Classical Lie subalgebras of the Lie algebra of matrix differential operators on the circle

Abstract

We give a complete description of the anti-involutions of the algebra DN of N X N-matrix differential operators on the circle, preserving the principal ℤ gradation. We obtain, up to conjugation, two families σ±,m with 1≤m≤N, getting two families DN±,m simple Lie subalgebras fixed by -σ±,m. We also give a geometric realization of σ±.m, concluding that DN+,m is a subalgebra of DN of type o(m,n) and DN-,m is a subalgebra of DN of type o s p(m,n) (ortho-symplectic). Finally, we study the conformal algebras associated with DN+,m and DN-,m © 2001 American Institute of Physics.