Second cohomology group of generalized Witt type Lie algebras and certain representations

We determine the second cohomology groups of Lie algebras of generalized Witt type which are some Lie algebras defined by Passman and Jordan, more general than those defined by Dokovic and Zhao, and slightly more general than those defined by Xu. Among all the 2-cocycles, there is a special one we think interesting. Using this 2-cocycle, we define the so-called Virasoro-like algebras. Then we give a class of their representations.