JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2026, Vol. 61 ›› Issue (4): 19-24.doi: 10.6040/j.issn.1671-9352.0.2024.189

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Classification of Z+-modules over the representation ring of group algebra kA4

SUN Hua, WANG Wei*, YIN Zetao   

  1. School of Mathematical Science, Yangzhou University, Yangzhou 225009, Jiangsu, China
  • Published:2026-04-08

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Abstract: Let k be an algebraically closed field with char(k)12, denoted r(kA4) the representation ring of group algebra r(kA4). All irreducible Z+-modules over r(kA4) are classified. We prove that there are 9 non-equivalent irreducible Z+-modules.

Key words: group algebra kA4, irreducible Z+-module, representation ring, matrix equation

CLC Number: 

  • O156
[1] 叶舒依. 量子偶D(kA4)的表示环[D]. 扬州:扬州大学,2020. YE Shuyi. The representation rings of Drinfeld double D(kA4)[D]. Yangzhou: Yangzhou University, 2020.
[2] SUN Hua, CHEN Huixiang. Green ring of the category of weight modules over the Hopf-Ore extensions of group algebras[J]. Communications in Algebra, 2018, 47(11):4441-4461.
[3] SUN Hua, CHEN Huixiang. Tensor product decomposition rules for weight modules over the Hopf-Ore extensions of group algebras[J]. Communications in Algebra, 2018, 46(4):1586-1613.
[4] SUN Hua, CHEN Huixiang, ZHANG Yinhuo. Representations of Hopf-Ore extensions of group algebras[J]. Algebra and Represent Theory, 2023, 26(5):1441-1463.
[5] WANG Zhihua, LI Libin, ZHANG Yinhuo. Green rings of pointed rank one Hopf algebras of nilpotent type[J]. Algebra and Represent Theory, 2014, 17(6):1901-1924.
[6] 王志华,李立斌. 一类Hopf代数的不可约表示[J]. 扬州大学学报(自然科学版),2013,16(2):1-3. WANG Zhihua, LI Libin. On the irreducible representations of a class of Hopf algebras[J]. Journal of Yangzhou University(Natural Science Edition), 2013, 16(2):1-3.
[7] OSTRIK V. Module categories, weak Hopf algebras and modular invariants[J]. Transformation Groups, 2003, 8(2):177-206.
[8] BEHREND R, PEARCE P, PETKOV V, et al. Boundary conditions in rational conformal field theories[J]. Nuclear Physics, 1999, 579(B):707-773.
[9] BOOKER T, DAVYDOV A. Commutative algebras in Fibonacci categories[J]. Journal of Algebra, 2012, 355(1):176-204.
[10] FUCHS J, SCHWEIGERT C. Category theory for conformal boundary conditions[J]. Fields Institute Communications, 2003, 39(25):25-71.
[11] GANNON T. Boundary conformal field theory and fusion ring representations[J]. Nuclear Physics, 2002, 627(B):506-564.
[12] CHEN Zhichao, CAI Jiayi, MENG Lingchao, et al. Non-negative integer matrix representations of a Z+-ring[J]. Journal of Mathematical Study, 2021, 51(4):357-370.
[13] 苑呈涛. 近群融合环上的不可约Z+-模的分类[D]. 扬州:扬州大学,2018. YUAN Chengtao. The classification of the irreducible Z+-modules over near-group fusion rings[D]. Yangzhou: Yangzhou University, 2018.
[14] 周芯雨. 近群融合环上K(Q8,n)的Z+-模的分类[D]. 扬州:扬州大学,2022. ZHOU Xinyu. The classification of the irreducible Z+-modules over near-group fusion ring K(Q8,n)[D]. Yangzhou: Yangzhou University, 2022.
[15] 陈勇. 一类环上的不可约Z+-模[D]. 扬州:扬州大学,2024. CHEN Yong. Irreducible Z+-modules over a class of domain[D]. Yangzhou: Yangzhou University, 2024.
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