Structure and representation for a class of infinite-dimensional Lie algebras

Let be a commutative associative algebra over the complex field C, and be the complexification of the real Lie algebra so(3). For any fixed elements , we define a Lie algebra with Lie bracket given by (1.2). When the associative algebra is the Laurent polynomial algebra , we determine its derivation Lie algebra , and universal central extension . We also give a vertex operator representation for the Lie algebra . This new class of Lie algebras includes the affine Lie algebra and the toroidal Lie algebras of type A1. We note that in general this kind of Lie algebras is not Zν-graded.