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

Abstract: Let be a commutative associative algebra over the complex field C, and be the complexification of the real Lie algebra . 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 . We note that in general this kind of Lie algebras is not -graded. [Copyright &y& Elsevier]