李超代数P^~(2)到Kac模的一阶上同调

在p≥3的域上,根据P^~(2)的Z-阶化结构以及零阶化项P^~(2)₀的标准Cartan分解,对于给定的权λ以及极大向量v_λ,构作出有限维不可约P^~(2)₀-模M(λ),进而给出其Kac结构.其次,利用一阶上同调的定义,将P^~(2)到其Kac模的一阶上同调转化为计算P^~(2)到Kac模的非内导子的权导子.最后,本文确定了P^~(2)到一类Kac模的一阶上同调.

First cohomology of Lie superalgebra P^~(2)with coefficients to Kac modules

Over a field of characteristic p≥3,firstly,according to the Z-degree structure of P^~(2)and the standard Cartan decomposition of the 0-degree P^~(2)₀,for a given weight λ and the maximal vector v_λ, finite-dimensional irreducible modules M(λ)of P^~(2)₀ are constructed, and then its Kac structure is given. Secondly, using the definition of the first cohomology group, the first cohomology group from P^~(2)to its Kac module is transformed into the weight-derivation of the non-inner derivation of P^~(2)to Kac module. Finally, the first cohomology group of P^~(2)with coefficients to a class of Kac modules is determined.