Vertex operator representation of some quantum tori Lie algebras

We are defining the trigonometric Lie subalgebras in anti X∞=anti A∞(anti B∞, anti C∞, anti D∞) which are than the natural generalization of the well known Sin-Lie algebra. The embedding formulas into anti X∞ are introduced. These algebras can be considered as some Lie algebras of quantum tori. An irreducible representation of A, B series of trigonometric Lie algebras is constructed. Special cases of the trigonometric Lie algebras, which can be considered as a quantum (preserving Lie algebra structure) deformation of the Kac-Moody algebras are considered. (orig.)