7 N ov 2 00 4 Representations of Centrally - Extended Lie Algebras over Differential Operators and Vertex Algebras

We construct irreducible modules of centrally-extended classical Lie algebras over left ideals of the algebra of differential operators on the circle, through certain irreducible modules of centrally-extended classical Lie algebras of infinite matrices with finite number of nonzero entries. The structures of vertex algebras associated with the vacuum representations of these algebras are determined. Moreover, we prove that under certain conditions, the highest weight irreducible modules of centrally-extended classical Lie algebras of infinite matrices with finite number of nonzero entries naturally give rise to the irreducible modules of the simple quotients of these vertex algebras. Our results are natural generalizations of the well-known WZW models in conformal field theory associated with affine Kac-Moody algebras. They can also be viewed as quadratic generalizations of free field theory. In the very special case of W1+∞, our results with integral central charges are more direct and explicit than those of Kac and Radul. 1991 Mathematical Subject Classification. Primary 17B10, 17B69; Secondary 81Q40 Research Supported by China NSF 10371121 ACKNOWLEDGEMENT: Part of this work was done during the author’s visit to The University of Sydney, under the financial support from Prof. Ruibin Zhang’s ARC research grant. The author thanks Prof. Zhang for his invitation and hospitality.