Cluster multiplication theorem in the quantum cluster algebra of type A2(2) and the triangular basis

Recursive formulas for the Kronecker quantum cluster algebra with principal coefficients

“…Let C Q be the cluster category of the valued quiver Q with the shift functor [1] and the Aulander-Reiten translation functor τ . For more details about cluster category, the readers can refer to [2].…”

“…(3) The first equality can be calculated as follows 1) x −2 =X δ (x −3 + X (0,0,1,1) x −1 ) − 2X (0,0,1,1) x −2 =x −4 + X (0,0,1,1) x −2 + X (0,0,1,1) (x −2 + X (0,0,1,1) x 0 ) − 2X (0,0,1,1) x −2 =x −4 + X (0,0,2,2) x 0 .…”

“…For the classical cluster algebras of acyclic quivers, Sherman and Zelevinsky [23] firstly provided the cluster multiplication formulas in rank 2 cluster algebras of finite and affine types. This result was generalized to rank 3 cluster algebra of affine type A (1) 2 by Cerulli [7]. Caldero and Keller [4] constructed the cluster multiplication formulas between two generalized cluster variables for simply laced Dynkin quivers, which was generalized to affine types in [17] and to acyclic types in [24,25].…”

“…Caldero and Keller [4] constructed the cluster multiplication formulas between two generalized cluster variables for simply laced Dynkin quivers, which was generalized to affine types in [17] and to acyclic types in [24,25]. In the quantum setting, Ding and Xu [10] firstly gave the cluster multiplication formulas of the quantum cluster algebra of type A 2 by Bai, Chen, Ding and Xu in [1]. Recently, Chen, Ding and Zhang [9] constructed the cluster multiplication formulas in the acyclic quantum cluster algebras with arbitrary coefficients through some quotients of derived Hall algebras of acyclic valued quivers.…”

“…2 and A 2n−1,1 without coefficients were constructed in [10,1,12] by using the explicit cluster multiplication formulas. However, most of quantum cluster algebras must have coefficients involved.…”