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Hu, Desheng; Obul, Abdukadir

doi:https://doi.org/10.1155/2023/6820173

Journal of Mathematics

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Abstract Introduction Preliminaries Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 6820173 | https://doi.org/10.1155/2023/6820173

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

Desheng Hu¹and Abdukadir Obul¹

Academic Editor: Naihuan Jing

Received26 Sept 2022

Revised08 Dec 2022

Accepted02 Mar 2023

Published14 Mar 2023

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 type 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 .

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 and defined an algebra (now) called the Bridgeland Hall algebra of . He proved that Ringel–Hall algebra of 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 , 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 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 and are constructed in detail. In this paper, we generalize this result to the modified Ringel–Hall algebra of type . 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 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 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].

Let be a well-ordered nonempty set, the well-ordering on and the free semigroup generated by and assuming that . Let be a field and denote by the free associative -algebra generated by . Then, the set is a -linear basis of . For any , we denote by the length of , i.e., the number of elements contained in . The well-ordering on the set induces a deg-lex ordering on , i.e., for any two elements , we define if and only if or and . It is clear that is a monomial ordering, i.e., compatible with the multiplication of . We denote by the leading monomial of a polynomial with respect to the ordering , and if the coefficient of is 1, then the polynomial is called monic.

For any two monic polynomials and in , we define the composition as follows:(1)If there are such that , and , then is called the composition of intersection of and with respect to (2)If for some unique elements and in , then is called the composition of inclusion of and with respect to

The composition of intersection and the composition of inclusion are referred to as the compositions.

Let be a nonempty set of monic polynomials. For any , if , where and , then we say is trivial modulo and write

For any , if , where and , then we write

We denote by the ideal of generated by .

Definition 1. If all the compositions among the polynomials in are trivial modulo , then we call a Gröbner–Shirshov basis in of the ideal . If there is no composition of inclusion, then 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]). Let be nonempty set of monic polynomials, the monomial ordering on . The following statements are equivalent:(1) is a Gröbner–Shirshov basis(2)If , then there are and such that (3) is a -linear basis of the factor algebra

2.2. The Modified Ringel–Hall Algebra

Now, we recall some notions about the modified Ringel–Hall algebra from [14].

Let be a finite field with elements, and we set , and a finitary hereditary abelian -linear category (for definition see [6]). We denote by and the set of isoclasses of objects in and the Grothendieck group of , respectively. For any object in , we denote by the isoclass of ; that is , and for a finite set , we denote by the cardinality of . For each object in , the class of in is denoted by , and the automorphism group of is denoted by .

For any three objects in , we let be the number of subobjects of such that and . For any two objects , we define

Then, it induces a bilinear formknown as the Euler form. We define the symmetric Euler formas for all .

For four objects in , we let be the set of exact sequences

Then, is a finite set, and from [16], we have

From [14], the modified Ringel–Hall algebra is an associative algebra with 1 generated bysubject to the relations

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

From now on, we assume that is the category of finite-dimensional representations of the quiver of type . In this section, we compute all the commutator relations between the generators (where is an isoclass of indecomposable representation of the quiver of type ) and (where is the standard unit vectors in ).

We choose the following orientation for the Dynkin graph of type :

We know that the Cartan matrix and the Auslander–Reiten quiver of areand

respectively, where is the following indecomposable representation or, equivalently, is the factor representation , where is the indecomposable projective representation corresponding to the vertex .

For convenience, from [17], we set

By [18, 19] and making some modifications if necessary, we get the relations

By direct computation, we also have following formulas:

Finally, by using the Auslander–Reiten quiver previously, we compute the relations

To compute , we consider the exact sequence

Note that since the Cartan matrix is symmetric here, so , for each indecomposable module .

If , then , and we let be the basis of . Then, from the Auslander–Reiten quiver, we know that is a right almost split epimorphism, so , and the kernel of is indecomposable. By direct computation, we know that . So, if , then

From [19], we know that , so .

If , then and . So

Hence,

In a similar way, we get the relations

We set

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

In this section, we prove that the set is a minimal Gröbner–Shirshov basis of the modified Ringel–Hall algebra . We denote by the set of isoclasses of indecomposable representations of the quiver of type .

Letthen

We define a well-ordering on as follows: for any , define

Then, this ordering induces a deg-lex ordering on , the free semigroup generated by .

Now, we state the main result in this paper.

Theorem 1. The set is a minimal Gröbner–Shirshov basis of the modified Ringel–Hall algebra .

Proof. To prove the theorem, we had to compute all the possible compositions between the elements of .(i)Let then So We consider several different cases. If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then(ii)Let then So Again, we consider several different cases. If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , then If and and , thenBy similar computations, we know that all the possible compositions between the elements of are trivial. To save space, we omitted the details.
Now, it is easy to see thatFrom Lemma 1, we know the following:

Corollary 1. The set is a PBW basis of the modified Ringel–Hall algebra .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was funded by the National Natural Science Foundation of China (Grant no. 11861061).

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Copyright

Copyright © 2023 Desheng Hu and Abdukadir Obul. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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