VERTEX OPERATOR REPRESENTATION OF SOME QUANTUM TORI LIE-ALGEBRAS

We are defining the trigonometric Lie subalgebras in XBAR(infinity) = ABAR(infinity)(BBAR(infinity), CBAR(infinity), DBAR(infinity)) which are the natural generalization of the well known Sin-Lie algebra. The embedding formulas into XBAR(infinity) 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 factor algebras, which can be considered as a quantum (preserving Lie algebra structure) deformation of the Kac-Moody algebras are considered.