一般线性李超代数及其超群的无穷小子群的表示  

On Representations of gl(m|n) and Infinitesimal Subgroups of GL(m|n)

作　　者：郑立笋 舒斌[2] 

机构地区：[1]海应用技术学院数理教学部,上海200235 [2]华东师范大学数学系,上海200241

出　　处：《数学年刊（A辑）》2010年第2期129-142,共14页Chinese Annals of Mathematics

基　　金：国家自然科学基金(No.10871067)资助的项目

摘　　要：在素特征代数闭域上考虑一般线性李超代数g(?)(m|n)的限制表示与超群GL(m|n)的有理表示以及它们的关系.主要结果为:(1)对g(?)(m|n)的不可约限制表示进行分类,其中某些单模恰是Kac-模.类似于复数域情形,给出了Kac-模不可约的充要条件;(2)当m(?)n(mod p)以及p≥2h-2(h=max{m,n})时,g(?)(m|n)的限制投射模可以被提升为有理GL(m|n)-模,并且证明了不可约表示的投射覆盖具有Z-滤过,即滤过中的每个子商同构于"baby Verma模";(3)得到了一般线性超群G=GL(m|n)的r阶Frobenius核的反转公式,它反映了单G_r-模的投射覆盖的Z-滤过重数与广义baby Verma模的合成因子数之间的关系.Let k be an algebraically closed field of characteristic p 〉 2. In this paper, the authors study the restricted representations of the general linear Lie superalgebras gl（m ｜ n）, and take the connection into account with rational representations of the corresponding algebraic supergroup GL（m ｜ n）. The main results include （1） The iso-classes of restricted irreducible modules for gl（m ｜ n） are easily parameterized. Some of those modules are just Kac-modules, for judgement of which a necessary and sufficient condition is given, parallel to the case in the field of complex number; （2） The restricted projective modules of gl（m ｜ n）-module can be lifted to a rational GL（m I n）-module when m ≠ n （modp） and p ≥ 2h - 2 （h -- max{m, n}）. Furthermore, such projective modules have Z-filtrations, i.e., each subquotient of such a filtration is isomorphic to some ＂baby Verma modules＂; （3） A reciprocity formula for the rth Frobenius kernels Gτ of G is obtained, which reflects the relation between the multiplicities in such a filtration of the projective cover of simple Gτ-modules and the composite numbers of baby Verma modules.

关 键 词：限制李超代数 广衍代数 Z-滤过 

分 类 号：O152.5[理学—数学]