Triangular Decomposition Of Lie Superalgebras Assignment Mode

Recently the method of evaluation modules plays an active role in the study of affine Lie algebras and related topics [1-2]. In [3], Professor Wang Xian-dong discussed the evaluation modules of Loop algebras of Lie algebras with a triangular decomposition; particularly, irreducibility of the tensor product of finitely many such modules was partially determined.The main idea of this paper is to persuade [3] into Lie superalgebra case. For this purpose, the concepts of Lie superalgebras with a triangular decomposition, their Verma modules and highest weight modules are introduced in Section 3. The process of persuading is in no way smooth, and the major obstacle lies in the fact that irreducible finite dimensional representations of solvable (nilpotent) Lie superalgebras are not necessarily 1-dimensional. As two important tools, the weight space decomposition of modules of a nilpotent Lie algebra (Section 1) and irreducible finite dimensional representations of solvable (thus nilpotent) Lie superalgebras (Section 2.3) are presented in detail. Finally, after defining an evaluation module of the corresponding Loop superalgebra (Section 4), two major results of the paper -Theorem 4.land Theorem 4.2 are proved: Theorem 4.1 reduces the irreducibility of the tensor product of finitely many evaluation modules to the irreducibility of the tensor product of finitely many irreducible modules of a nilpotent Lie superalgebra; Theorem 4.2 gives (by irrindices) a criterion for the tensor product of such modules to be irreducible.