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In a recent paper by H. X. Cao, J. H. Zhang and Z. B. Xu an $\alpha$-Lipschitz operator from a compact metric space into a Banach space $A$ is defined and characterized in a natural way in the sence that $F:K\to A$ is a $\alpha$-Lipschitz operator if and only if for each $\sigma \in X^{*}$ the mapping $\sigma \circ F$ is a $\alpha$-Lipschitz function. The Lipschitz operators algebras $L^{\alpha }(K,A)$ and $l^{\alpha }(K,A)$ are developed here further, and we study their amenability and weak amenability of these algebras. Moreover, we prove an interesting result that $L^{\alpha }(K,A)$ and $l^{\alpha }(K,A)$ are isometrically...