Two-Parameter Quantum Divided Power Algebra And Structure Of Its Realization As U_r,S（sl_n）-module

This paper gives the definition of two-parameter quantum divided power algebra and realizes it as a two-parameter quantum enveloping module algebra of type A, thus the case of the Single parameter quantum group is successfully extended to the case of two-parameter quantum group. By use of the conception’level’, we can complete the research on each graded subspace of two-parameter quantum divided power head-cut submod-ule, we have the conclusion that each submodule (corresponding to the grade) are simple or indecomposable. When its Socle isn’t simple, the graded submodule is indecomposable. Namely, every arbitrary element of the submodule can be intertwined with a constantly up to the element corresponding to the highest level by module actions. At the end of this paper, we have full discussions on Socle and Loewy filtration for each module. The concrete forms of each grade are given by proof in detail and we give a more comprehensive analysis of the submodule structure.