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 - <p style="text-align: justify;">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.</p>