We construct bar-invariant $\mathbb{Z}\left[q^{\pm 1/2}\right]$-bases of the quantum cluster algebra of the valued quiver $A_2^{\left(2\right)}$, one of which coincides with the quantum analogue of the basis of the corresponding cluster algebra discussed in P. Sherman, A. Zelevinsky: Positivity and canonical bases in rank 2 cluster algebras of finite and affine types, Moscow Math. J., 4, 2004, 947–974.

Motivated by our attempts to construct an analogue of the Dirac operator in the setting of $U_q\left({\mathrm{𝔰𝔩}}_n\right)$, we write down explicitly the braided coproduct, antipode, and adjoint action for quantum algebra $U_q\left({\mathrm{𝔰𝔩}}_2\right)$. The braided adjoint action is seen to coincide with the ordinary quantum adjoint action, which also follows from the general results of S. Majid.