Structure of a class of Lie algebras of Block type

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