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Abstract

We define an operator $\alpha$ on $\mathbb{C}^3\otimes \mathbb{C}^3$ associated with the quantum group $U_q(2)$, which satisfies the Yang-Baxter equation and a cubic equation $(\alpha^2-1)(\alpha+q^2)=0$. This operator can be extended to a family of operators $h_j:=I_j\otimes \alpha \otimes I_{n-2-j}$ on $(\mathbb{C}^3)^{\otimes n}$ with $0\le j \le n-2$. These operators generate the cubic Hecke algebra ${\cal H}_{q, n}(2)$ associated with the quantum group $U_q(2)$. The purpose of this note is to present the construction.