A Number Of Issues In The Theory Of Infinite-dimensional Lie Algebra Structure

There are three contains in the paper: pairing problem of generators in lKac-Moody algebras; classification of irreducible finite dimensional modules over sln(Cq); Lie algebras of derivations of n-differential operator algebra. The minimal number of the generators or the minimal set of the genera- tors of an algebraic system, is a very fundamental problem. In the theory of Lie algebras, the minimal number of the generators of finite dimensional semi-simple Lie algebras and Kac-Moodv algebras were solved. A further question is the pairing problem of generators, i.e. given an element in an algebra, if there exists another element such that the two elements can generate the algebra. For a finite dimensional simple algebra. the pairing problem of generators was solved. In Kac-Moody algebra, if the given element is a real root vector or in Cartari subalgebra, there exists an element such that they can generate the algebra. As to the given element is a imaginary root vector there are less result mainly because the structure of imaginary root space and the kac-Moody algebras are not clear. The paper gives a sufficient condition for pairing problem of generators in a symmetrizable Nac- Moody algebras. For a given imaginary root vector satisfying some conditions. there exists its pairing generator. In four non-twisted afflne Lie algebras, the paper completly solved the pairing problem of generators about imaginary root vector, that is, for a given imaginary root vector, there exists another vector such that the two vectors can generate the algebra. In chapter 4, we discuss the classification of finite dimensional irreducible modules over sln(Cq). C is the quantum torus associated to matrix q. sln(Cq) is the subalgebra of gln(Cq) which generate by E^,3(s). where .s Cq. The representation theory of aln(Cq) is very important to the classification and representation of Quasi-simple algebras. For convenience, we discuss 2 x 2 matrix q . The same result holds to ii x n matrix. There are two cases: When ptm # 1,Vm N, where p is in (1.2)-position in q. It is easy to prove that the ? modules over Sln(Cq) are all trivial modules; When pm i, we get an algebra of quotients by .s1^C7 modulo an ideal, which is a semi-simple Lie algebra. The classification of finite dimensional irreducible modules over semi-simple Lie algebra was known, so we obtain the result. In chapter 5, we discuss the Lie algebra of derivations of n-differential operator algebra. In recent years, the generalization of Cartan type Lie alge- bra was verY active. WevI type Lie algebra was recently raised. There are only elementary theory of Wev I type Lie algebra such as simplicity and automor- phism etc. n-differential operator algebra is one kind of Weyl type algebra. Ref. [Z3] give the structure of Lie algebra of derivations of differential operator algebra when n = 1. we get the structure of the Lie algebra of derivations of n-differential operator algebra when n > 2: That is the direct sum of a fixed outer derivation and all inner derivation.