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

摘要

We give a complete description of the anti-involutions of the algebra D-N of NxN-matrix differential operators on the circle, preserving the principal Z gradation. We obtain, up to conjugation, two families sigma (+/- ,m) with 1 less than or equal tom less than or equal toN, getting two families D-+/- ,m(N) of simple Lie subalgebras fixed by -sigma (+/- ,m). We also give a geometric realization of sigma (+/- ,m), concluding that D-+,m(N) is a subalgebra of D-N of type o(m,n) and D-,m(N) is a subalgebra of D-N of type osp(m,n) (ortho-symplectic). Finally, we study the conformal algebras associated with D-+,m(N) and D--,m(N). (C) 2001 American Institute of Physics. [References: 5]