Quasifinite highest weight modules over the Lie algebra of matrix differential operators on the circle

BOYALLIAN, CARINA; KAC, VICTOR; LIBERATI, JOSE I.; YAN, CATHERINE H.

JOURNAL OF MATHEMATICAL PHYSICS

AMER INST PHYSICS

Lugar: New York; Año: 1998 vol. 39 p. 2910 - 2928

0022-2488

We give a complete description of the quasifinite highest weight modules over the central extension of the Lie algebra of MxM matrix differential operators on the circle and obtain them in terms of representation theory of the Lie algebra ĝl(∞, R_m) of infinite matrices with only finitely many nonzero diagonals over the algebra R_m=C[t]/(t^{m+1}). We also classify the unitary ones, and construct them in terms of charged free fermions. This construction provides a large (and conjecturally complete) family of irreducible modules over the associated vertex algebra W.