Structure of a class of Lie algebras of Block type

Let $\BB$ be a class of Lie algebras of Block type with basis $\{L_{\a,i}|\a,i\in\Z, i\geq 0\}$ and relations $[L_{\a,i},L_{\b,j}]=(\b(i+q)-\a(j+q))L_{\a+\b,i+j}$, where $q$ is a positive integer. In this paper, it is shown that $\BB$ are different from each other for distinct positive integers $q$'s. The automorphism groups, the derivation algebras and the central extensions of all $\BB$ are also uniformly and explicitly described, which generalize some previous results.