Volume 54, Issue 4
Non-Negative Integer Matrix Representations of a $\mathbb{Z}_{+}$-Ring

Zhichao Chen, Jiayi Cai, Lingchao Meng & Libin Li

J. Math. Study, 54 (2021), pp. 357-370.

Published online: 2021-06

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  • Abstract

The $\mathbb{Z}_{+}$-ring is an important invariant in the theory of tensor category. In this paper, by using matrix method, we describe all irreducible $\mathbb{Z}_{+}$-modules over a $\mathbb{Z}_{+}$-ring $\mathcal{A}$, where $\mathcal{A}$ is a commutative ring with a $\mathbb{Z}_{+}$-basis{$1$, $x$, $y$, $xy$} and relations: $$ x^{2}=1,\;\;\;\;\; y^{2}=1+x+xy.$$We prove that when the rank of $\mathbb{Z}_{+}$-module $n\geq5$, there does not exist irreducible $\mathbb{Z}_{+}$-modules and when the rank $n\leq4$, there exists finite inequivalent irreducible $\mathbb{Z}_{+}$-modules, the number of which is respectively 1, 3, 3, 2 when the rank runs from 1 to 4.

  • Keywords

Non-negative integer, matrix representation, irreducible $\mathbb{Z}_{+}$-module, $\mathbb{Z}_{+}$-ring.

  • AMS Subject Headings

13C05, 16W20, 19A22

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

351996442@qq.com (Zhichao Chen)

563672447@qq.com (Jiayi Cai)

1072488663@qq.com (Lingchao Meng)

lbli@yzu.edu.cn (Libin Li)

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@Article{JMS-54-357, author = {Zhichao and Chen and 351996442@qq.com and 16769 and School of Mathematical Sciences, Yangzhou University, Yangzhou 225002, China and Zhichao Chen and Jiayi and Cai and 563672447@qq.com and 16770 and School of Mathematical Sciences, Yangzhou University, Yangzhou 225002, China and Jiayi Cai and Lingchao and Meng and 1072488663@qq.com and 16771 and School of Mathematical Sciences, Yangzhou University, Yangzhou 225002, China and Lingchao Meng and Libin and Li and lbli@yzu.edu.cn and 16772 and School of Mathematical Sciences, Yangzhou University, Yangzhou 225002, China and Libin Li}, title = {Non-Negative Integer Matrix Representations of a $\mathbb{Z}_{+}$-Ring}, journal = {Journal of Mathematical Study}, year = {2021}, volume = {54}, number = {4}, pages = {357--370}, abstract = {

The $\mathbb{Z}_{+}$-ring is an important invariant in the theory of tensor category. In this paper, by using matrix method, we describe all irreducible $\mathbb{Z}_{+}$-modules over a $\mathbb{Z}_{+}$-ring $\mathcal{A}$, where $\mathcal{A}$ is a commutative ring with a $\mathbb{Z}_{+}$-basis{$1$, $x$, $y$, $xy$} and relations: $$ x^{2}=1,\;\;\;\;\; y^{2}=1+x+xy.$$We prove that when the rank of $\mathbb{Z}_{+}$-module $n\geq5$, there does not exist irreducible $\mathbb{Z}_{+}$-modules and when the rank $n\leq4$, there exists finite inequivalent irreducible $\mathbb{Z}_{+}$-modules, the number of which is respectively 1, 3, 3, 2 when the rank runs from 1 to 4.

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v54n4.21.02}, url = {http://global-sci.org/intro/article_detail/jms/19288.html} }
TY - JOUR T1 - Non-Negative Integer Matrix Representations of a $\mathbb{Z}_{+}$-Ring AU - Chen , Zhichao AU - Cai , Jiayi AU - Meng , Lingchao AU - Li , Libin JO - Journal of Mathematical Study VL - 4 SP - 357 EP - 370 PY - 2021 DA - 2021/06 SN - 54 DO - http://doi.org/10.4208/jms.v54n4.21.02 UR - https://global-sci.org/intro/article_detail/jms/19288.html KW - Non-negative integer, matrix representation, irreducible $\mathbb{Z}_{+}$-module, $\mathbb{Z}_{+}$-ring. AB -

The $\mathbb{Z}_{+}$-ring is an important invariant in the theory of tensor category. In this paper, by using matrix method, we describe all irreducible $\mathbb{Z}_{+}$-modules over a $\mathbb{Z}_{+}$-ring $\mathcal{A}$, where $\mathcal{A}$ is a commutative ring with a $\mathbb{Z}_{+}$-basis{$1$, $x$, $y$, $xy$} and relations: $$ x^{2}=1,\;\;\;\;\; y^{2}=1+x+xy.$$We prove that when the rank of $\mathbb{Z}_{+}$-module $n\geq5$, there does not exist irreducible $\mathbb{Z}_{+}$-modules and when the rank $n\leq4$, there exists finite inequivalent irreducible $\mathbb{Z}_{+}$-modules, the number of which is respectively 1, 3, 3, 2 when the rank runs from 1 to 4.

Zhichao Chen, Jiayi Cai, Lingchao Meng & Libin Li. (2021). Non-Negative Integer Matrix Representations of a $\mathbb{Z}_{+}$-Ring. Journal of Mathematical Study. 54 (4). 357-370. doi:10.4208/jms.v54n4.21.02
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