李超代数P(2)到Kac模的一阶上同调

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

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vλ,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.

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 weightλand the maximal vector vλ,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.