Commutant Algebras Of Lie Algebras Of Vector Fields

Kyo Nishiyama,a Japanese mathematician,has got the results of the commutant algebra of natural representation of Lie algebra W(n) of vector fields of infinite dimension in his paper published in 1995.In this article,we try to generalize the results of Nishiyama.In the case of natural representation of char F=p＞0,we get the complete result of the commutant algebra of Lie algebra W(n,(?)) of vector fields(Theorem 3.2.2).In the case of adjoint representation of char F = 0,we give a satisfactory description of generators of W(n×m) as a W(n)—module(Theorem 4.2.3),and in addition,we give an estimation of the commutant algebra of W(n)(Proposition 4.4.2),and we give a discussion of a kind of specific generators of W(n×m)(Proposition 4.5.1).