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We give a complete description of the anti-involutions of the algebra DN of N×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 D±,mN of simple Lie subalgebras fixed by −σ±,m. We also give a geometric realization of σ±,m, concluding that D+,mN is a subalgebra of DN of type o(m,n) and D−,mN is a subalgebra of DN of type osp(m,n) (ortho-symplectic). Finally, we study the conformal algebras associated with D+,mN and D−,mN.

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