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学者姓名：杨士林
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Abstract ：
In this paper, we discuss the group consisting of some good automorphisms of representation ring of the non-pointed Hopf algebra D(n)D(n), the quotient of the non-pointed prime Hopf algebras of GK-dimension one, which is generated by x,y,zx,y,z with the relations: x(2)n=1,y(2)=0,z(2)=xn,xy=-yx, xz=zx(-1), yz=omega zy, where omega omega is a 44th primitive root of unity. First, we describe the group formed by permutations of the isomorphism classes of indecomposable modules of D(n)D(n), which can be extended to automorphisms of the representation ring of D(n)D(n). An element within this group is regarded as a permutation of the set of points of AR-quiver of D(n)D(n) such that the tensor product of indecomposable modules corresponding to these points are isomorphic. Then, we try to understand automorphism group of representation ring of D(n)D(n). By straightforward computation, it is shown that the automorphism group of the representation ring of D(3)D(3) is isomorphic to Klein four group K4K4.
Keyword ：
Automorphism group Automorphism group representation ring representation ring Hopf algebra Hopf algebra permutation permutation
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| GB/T 7714 | Wang, Wanxia , Yang, Shilin . Automorphism groups associated to a family of Hopf algebras [J]. | JOURNAL OF ALGEBRA AND ITS APPLICATIONS , 2025 . |
| MLA | Wang, Wanxia 等. "Automorphism groups associated to a family of Hopf algebras" . | JOURNAL OF ALGEBRA AND ITS APPLICATIONS (2025) . |
| APA | Wang, Wanxia , Yang, Shilin . Automorphism groups associated to a family of Hopf algebras . | JOURNAL OF ALGEBRA AND ITS APPLICATIONS , 2025 . |
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Abstract ：
The aim of this paper is to characterize the representation ring of a non-pointed and noncocommutative bialgebra. First, the isomorphism classes of its indecomposable modules are classified. Then the tensor product of modules is established. Finally, its representation ring is described.
Keyword ：
representation ring representation ring bound quiver bound quiver bialgebra bialgebra
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| GB/T 7714 | Gong, Huaqing , Yang, Shilin . The representation ring of a non-pointed bialgebra [J]. | AIMS MATHEMATICS , 2025 , 10 (3) : 5110-5123 . |
| MLA | Gong, Huaqing 等. "The representation ring of a non-pointed bialgebra" . | AIMS MATHEMATICS 10 . 3 (2025) : 5110-5123 . |
| APA | Gong, Huaqing , Yang, Shilin . The representation ring of a non-pointed bialgebra . | AIMS MATHEMATICS , 2025 , 10 (3) , 5110-5123 . |
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Abstract ：
In this paper, all string modules of one class of basic Hopf algebras of tame type, which are denoted by H2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {H}_2$$\end{document} are studied. Firstly, the isomorphism classes of all string modules of H2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {H}_2$$\end{document} are classified. Then the projective class ring of H2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {H}_2$$\end{document} is described. Finally, the McKay matrix and the corresponding McKay quiver of H2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {H}_2$$\end{document} are discussed.
Keyword ：
McKay matrix McKay matrix String modules String modules Basic Hopf algebra Basic Hopf algebra
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| GB/T 7714 | Lin, Shiyu , Yang, Shilin . Representations of Basic Hopf Algebras of Tame Type [J]. | BULLETIN OF THE IRANIAN MATHEMATICAL SOCIETY , 2024 , 50 (5) . |
| MLA | Lin, Shiyu 等. "Representations of Basic Hopf Algebras of Tame Type" . | BULLETIN OF THE IRANIAN MATHEMATICAL SOCIETY 50 . 5 (2024) . |
| APA | Lin, Shiyu , Yang, Shilin . Representations of Basic Hopf Algebras of Tame Type . | BULLETIN OF THE IRANIAN MATHEMATICAL SOCIETY , 2024 , 50 (5) . |
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Abstract ：
In this paper, we focus on studying two classes of finite dimensional triangle-associative algebras, which are extensions of a family of 2n2 2-dimensional Kac-Paljutkin type semi-simple Hopf algebras H2n2. 2n 2 . All their indecomposable modules are classified. Furthermore, their representation rings are described by generators with suitable relations.
Keyword ：
tensor product tensor product triangle -associative algebra triangle -associative algebra representation ring representation ring
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| GB/T 7714 | Su, Dong , Yang, Shilin . Representation rings of extensions of Hopf algebra of Kac-Paljutkin type [J]. | ELECTRONIC RESEARCH ARCHIVE , 2024 , 32 (9) : 5201-5230 . |
| MLA | Su, Dong 等. "Representation rings of extensions of Hopf algebra of Kac-Paljutkin type" . | ELECTRONIC RESEARCH ARCHIVE 32 . 9 (2024) : 5201-5230 . |
| APA | Su, Dong , Yang, Shilin . Representation rings of extensions of Hopf algebra of Kac-Paljutkin type . | ELECTRONIC RESEARCH ARCHIVE , 2024 , 32 (9) , 5201-5230 . |
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Abstract ：
In the paper, the algebra A (n), which is generated by an upper triangular generating matrix with triple relations, is introduced. It is shown that there exists an isomorphism between the algebra A (n) and the higher-rank Askey-Wilson algebra aw(n) introduced by Crampe et al. Furthermore, we establish a series of automorphisms of A (n), which satisfy braid group relations and coincide with those in aw(n).
Keyword ：
Askey-Wilson algebra Askey-Wilson algebra braid group braid group
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| GB/T 7714 | Wang, Wanxia , Yang, Shilin . On the Higher-Rank Askey-Wilson Algebras [J]. | SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS , 2024 , 20 . |
| MLA | Wang, Wanxia 等. "On the Higher-Rank Askey-Wilson Algebras" . | SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS 20 (2024) . |
| APA | Wang, Wanxia , Yang, Shilin . On the Higher-Rank Askey-Wilson Algebras . | SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS , 2024 , 20 . |
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Abstract ：
In this paper, all simple Yetter-Drinfeld modules and indecomposable projective Yetter-Drinfeld modules over a family of non-pointed 8m-dimension Hopf algebras of tame type with rank two, are construted and classified. The technique is Radford's method of constructing Yetter-Drinfeld modules over a Hopf algebra. Furthermore, the projective class rings of the category of Yetter-Drinfeld modules over this class of Hopf algebras are described explicitly by generators and relations.
Keyword ：
tensor product tensor product the projective class ring the projective class ring Yetter-Drinfeld module Yetter-Drinfeld module simple subcomodule simple subcomodule
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| GB/T 7714 | Guo, Yaguo , Yang, Shilin . Projective class rings of a kind of category of Yetter-Drinfeld modules [J]. | AIMS MATHEMATICS , 2023 , 8 (5) : 10997-11014 . |
| MLA | Guo, Yaguo 等. "Projective class rings of a kind of category of Yetter-Drinfeld modules" . | AIMS MATHEMATICS 8 . 5 (2023) : 10997-11014 . |
| APA | Guo, Yaguo , Yang, Shilin . Projective class rings of a kind of category of Yetter-Drinfeld modules . | AIMS MATHEMATICS , 2023 , 8 (5) , 10997-11014 . |
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Abstract ：
As is well known, a class of 2n(2)-dimensional Kac-Paljutkin Hopf algebras H(2)n(2) was introduced by Pansera. It is the generalization of the 8-dimensional Kac-Paljutkin Hopf algebra H-8. In this paper, the classification of Yetter-Drinfeld modules over H-2n(2) is given by Radford's method of constructing Yetter-Drinfeld modules for a Hopf algebra. Furthermore, the tensor product of Yetter-Drinfeld modules over H-2n(2) is established, and the Grothendieck ring r((2nH2nYD)-H-H-H-2-Y-2) is described explicitly by generators and relations.Communicated by Julia Plavnik
Keyword ：
tensor product tensor product Kac-Paljutkin algebra Kac-Paljutkin algebra Grothendieck ring Grothendieck ring Yetter-Drinfeld module Yetter-Drinfeld module
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| GB/T 7714 | Guo, Yaguo , Yang, Shilin . The Grothendieck ring of Yetter-Drinfeld modules over a class of 2n(2)-dimensional Kac-Paljutkin Hopf algebras [J]. | COMMUNICATIONS IN ALGEBRA , 2023 , 51 (11) : 4517-4566 . |
| MLA | Guo, Yaguo 等. "The Grothendieck ring of Yetter-Drinfeld modules over a class of 2n(2)-dimensional Kac-Paljutkin Hopf algebras" . | COMMUNICATIONS IN ALGEBRA 51 . 11 (2023) : 4517-4566 . |
| APA | Guo, Yaguo , Yang, Shilin . The Grothendieck ring of Yetter-Drinfeld modules over a class of 2n(2)-dimensional Kac-Paljutkin Hopf algebras . | COMMUNICATIONS IN ALGEBRA , 2023 , 51 (11) , 4517-4566 . |
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Abstract ：
In this paper, representations of the 8m-dimensional non-pointed Hopf algebra H-8m of tame type are studied, where m is even. First, the isomorphism classes of all indecomposable modules of H-8m are classified and the components of Auslander-Reiten quivers are constructed. Second, the tensor products of arbitrary indecomposable modules and simple (or projective) modules are established and the projective class rings and Grothendieck rings of H-8m are characterized.
Keyword ：
projective class ring projective class ring Non-pointed Hopf algebra Non-pointed Hopf algebra bound quiver bound quiver
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| GB/T 7714 | Guo, Yaguo , Yang, Shilin . Representations of an 8m-dimensional non-pointed Hopf algebra of tame type [J]. | JOURNAL OF ALGEBRA AND ITS APPLICATIONS , 2023 . |
| MLA | Guo, Yaguo 等. "Representations of an 8m-dimensional non-pointed Hopf algebra of tame type" . | JOURNAL OF ALGEBRA AND ITS APPLICATIONS (2023) . |
| APA | Guo, Yaguo , Yang, Shilin . Representations of an 8m-dimensional non-pointed Hopf algebra of tame type . | JOURNAL OF ALGEBRA AND ITS APPLICATIONS , 2023 . |
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Abstract ：
We first describe the Sekine quantum groups A(k) (the finite-dimensional Kac algebra of Kac-Paljutkin type) by generators and relations explicitly, which maybe convenient for further study. Then we classify all irreducible representations of A(k) and describe their representation rings r(A(k)). Finally, we compute the the Frobenius-Perron dimension of the Casimir element and the Casimir number of r(A(k)).
Keyword ：
representation ring representation ring Sekine quantum group Sekine quantum group Casimir number Casimir number
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| GB/T 7714 | Chen, Jialei , Yang, Shilin . REMARKS ON SEKINE QUANTUM GROUPS [J]. | CZECHOSLOVAK MATHEMATICAL JOURNAL , 2022 , 72 (3) : 695-707 . |
| MLA | Chen, Jialei 等. "REMARKS ON SEKINE QUANTUM GROUPS" . | CZECHOSLOVAK MATHEMATICAL JOURNAL 72 . 3 (2022) : 695-707 . |
| APA | Chen, Jialei , Yang, Shilin . REMARKS ON SEKINE QUANTUM GROUPS . | CZECHOSLOVAK MATHEMATICAL JOURNAL , 2022 , 72 (3) , 695-707 . |
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Abstract ：
Let w(2,2)(s) (s = 0, 1) be two classes of weak Hopf algebras corresponding to the Sweedler Hopf algebra, and r(w(2,2)(s)) be the representation rings of w(2,2)(s). In this paper, we investigate the automorphism groups Aut(r(w)(2,2)(s))) of r(w(2,2)(s)), and discuss some properties of Aut(r(w(2,2)(s))). We obtain that Aut(r(w(2,2)(0))) is isomorphic to K-4, where K-4 is the Klein four-group. It is shown that Aut(r(w(2,2)(1))) is a non-commutative infinite solvable group, but it is not nilpotent. In addition, Aut(r(w(2,2)(1))) is isomorphic to (Z x Z(2)) (sic) Z(2), and its centre is isomorphic to Z(2).
Keyword ：
representation ring representation ring weak Hopf algebra weak Hopf algebra Sweedler Hopf algebra Sweedler Hopf algebra automorphism group automorphism group
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| GB/T 7714 | Su, Dong , Yang, Shilin . Automorphism groups of representation rings of the weak Sweedler Hopf algebras [J]. | AIMS MATHEMATICS , 2022 , 7 (2) : 2318-2330 . |
| MLA | Su, Dong 等. "Automorphism groups of representation rings of the weak Sweedler Hopf algebras" . | AIMS MATHEMATICS 7 . 2 (2022) : 2318-2330 . |
| APA | Su, Dong , Yang, Shilin . Automorphism groups of representation rings of the weak Sweedler Hopf algebras . | AIMS MATHEMATICS , 2022 , 7 (2) , 2318-2330 . |
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