模李超代数在理论物理和数学领域中扮演重要角色，而Cartan型模李超代数在模李超代数中又占有重要的地位。现阶段Cartan型模李超代数的表示理论还处于起步阶段，因此Cartan型模李超代数的表示理论亟待研究。本项目拟对某些限制Cartan型模李超代数的限制表示与非限制表示以及非限制Cartan型模李超代数的表示做一系列的研究。项目的预期结果对模李超代数表示理论的发展具有促进作用。项目的工作具体陈述如下：.① 计算限制Cartan型模李超代数S，H，K，HO，SHO，KO，SKO的Cartan不变量，刻画这些代数的不可分解投射表示并给出它们的维数；.② 研究限制Cartan型模李超代数K，HO，KO，SKO的非限制Kac模的不可约性并得到非限制不可约表示的分类；.③ 利用广义限制李超代数研究某些小维数的非限制Cartan型模李超代数的表示理论，给出其不可约表示的分类。

Modular Lie superalgebras play an important role in the theoretical physics and mathematics, and modular Lie superalgebras of Cartan type play an important roles in modular Lie superalgebras. At the present stage, representation theory of modular Lie superalgebras of Cartan type still in the early stage. So the representations theory of modular Lie superalgebras of Cartan type needs to be researched. Thus we will study the restricted and non-restricted representions for some restricted modular Lie superalgebras of Cartan type and representions for some non-restricted modular Lie superalgebras of Cartan type. The expected results will promote the development of the representation theory of modular Lie superalgebras. The contents of the project are listed as follows: .① Get the Cartan invariants for restricted modular Lie superalgebras of Cartan type S，H，K，HO，SHO，KO，SKO, and characterize the projective modules for these superalgebras and give their dimensions;.② Study the irreducibility of non-restricted Kac modules for restricted modular Lie superalgebras of Cartan type K, HO, KO, SKO, and give the classification of non-restricted irreducible representation;.③ Study the representation theory of some non-restricted modular Lie superalgebras of Cartan type with small dimension by use of the generalized restricted Lie superalgebras and give the classification of irreducible representation.