij,m+1UMij,m−v2−1Kαij,m+1,1⩽i<j⩽n,t7=UMrs,mUMij,m+1−UMij,m+1UMrs,m−1−v−2Kαrj,m+1UMjs,m,ij,rs∈C1,t8=UMrs,mUMij,m+1−UMij,m+1UMrs,m−v2−1Kαrs,m+1UMir,m+1,ij,rs∈C3,t9=UMrs,mUMij,m+1−UMij,m+1UMrs,m,ij,rs∈C2∪C5∪C6,t10=UMrs,mUMij,m+1−UMij,m+1UMrs,m−1−v−2Kαrj,m+1UMir,m+1UMjs,m,ij,rs∈C4,t11=UMrs,mUMij,m′−UMij,m′UMrs,m,m−m′⩾2,k12=Kαrs,mKαij,m−v2Kαij,mKαrs,m,ij,rs∈C1∪C3∪C5,k13=Kαrs,mKαij,m−Kαij,mKαrs,m,ij,rs∈C2∪C6,k14=Kαrs,mKαij,m−v4Kαij,mKαrs,m,ij,rs∈C4,k15=Kαij,mKαrs,m+1−v2Kαrs,m+1Kαij,m,ij,rs∈C1∪C3∪C4,k16=Kαij,mKαrs,m+1−Kαrs,m+1Kαij,m,ij,rs∈C2∪C5∪C6,k17=Kαrs,mKαij,m+1−Kαij,m+1Kαrs,m,ij,rs∈C1∪C2∪C3∪C6k18=Kαrs,mKαij,m+1−v−2Kαij,m+1Kαrs,m,ij,rs∈C4∪C5,k19=Kαrs,mKαij,m′−Kαij,m′Kαrs,m,m−m′⩾2,k20=Kαij,mKαij,m+1−v2Kαij,m+1Kαij,m,1⩽i<j⩽n,q21=UMij,mKαrs,m−Kαrs,mUMij,m,ij,rs∈C1∪C2∪C3∪C6,q22=UMij,mKαrs,m−v2Kαrs,mUMij,m,ij,rs∈C4∪C5,q23=UMrs,mKαij,m−v−2Kαij,mUMrs,m,ij,rs∈C1∪C3∪C4,q24=UMrs,mKαij,m−Kαij,mUMrs,m,ij,rs∈C2∪C5∪C6,q25=UMij,mKαij,m−v−2Kαij,mUMij,m,1⩽i<j⩽n,q26=UMij,mKαrs,m+1−v2Kαrs,m+1UMij,m,ij,rs∈C1∪C3∪C4,q27=UMij,mKαrs,m+1−Kαrs,m+1UMij,m,ij,rs∈C2∪C5∪C6,q28=UMrs,mKαij,m+1−Kαij,m+1UMrs,m,ij,rs∈C1∪C2∪C3∪C6,q29=UMrs,mKαij,m+1−v−2Kαij,m+1UMrs,m,ij,rs∈C4∪C5,q30=UMij,mKαij,m+1−v2Kαij,m+1UMij,m,1⩽i<j⩽n,q31=UMij,m+1Kαrs,m−Kαrs,mUMij,m+1,ij,rs∈C1∪C2∪C3∪C4∪C5∪C6,q32=UMrs,m+1Kαij,m−Kαij,mUMrs,m+1,ij,rs∈C1∪C2∪C3∪C4∪C5∪C6,q33=UMij,m+1Kαij,m−Kαij,mUMij,m+1,1⩽i<j⩽n,q34=UMij,mKαrs,m′−Kαrs,m′UMij,m,m−m′⩾2,q35=UMrs,mKαij,m′−Kαij,m′UMrs,m,m−m′⩾2,q36=UMij,mKαij,m′−Kαij,m′UMij,m,m−m′⩾2.We set(21)S=s1,s2,s3,s4,T=t5,…,t11,K=k12,…,k20,Q=q21,…,q36.


## 4. The Gröbner–Shirshov Basis of the Modified Ringel–Hall AlgebraDℳℋ

In this section, we prove that the setS1≔S∪K∪T∪Q is a minimal Gröbner–Shirshov basis of the modified Ringel–Hall algebra Dℳℋ. We denote by IndecisoAn the set of isoclasses of indecomposable representations of the quiver of type An.Let(22)X=UMij,m,Kαij,m|Mij∈IndecisoAn,αij=M^ij,1⩽i<j⩽n+1,m∈Z,then(23)Dℳℋ=KXIdS1.We define a well-ordering〈X on X as follows: for any 1⩽i<j⩽n+1,1⩽r<s⩽n+1, define(24)UMij,m<XUMrs,m⇔ij<rslexicographicalyand for anym∈ZUMij,m<XUMrs,m′⇔m′<mand for any1⩽i<j⩽n+1,1⩽r<s⩽n+1,Kαij,m<XKαrs,m⇔ij<rslexicographicalyand for anym∈Z,Kαij,m<XKαrs,m′⇔m′<mand for any1⩽i<j⩽n+1,1⩽r<s⩽n+1,Kαij−<XKαrs−⇔ij<rslexicographicaly,ωMij−<XωMrs−⇔ij<rslexicographicaly,Kαrs,m′<XUMrs,mfor anym,m′∈Zand for any1⩽i<j⩽n+1,1⩽r<s⩽n+1.Then, this ordering induces a deg-lex ordering< on X∗, the free semigroup generated by X.Now, we state the main result in this paper.Theorem 1.
The setS1 is a minimal Gröbner–Shirshov basis of the modified Ringel–Hall algebra Dℳℋ.Proof.
To prove the theorem, we had to compute all the possible compositions between the elements ofS1.(i)
Let(25)s1=UMrs,mUMij,m−υ−2UMij,mUMrs,m∈S,ij,rs∈C1∪C3;,t5=UMij,mUMkl,m+1−UMkl,m+1UMij,m∈T,ij,kl∈C1∪C2∪C3∪C4∪C5∪C6;, then(26)s¯1=UMrs,mUMij,m,t¯5=UMij,mUMkl,m+1,andω=UMrs,mUMij,mUMkl,m+1.So(27)s1,t5ω=−υ−2UMij,mUMrs,mUMkl,m+1+UMrs,mUMkl,m+1UMij,m.We consider several different cases.Ifij,rs∈C1 and ij,kl∈C1 and k=i=r<j<l<s, then(28)s1,t5ω≡−υ−2UMij,mUMkl,m+1UMrs,m−υ−21−υ−2UMij,mKαrl,m+1UMls,m+UMkl,m+1UMrs,mUMij,m+1−υ−2Kαrl,m+1UMls,mUMij,m≡−υ−2UMkl,m+1UMij,mUMrs,m−1−υ−2Kαrl,m+1UMij,mUMls,m+υ−2UMkl,m+1UMij,mUMrs,m+1−υ−2Kαrl,m+1UMij,mUMls,m≡0modS1;ω.Ifij,rs∈C1 and ij,kl∈C1 and k=i=r<j<l=s, then(29)s1,t5ω≡−υ−2UMij,mUMkl,m+1UMrs,m−υ−2υ2−1UMij,mKαrs,m+1+UMkl,m+1UMrs,mUMij,m+υ2−1Kαrs,m+1UMij,m≡−υ−2UMkl,m+1UMij,mUMrs,m−υ2−1Kαrs,m+1UMij,m+υ−2UMkl,m+1UMij,mUMrs,m+υ2−1Kαrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C1 and ij,kl∈C1 and k=i=r<j<s<l, then(30)s1,t5ω≡−υ−2UMij,mUMkl,m+1UMrs,m+UMkl,m+1UMrs,mUMij,m≡−υ−2UMkl,m+1UMij,mUMrs,m+υ−2UMkl,m+1UMij,mUMrs,m≡0modS1;ω.Ifij,rs∈C1 and ij,kl∈C2 and i=r<k<l<j<s, then(31)s1,t5ω≡−υ−2UMij,mUMkl,m+1UMrs,m+UMkl,m+1UMrs,mUMij,m≡−υ−2UMkl,m+1UMij,mUMrs,m+υ−2UMkl,m+1UMij,mUMrs,m≡0modS1;ω.Ifij,rs∈C1 and ij,kl∈C3 and i=r<k<l=j<s, then(32)s1,t5ω≡−υ−2UMij,mUMkl,m+1UMrs,m+UMkl,m+1UMrs,mUMij,m≡−υ−2UMkl,m+1UMij,mUMrs,m+υ−2UMkl,m+1UMij,mUMrs,m≡0modS1;ω.Ifij,rs∈C1 and ij,kl∈C4 and i=r<k<j<l<s, then(33)s1,t5ω≡−υ−2UMij,mUMkl,m+1UMrs,m+UMkl,m+1UMrs,mUMij,m≡−υ−2UMkl,m+1UMij,mUMrs,m+υ−2UMkl,m+1UMij,mUMrs,m≡0modS1;ω.Ifij,rs∈C1 and ij,kl∈C4 and i=r<k<j<l=s, then(34)s1,t5ω≡−υ−2UMij,mUMkl,m+1UMrs,m+UMkl,m+1UMrs,mUMij,m≡−υ−2UMkl,m+1UMij,mUMrs,m+υ−2UMkl,m+1UMij,mUMrs,m≡0modS1;ω.Ifij,rs∈C1 and ij,kl∈C4 and i=r<k<j<s<l, then(35)s1,t5ω≡−υ−2UMij,mUMkl,m+1UMrs,m+UMkl,m+1UMrs,mUMij,m≡−υ−2UMkl,m+1UMij,mUMrs,m+υ−2UMkl,m+1UMij,mUMrs,m≡0modS1;ω.Ifij,rs∈C1 and ij,kl∈C5 and i=r<k=j<l<s, then(36)s1,t5ω≡−υ−2UMij,mUMkl,m+1UMrs,m+UMkl,m+1UMrs,mUMij,m≡−υ−2UMkl,m+1UMij,mUMrs,m+υ−2UMkl,m+1UMij,mUMrs,m≡0modS1;ω.Ifij,rs∈C1 and ij,kl∈C5 and i=r<k=j<l=s, then(37)s1,t5ω≡−υ−2UMij,mUMkl,m+1UMrs,m+UMkl,m+1UMrs,mUMij,m≡−υ−2UMkl,m+1UMij,mUMrs,m+υ−2UMkl,m+1UMij,mUMrs,m≡0modS1;ω.Ifij,rs∈C1 and ij,kl∈C5 and i=r<k=j<s<l, then(38)s1,t5ω≡−υ−2UMij,mUMkl,m+1UMrs,m+UMkl,m+1UMrs,mUMij,m≡−υ−2UMkl,m+1UMij,mUMrs,m+υ−2UMkl,m+1UMij,mUMrs,m≡0modS1;ω.Ifij,rs∈C1 and ij,kl∈C6 and i=r<j<s<k<l, then(39)s1,t5ω≡−υ−2UMij,mUMkl,m+1UMrs,m+UMkl,m+1UMrs,mUMij,m≡−υ−2UMkl,m+1UMij,mUMrs,m+υ−2UMkl,m+1UMij,mUMrs,m≡0modS1;ω.Ifij,rs∈C1 and ij,kl∈C6 and i=r<j<s=k<l, then(40)s1,t5ω≡−υ−2UMij,mUMkl,m+1UMrs,m+UMkl,m+1UMrs,mUMij,m≡−υ−2UMkl,m+1UMij,mUMrs,m+υ−2UMkl,m+1UMij,mUMrs,m≡0modS1;ω.Ifij,rs∈C1 and ij,kl∈C6 and i=r<j<k<s<l, then(41)s1,t5ω≡−υ−2UMij,mUMkl,m+1UMrs,m+UMkl,m+1UMrs,mUMij,m≡−υ−2UMkl,m+1UMij,mUMrs,m+υ−2UMkl,m+1UMij,mUMrs,m≡0modS1;ω.Ifij,rs∈C1 and ij,kl∈C6 and i=r<j<k<s=l, then(42)s1,t5ω≡−υ−2UMij,mUMkl,m+1UMrs,m+UMkl,m+1UMrs,mUMij,m≡−υ−2UMkl,m+1UMij,mUMrs,m+υ−2UMkl,m+1UMij,mUMrs,m≡0modS1;ω.Ifij,rs∈C1 and ij,kl∈C6 and i=r<j<k<l<s, then(43)s1,t5ω≡−υ−2UMij,mUMkl,m+1UMrs,m+UMkl,m+1UMrs,mUMij,m≡−υ−2UMkl,m+1UMij,mUMrs,m+υ−2UMkl,m+1UMij,mUMrs,m≡0modS1;ω.Ifij,rs∈C3 and ij,kl∈C1 and i=k<r<j=s<l, then(44)s1,t5ω≡−υ−2UMij,mUMkl,m+1UMrs,m+UMkl,m+1UMrs,mUMij,m≡−υ−2UMkl,m+1UMij,mUMrs,m+υ−2UMkl,m+1UMij,mUMrs,m≡0modS1;ω.Ifij,rs∈C3 and ij,kl∈C2 and i<r<k<l<j=s, then(45)s1,t5ω≡−υ−2UMij,mUMkl,m+1UMrs,m+UMkl,m+1UMrs,mUMij,m≡−υ−2UMkl,m+1UMij,mUMrs,m+υ−2UMkl,m+1UMij,mUMrs,m≡0modS1;ω.Ifij,rs∈C3 and ij,kl∈C2 and i<k=r<l<j=s, then(46)s1,t5ω≡−υ−2UMij,mUMkl,m+1UMrs,m−υ−21−υ−2UMij,mKαrl,m+1UMls,m+UMkl,m+1UMrs,mUMij,m+1−υ−2Kαrl,m+1UMls,mUMij,m≡−υ−2UMkl,m+1UMij,mUMrs,m−υ−21−υ−2Kαrl,m+1UMij,mUMls,m+υ−2UMkl,m+1UMij,mUMrs,m+υ−21−υ−2Kαrl,m+1UMij,mUMls,m≡0modS1;ω.Ifij,rs∈C3 and ij,kl∈C2 and i<k<r<l<j=s, then(47)s1,t5ω≡−υ−2UMij,mUMkl,m+1UMrs,m−υ−21−υ−2UMij,mKαrl,m+1UMkr,m+1UMls,m+UMkl,m+1UMrs,mUMij,m+1−υ−2Kαrl,m+1UMkr,m+1UMls,mUMij,m≡−υ−2UMkl,m+1UMij,mUMrs,m−υ−21−υ−2Kαrl,m+1UMij,mUMkr,m+1UMls,m+υ−2UMkl,m+1UMij,mUMrs,m+υ−21−υ−2Kαrl,m+1UMkr,m+1UMij,mUMls,m≡−υ−21−υ−2Kαrl,m+1UMkr,m+1UMij,mUMls,m≡0modS1;ω.Ifij,rs∈C3 and ij,kl∈C2 and i<k<r=l<j=s, then(48)s1,t5ω≡−υ−2UMij,mUMkl,m+1UMrs,m+UMkl,m+1UMrs,mUMij,m≡−υ−2UMkl,m+1UMij,mUMrs,m+υ−2UMkl,m+1UMij,mUMrs,m≡0modS1;ω.Ifij,rs∈C3 and ij,kl∈C2 and i<k<l<r<j=s, then(49)s1,t5ω≡−υ−2UMij,mUMkl,m+1UMrs,m+UMkl,m+1UMrs,mUMij,m≡−υ−2UMkl,m+1UMij,mUMrs,m+υ−2UMkl,m+1UMij,mUMrs,m≡0modS1;ω.Ifij,rs∈C3 and ij,kl∈C3 and i<r<k<l=j=s, then(50)s1,t5ω≡−υ−2UMij,mUMkl,m+1UMrs,m+UMkl,m+1UMrs,mUMij,m≡−υ−2UMkl,m+1UMij,mUMrs,m+υ−2UMkl,m+1UMij,mUMrs,m≡0modS1;ω.Ifij,rs∈C3 and ij,kl∈C3 and i<k=r<l=j=s, then(51)s1,t5ω≡−υ−2UMij,mUMkl,m+1UMrs,m−υ−2υ2−1UMij,mKαrs,m+1+UMkl,m+1UMrs,mUMij,m+υ2−1Kαrs,m+1UMij,m≡−υ−2UMkl,m+1UMij,mUMrs,m−υ2−1Kαrs,m+1UMij,m+υ−2UMkl,m+1UMij,mUMrs,m+υ2−1Kαrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C3 and ij,kl∈C3 and i<k<r<l=j=s, then(52)s1,t5ω≡−υ−2UMij,mUMkl,m+1UMrs,m−υ−2υ2−1UMij,mKαrs,m+1UMkr,m+1+UMkl,m+1UMrs,mUMij,m+υ2−1Kαrs,m+1UMkr,m+1UMij,m≡−υ−2UMkl,m+1UMij,mUMrs,m−υ2−1Kαrs,m+1UMij,mUMkr,m+1+υ−2UMkl,m+1UMij,mUMrs,m+υ2−1Kαrs,m+1UMkr,m+1UMij,m≡−υ2−1Kαrs,m+1UMkr,m+1UMij,m+υ2−1Kαrs,m+1UMkr,m+1UMij,m≡0modS1;ω.Ifij,rs∈C3 and ij,kl∈C4 and i<r<k<j=s<l, then(53)s1,t5ω≡−υ−2UMij,mUMkl,m+1UMrs,m+UMkl,m+1UMrs,mUMij,m≡−υ−2UMkl,m+1UMij,mUMrs,m+υ−2UMkl,m+1UMij,mUMrs,m≡0modS1;ω.Ifij,rs∈C3 and ij,kl∈C4 and i<r=k<j=s<l, then(54)s1,t5ω≡−υ−2UMij,mUMkl,m+1UMrs,m+UMkl,m+1UMrs,mUMij,m≡−υ−2UMkl,m+1UMij,mUMrs,m+υ−2UMkl,m+1UMij,mUMrs,m≡0modS1;ω.Ifij,rs∈C3 and ij,kl∈C4 and i<k<r<j=s<l, then(55)s1,t5ω≡−υ−2UMij,mUMkl,m+1UMrs,m+UMkl,m+1UMrs,mUMij,m≡−υ−2UMkl,m+1UMij,mUMrs,m+υ−2UMkl,m+1UMij,mUMrs,m≡0modS1;ω.Ifij,rs∈C3 and ij,kl∈C5 and i<r<k=j=s<l, then(56)s1,t5ω≡−υ−2UMij,mUMkl,m+1UMrs,m+UMkl,m+1UMrs,mUMij,m≡−υ−2UMkl,m+1UMij,mUMrs,m+υ−2UMkl,m+1UMij,mUMrs,m≡0modS1;ω.Ifij,rs∈C3 and ij,kl∈C6 and i<r<j=s<k<l, then(57)s1,t5ω≡−υ−2UMij,mUMkl,m+1UMrs,m+UMkl,m+1UMrs,mUMij,m≡−υ−2UMkl,m+1UMij,mUMrs,m+υ−2UMkl,m+1UMij,mUMrs,m≡0modS1;ω.(ii)
Let(58)t5=UMij,mUMrs,m+1−UMrs,m+1UMij,m∈T,ij,rs∈C1∪C2∪C3∪C4∪C5∪C6;,q31=UMrs,m+1Kαkl,m−Kαkl,mUMrs,m+1∈Q,rs,kl∈C1∪C2∪C3∪C4∪C5∪C6;, then(59)t¯5=UMij,mUMrs,m+1,q¯31=UMrs,m+1Kαkl,m,andω=UMij,mUMrs,m+1Kαkl,m.So(60)t5,q31ω=−UMrs,m+1UMij,mKαkl,m+UMij,mKαkl,mUMrs,m+1.Again, we consider several different cases.Ifij,rs∈C1 and rs,kl∈C1 and i=r=k<j<s<l, then(61)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C1 and rs,kl∈C2 and i=r<j<k<l<s, then(62)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C1 and rs,kl∈C2 and i=r<k=j<l<s, then(63)t5,q31ω≡−υ2UMrs,m+1Kαkl,mUMij,m+υ2Kαkl,mUMij,mUMrs,m+1≡−υ2Kαkl,mUMrs,m+1UMij,m+υ2Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C1 and rs,kl∈C2 and i=r<k<j<l<s, then(64)t5,q31ω≡−υ2UMrs,m+1Kαkl,mUMij,m+υ2Kαkl,mUMij,mUMrs,m+1≡−υ2Kαkl,mUMrs,m+1UMij,m+υ2Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C1 and rs,kl∈C2 and i=r<k<j=l<s, then(65)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C1 and rs,kl∈C2 and i=r<k<l<j<s, then(66)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C1 and rs,kl∈C3 and i=r<j<k<l=s, then(67)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C1 and rs,kl∈C3 and i=r<k=j<l=s, then(68)t5,q31ω≡−υ2UMrs,m+1Kαkl,mUMij,m+υ2Kαkl,mUMij,mUMrs,m+1≡−υ2Kαkl,mUMrs,m+1UMij,m+υ2Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C1 and rs,kl∈C3 and i=r<k<j<l=s, then(69)t5,q31ω≡−υ2UMrs,m+1Kαkl,mUMij,m+υ2Kαkl,mUMij,mUMrs,m+1≡−υ2Kαkl,mUMrs,m+1UMij,m+υ2Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C1 and rs,kl∈C4 and i=r<j<k<s<l, then(70)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C1 and rs,kl∈C4 and i=r<k=j<s<l, then(71)t5,q31ω≡−υ2UMrs,m+1Kαkl,mUMij,m+υ2Kαkl,mUMij,mUMrs,m+1≡−υ2Kαkl,mUMrs,m+1UMij,m+υ2Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C1 and rs,kl∈C4 and i=r<k<j<s<l, then(72)t5,q31ω≡−υ2UMrs,m+1Kαkl,mUMij,m+υ2Kαkl,mUMij,mUMrs,m+1≡−υ2Kαkl,mUMrs,m+1UMij,m+υ2Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C1 and rs,kl∈C5 and i=r<j<k=s<l, then(73)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C1 and rs,kl∈C6 and i=r<j<s<k<l, then(74)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C2 and rs,kl∈C1 and i<r=k<s<l<j, then(75)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C2 and rs,kl∈C1 and i<r=k<s<l=j, then(76)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C2 and rs,kl∈C1 and i<r=k<s<j<l, then(77)t5,q31ω≡−υ2UMrs,m+1Kαkl,mUMij,m+υ2Kαkl,mUMij,mUMrs,m+1≡−υ2Kαkl,mUMrs,m+1UMij,m+υ2Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C2 and rs,kl∈C2 and i<r<k<l<s<j, then(78)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C2 and rs,kl∈C3 and i<r<k<l=s<j, then(79)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C2 and rs,kl∈C4 and i<r<k<s<j<l, then(80)t5,q31ω≡−υ2UMrs,m+1Kαkl,mUMij,m+υ2Kαkl,mUMij,mUMrs,m+1≡−υ2Kαkl,mUMrs,m+1UMij,m+υ2Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C2 and rs,kl∈C4 and i<r<k<s<j=l, then(81)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C2 and rs,kl∈C4 and i<r<k<s<l<j, then(82)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C2 and rs,kl∈C5 and i<r<k=s<l<j, then(83)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C2 and rs,kl∈C5 and i<r<k=s<l=j, then(84)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C2 and rs,kl∈C5 and i<r<k=s<j<l, then(85)t5,q31ω≡−υ2UMrs,m+1Kαkl,mUMij,m+υ2Kαkl,mUMij,mUMrs,m+1≡−υ2Kαkl,mUMrs,m+1UMij,m+υ2Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C2 and rs,kl∈C6 and i<r<s<j<k<l, then(86)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1+≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C2 and rs,kl∈C6 and i<r<s<j=k<l, then(87)t5,q31ω≡−υ2UMrs,m+1Kαkl,mUMij,m+υ2Kαkl,mUMij,mUMrs,m+1≡−υ2Kαkl,mUMrs,m+1UMij,m+υ2Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C2 and rs,kl∈C6 and i<r<s<k<j<l, then(88)t5,q31ω≡−υ2UMrs,m+1Kαkl,mUMij,m+υ2Kαkl,mUMij,mUMrs,m+1≡−υ2Kαkl,mUMrs,m+1UMij,m+υ2Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C2 and rs,kl∈C6 and i<r<s<k<j=l, then(89)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C2 and rs,kl∈C6 and i<r<s<k<l<j, then(90)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C3 and rs,kl∈C1 and i<r=k<s=j<l, then(91)t5,q31ω≡−υ2UMrs,m+1Kαkl,mUMij,m+υ2Kαkl,mUMij,mUMrs,m+1≡−υ2Kαkl,mUMrs,m+1UMij,m+υ2Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C3 and rs,kl∈C2 and i<r<k<l<s=j, then(92)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C3 and rs,kl∈C3 and i<r<k<l=s=j, then(93)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C3 and rs,kl∈C4 and i<r<k<s=j<l, then(94)t5,q31ω≡−υ2UMrs,m+1Kαkl,mUMij,m+υ2Kαkl,mUMij,mUMrs,m+1≡−υ2Kαkl,mUMrs,m+1UMij,m+υ2Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C3 and rs,kl∈C5 and i<r<k=s=j<l, then(95)t5,q31ω≡−υ2UMrs,m+1Kαkl,mUMij,m+υ2Kαkl,mUMij,mUMrs,m+1≡−υ2Kαkl,mUMrs,m+1UMij,m+υ2Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C3 and rs,kl∈C6 and i<r<s=j<k<l, then(96)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C4 and rs,kl∈C1 and i<r=k<j<s<l, then(97)t5,q31ω≡−υ2UMrs,m+1Kαkl,mUMij,m+υ2Kαkl,mUMij,mUMrs,m+1≡−υ2Kαkl,mUMrs,m+1UMij,m+υ2Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C4 and rs,kl∈C2 and i<r<j<k<l<s, then(98)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C4 and rs,kl∈C2 and i<r<j=k<l<s, then(99)t5,q31ω≡−υ2UMrs,m+1Kαkl,mUMij,m+υ2Kαkl,mUMij,mUMrs,m+1≡−υ2Kαkl,mUMrs,m+1UMij,m+υ2Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C4 and rs,kl∈C2 and i<r<k<j<l<s, then(100)t5,q31ω≡−υ2UMrs,m+1Kαkl,mUMij,m+υ2Kαkl,mUMij,mUMrs,m+1≡−υ2Kαkl,mUMrs,m+1UMij,m+υ2Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C4 and rs,kl∈C2 and i<r<k<j=l<s, then(101)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C4 and rs,kl∈C2 and i<r<k<l<j<s, then(102)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C4 and rs,kl∈C3 and i<r<k<j<l=s, then(103)t5,q31ω≡−υ2UMrs,m+1Kαkl,mUMij,m+υ2Kαkl,mUMij,mUMrs,m+1≡−υ2Kαkl,mUMrs,m+1UMij,m+υ2Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C4 and rs,kl∈C3 and i<r<k=j<l=s, then(104)t5,q31ω≡−υ2UMrs,m+1Kαkl,mUMij,m+υ2Kαkl,mUMij,mUMrs,m+1≡−υ2Kαkl,mUMrs,m+1UMij,m+υ2Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C4 and rs,kl∈C3 and i<r<j<k<l=s, then(105)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C4 and rs,kl∈C4 and i<r<k<j<s<l, then(106)t5,q31ω≡−υ2UMrs,m+1Kαkl,mUMij,m+υ2Kαkl,mUMij,mUMrs,m+1≡−υ2Kαkl,mUMrs,m+1UMij,m+υ2Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C4 and rs,kl∈C4 and i<r<k=j<s<l, then(107)t5,q31ω≡−υ2UMrs,m+1Kαkl,mUMij,m+υ2Kαkl,mUMij,mUMrs,m+1≡−υ2Kαkl,mUMrs,m+1UMij,m+υ2Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C4 and rs,kl∈C4 and i<r<j<k<s<l, then(108)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C4 and rs,kl∈C5 and i<r<j<k=s<l, then(109)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C4 and rs,kl∈C6 and i<r<j<s<k<l, then(110)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C5 and rs,kl∈C1 and i<r=k=j<s<l, then(111)t5,q31ω≡−υ2UMrs,m+1Kαkl,mUMij,m+υ2Kαkl,mUMij,mUMrs,m+1≡−υ2Kαkl,mUMrs,m+1UMij,m+υ2Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C5 and rs,kl∈C2 and i<r=j<k<l<s, then(112)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C5 and rs,kl∈C3 and i<r=j<k<l=s, then(113)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C5 and rs,kl∈C4 and i<r=j<k<s<l, then(114)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C5 and rs,kl∈C5 and i<r=j<k=s<l, then(115)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C5 and rs,kl∈C6 and i<r=j<s<k<l, then(116)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C6 and rs,kl∈C1 and i<j<r=k<s<l, then(117)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C6 and rs,kl∈C2 and i<j<r<k<l<s, then(118)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C6 and rs,kl∈C3 and i<j<r<k<l=s, then(119)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C6 and rs,kl∈C4 and i<j<r<k<s<l, then(120)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C6 and rs,kl∈C5 and i<j<r<k=s<l, then(121)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.Ifij,rs∈C6 and rs,kl∈C6 and i<j<r<s<k<l, then(122)t5,q31ω≡−UMrs,m+1Kαkl,mUMij,m+Kαkl,mUMij,mUMrs,m+1≡−Kαkl,mUMrs,m+1UMij,m+Kαkl,mUMrs,m+1UMij,m≡0modS1;ω.
By similar computations, we know that all the possible compositions between the elements ofS1 are trivial. To save space, we omitted the details.
Now, it is easy to see that(123)IrrS1=∏r=t′−tKαi1j1,rl1⋯Kαiaja,rla∏s=m′−mUMp1q1,sk1⋯UMpbqb,skb|m,t,a,b,t′,m′∈Z⩾0l1,…,la,k1,…,kb∈Z⩾0,i1j1<⋯<iaja,p1q1<⋯<pbqbiw<jw,pv<qv,w=1,…,a,v=1,…,b.
From Lemma1, we know the following:Corollary 1.
The setIrrS1 is a PBW basis of the modified Ringel–Hall algebra Dℳℋ.



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*Source: 6820173-2023-03-14.xml*